Apparent Resistivity

Overburden

GEOLOGICAL SURVEY PROFESSIONAL PAPER 365
Apparent Resistivity 

of a Single Uniform 

Overburden

By IRWIN ROMAN

GEOLOGICAL SURVEY PROFESSIONAL PAPER 365

An analysis of the two-boundary 

resistivity problem with tables 

for numerical applications

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1960
UNITED STATES DEPARTMENT OF THE INTERIOR 

FRED A. SEATON, Secretary

GEOLOGICAL SURVEY 

Thomas B. Nolan, Director

The U.S. Geological Survey Library has cataloged this publication as follows:

Roman, Irwin, 1893-

Apparent resistivity of a single uniform overburden. 

Washington, U.S. Govt. Print. Off., 1960.

iv, 99 p. diagrs., tables. 30 cm. (U.S. Geological Survey. 

Professional paper 365)

Bibliography: p. 99.

1. Electric resistance. 2. Prospecting-Geophysical methods. 

3. Electric prospecting. I. Title. (Series)

For sale by the Superintendent of Documents, U.S. Government Printing Office 

Washington 25, D.C. - Price 70 cents (paper cover)
CONTENTS

Abstract __ _______----_---_-__________

Introduction ___________________________

Physical problem. _ ____________________

Homogeneous space. _ _______________

Potential function- _____________

Four-electrode configuration __ _ 

Modified sum_____ _ _ _______

Generalized configurations __.--__ 

Form factor____________ ________

Uniform geophysical space___________

Definitions _ __________________

Configurations __________--_

Single-probe configuration. __ 

Collinear configuration. _____

Wenner configuration. ______

Lee partitioning configuration 

Basic configurations _ ______

Nonuniform geophysical space ___ ___

Normal resistivity. _____________________

Rigid configurations __ _ ________________

Apparent resistivity. _ _________________

Disturbing factor. ______________________

Resistivity surveying methods _ _____________

Horizontal surveying. ___________________

Vertical surveying. _____________________

Azimuthal surveying ____________________

Single overburden earth__--_-__-_-____-__--_--_-

N otation __________________________________

Normal potential.. _ ______________________

Modified potential. _________________________

Tables of modified potential ___ _____________

Interpolation _ _____ ___ _________________

Asymptotic expansions ______________________

Apparent resistivity____ ________ _________________

Disturbing factor. ______________________________

Wenner or Lee configuration. ________________

Superposition______ _ _____________________

Illustrative examples. _ _________________________

Superposition.. ____________________________

Four-electrode surface configuration _ ________

Interpolated values of the modified potential. _. 

Mathematical problem. ___________________________ _

Notation. ________________________ ___ _______

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Mathematical problem Continued

Potential function._____________________________ 16

Unit pole_______-__________-___---_----------_. 17

Boundary conditions-_____-_--__---_--_--__---_- 17

Simple-image method_-__-_---__---_---_-------- 17

Pole internal to a parallel plate___________________ 20

Pole external to a parallel plate___________________ 22

Some special cases_________--____-__--__---_-__- 23

Imbedded plate__________________________ 23

Insulated plate_________-________ _________ 23

Cover plate_____________________ 23

Geophysical case____________ _ _____ 23

Formulas for the evaluation of the modified potential. _ _ 25

Calculation of the reduced reciprocal distance._____ 25

Direct calculation ________________________ 25

Rescaling___ _____-__-_.____- ___ ______ 25

Control values__ ___________ ______________ 25

Asymptotic limits of W____________ _______ 27

Euler-Maclaurin summation formulas.__ _______ 27

Local expansions ______ ___ __________ 29

Local expansions for (?2 =l_______-___________ 29

Local expansions for Q2<1 --_-----_- -- - 32

Asymptotic expansions._______ ___________ 36

Asymptotic expansions for (?2<1._ __-____-__ 36

Asymptotic expansions for (?= + !____________ 37

Reduction formula from positive to negative reflection factor_-_____._____ ___ 40

Table smoothing and error detection._________ 41

Lagrangean interpolation ____________________ 41

Numerical evaluation of the modified potential.________ 44

Plan of calculation____________________________ 44

Formulas for calculations_________-_______-__--__ 45

Calculations for separate values of Q--~~.--------- 46

Calculations for £=+!______________ 46

Calculations for Q= 1-----.--------------- 48

Calculations for #=+0.1______________ 48

Calculations for <?= 0.1__________________ 49

Calculations for 9=+0.2___.__________ 50

Calculations for Q= 0.2 ________________ 51

Calculations for #=+0.3__________ 54

Calculations for Q= Q.B.------------------- 55

Calculations for 0.4<Q^0.9_._____________ 55

Numerical values of the modified potential ___________ 58

References cited________________ ________________ 99*

in
IV CONTENTS

ILLUSTRATIONS

FIGUEE 1. Four-electrode configuration____________________________________

2. Collinear configuration-________________________________________

3. Wenner and Lee partitioning configurations.._____________________

4. Single-overburden earth________________________________

5. Superposition curves, buried conductor, Wenner or Lee configuration.

6. Superposition curves, buried insulator, Wenner or Lee configuration. _

7. Interpolation chart, buried conductor.___________________________

8. Interpolation chart, buried insulator.____________________________

9. An illustrative example of superposition.__________________ ____

10. Two-boundary space, internal pole_____________________________

11. Simple images_______________________________________________

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TABLES

TABLE 1. Coefficients in asymptotic expansions-___ 

2. Disturbing factor, Wenner or Lee configuration, buried conductor._________________

3. Disturbing factor, Wenner or Lee configuration, buried insulator___________________

4. Two-boundary image sequences for an internal 

pole_ ________ _ _____ ______ 

5. Two-boundary image sequences for an external pole.______________________________

6. Error factors in direct evaluation of W__

7. Control values for g=0.9_____-___----_-___

8. Asymptotic limits of "FF____ ____________

9. Euler-Maclaurin coefficients _______________

10. Variations of (l D2k £?_ )_-_______-

11. Values of (l-D2k ____)_-----------___

12. Values of £> ____________________

13. Values of Ak,2*-- ___-__---_-_--______-_-

14. Values of ak.a _______---_-_--__--_____

15. Values of Hn________________________

16. Residue factors for » = 30-___-_------__--_-

17. Error patterns_-_--__-__---_--_------_-_-

18. Values of S% for g= + l____.______________

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TABLE 19. Local coefficients Cn _ and ratios Xn« for

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Values of <S% for g=0.1 ___ _- _ ____

Local coefficients Cn« for Q=+0.1 

Local ratios X n« for Q=+0.1___._ 

Local coefficients Cn _ for Q= 0.1 __ 

Local ratios X n« for Q= 0.1 __ __ 

Values of M«« for Q=+0.2 ___ _____

Local coefficients <? for Q=+0.2_____

Local ratios X»» for Q=+0.2 _ ______

Values of (-M»«) for Q=-0.2 __ ___

Local coefficients Cn« for Q= 0.2_ 

Local ratios Xnu for Q= 0.2 

Values of nnu for Q=+0.3 ___ ___ 

Local coefficients <? for Q=+0.3 

Local ratios Xn« for Q= + 0.3__ 

Values of (~Mn.) for Q= 0.3_-_ ____

Local coefficients <7n _ for Q= 0.3- 

Local ratios X»« for Q= 0.3 

Difference coefficients for interpolation 

Modified potential 

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59
APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

By IRWIN KOMAN

ABSTRACT

The interpretation of resistivity observations can be facilitated 

by the development of theoretical formulas and corresponding 

curves. A development for a single overburden of uniform 

thickness has been made and formulas derived that involve 

infinite series. The series have been evaluated for the geophysical case in which the measuring configuration is located on the 

surface of the earth. A set of curves that can be superimposed 

on the field observations has been prepared to permit direct 

determination of the resistivity and thickness of the overburden, 

and the resistivity of the underlying medium. Auxiliary tables 

are given to simplify the numerical evaluations for configurations 

not covered by the prepared curves.

INTRODUCTION

Geophysical methods of exploration in the United 

States have developed steadily since the importation of 

German instruments about 1923. Many methods have 

been tested and equipment has been developed to a high 

degree of proficiency. Theoretical development has 

paralleled field techniques of measuring and each has 

indicated improvements to be made in the other. 

When theoretical conclusions were available, interpretation of field data and techniques of observation have 

been guided by them. When theoretical conclusions 

were lacking, the interpreter has used empirical methods 

or intuitive and qualitative conclusions.

Specifically, in the use of electrical methods, the work 

of Fox, prior to 1830, and of Barus, in 1882, did not 

develop exploratory methods. In 1915 Wenner furnished the key to the exploratory difficulties and in 

1925 Kooney and Gish demonstrated the possibilities of 

the method of resistivity exploration. The theoretical 

analyses gave little promise until the work of Hummel 

(1929 a, b) which was followed by other theoretical 

papers, and the use of field measurements was accelerated thereafter. Although results have never been 

spectacular, successful resistivity measurements have 

been made by investigators in many parts of the world 

(Heiland, 1940, p. 28 and p. 707-763; Jakosky, 1950, 

p. 465-579; Koman, 1951).

In 1931, Koman published a mathematical analysis 

and a set of tables for use in the interpretation of 

resistivity measurements made at the surface of the 

earth, which was assumed to consist of a uniform overburden of constant thickness overlying a uniform bed 

of infinite thickness. The present report is the result 

of a study intended to improve the analysis and the 

tables of the earlier paper. Terminology has been 

clarified by a systematic description or definition of 

terms used. The upper limit of the argument in the 

tables has been increased in extent from 5 to 30, corresponding to lateral separations of current poles from 

potential electrodes of 10 and 60 times, respectively, 

the thickness of the overburden. Local expansions for 

small arguments have been augmented by asymptotic 

expansions for large arguments. The algebraic determination of the image series has been replaced by a 

geometric method, without sacrificing adherence to 

fundamental relations. The modified potential has 

been tabulated to eight decimals instead of five.

The increase in the range of the argument was the 

result of inadequacy of the earlier tables in investigating new methods of exploration. For the Wenner 

and the Lee partitioning methods, they usually were 

adequate. Over the years, problems arose in which 

greater ranges were needed. Among these were the 

single-probe method and multiple layering. In the 

former, the potential-reference electrode or that electrode and one current stake are remotely located. 

Although the contributions of these points to the 

measured potential differences are small, and sensibly 

constant over the area of survey, their being considered 

as negligible is not justified for all purposes, especially 

if the measured potential difference is small. In the 

latter, a distance that is a few times the depth of the 

lowest contact can be many times the thickness of the 

top layer.

The need for more significant figures also arose in the 

investigation of new methods of exploration, especially 

when the potential differences are much smaller than 

the separate potentials involved. This condition arises 

when the potential difference is measured between two 

points that are closely spaced with respect to their 

distances from the current poles, as can happen in the 

single-probe method or the potential-gradient method.

The increase in range of argument led to investigations resulting in greater accuracy of the calculated
APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

values. New formulas and methods of calculation 

were developed and these made the inclusion of additional significant digits relatively simple and rapid.

In numbering equations, decimal grouping is used. 

For example, the term "equations 2.4" refers to both 

of the equations 2.41 and 2.42. Likewise "equation 1" 

refers to the entire set from 1.1 to 1.552, whereas 

"equations 1.5" refers to the set from 1.51 to 1.552.

As bibliographies are available in the references 

cited, references are limited to those specifically needed.

The author thanks his colleagues who assisted in the 

execution of the project; in particular, Mrs. Mary E. 

Knettles and Mr. W. W. Schwendinger, each of whom 

checked some of the algebraic and numerical details 

and assisted in testing alternative methods from which 

the present methods were selected. Mr. Vincent 

Latorre wired the control panels of the punched card 

computing machines used in differencing tables calculated near the end of the project. Mrs. V. N. Rice 

cooperated enthusiastically in the application of 

punched-card equipment as an aid in the numerical 

calculations. Punched-card equipment was not available until the project was almost completed.

PHYSICAL PROBLEM 

HOMOGENEOUS SPACE

If electric current is introduced into a medium, an 

electric field is created. In this field, a difference of 

electrical potential exists between two points. The 

character of this field depends on the properties of the 

space and on the current introduced.

POTENTIAL FUNCTION

In a homogeneous isotropic space, the potential due 

to an isolated current source of strength / has the 

value

(1.1)

where r is the distance from the current source to the 

field point at which the potential is evaluated, and 

where k and B are constants for the field. As r increases indefinitely, U approaches the value B, which 

is usually considered as zero. If p is the resistivity of 

the medium, the constant k is p/(47r) so that the potential function may be written

(1.2)

As the potential can not be determined at an infinite 

distance, actual observation usually involves the 

measurement of the excess of the potential at the test 

point, H, over the potential at a reference point, L. 

If the reference point is at the distance s from the current

source, equation 1.1 shows that the potential at the 

test point is

U.=. +B, (1.3) 

s

where k and B have the same values as in equation 1.1. 

Accordingly, the excess of the potential at H over that 

at L is

-l/s), (1-4)

independently of B.

FOUR-ELECTRODE CONFIGURATION

In actual measurements, in addition to two potential 

points, test-point and reference point, there must be 

at least one current source, at which the current enters 

the medium, and at least one sink, at which the current 

leaves the medium. In the simplest form (see fig. 1)

Current 

sink

Reference 

potential

FIGTJKE 1. Four-electrode configuration. A current of I amperes enters the medium 

at P and leaves it at N under the influence of a source S. A drop in potential of 

V volts from H to £ results. The four points P, N, H, and L may have arbitrary 

locations and need not be coplanar.

there is one current source, P, one current sink, N, 

and two potential points, H and L; this arrangement 

is usually designated as a four-electrode configuration. 

The four points may be located arbitrarily and need 

not be in the same plane.

MODIFIED SUM

It is convenient to use the concept of a modified sum 

in discussing potential problems. Let x be an arbitrary 

value associated with a current pole and a potential 

point, jointly. The modified sum is denoted by S{x], 

which represents the sum of all possible values of x, each 

with its own sign if a current source is paired with a 

potential testpoint or a current sink is paired with the 

potential reference point, and with its sign reversed in 

the opposite pairings. This notation is convenient because a current source corresponds to a positive pole or 

a positive current strength, a current sink corresponds 

to a negative pole or a negative current strength, and 

the total potential at the potential reference point is 

subtracted from the total potential at the testpoint to 

obtain a measured value.
PHYSICAL PROBLEM

GENERALIZED CONFIGURATIONS

Either the source or the sink or both may consist of 

multiple contacts. Thus the current may be fed into 

the opposite corners of a quadrilateral and drained from 

the remaining corners, or contacts may be made at 

several points along a line. The only restriction is 

that no two points may coincide. Each observation 

must be of a single potential difference, so that the 

potential points cannot be multiple.

FORM FACTOR

The potential difference, V, for a selected configuration has the value S{U] where U is given by equation 

1.1, so that it may be written as

(1.51)

As B occurs an equal number of times with each sign, 

it disappears in the sum. As p and 4r are common to 

all pairs, equation 1.51 may be written

V=-f S[< (1.52)

For a four-electrode configuration, there are two current poles of the same strength so that / is common to 

all pairs and

V=^sl-\- (1.53)

(1.54)

(1.551)

(1.552)

If a "form factor", F, is defined by 

1

equation 1.53 may be written as

or

UNIFORM GEOPHYSICAL SPACE

DEFINITIONS

For purposes of the present paper, a "geophysical 

space" is defined as a space in which a medium of 

infinite resistivity is separated by a plane from a 

medium of finite resistivity, which may be constant or 

may vary in an arbitrary manner; it may even include 

one or more sections of infinite resistivity, if these 

sections are not in contact with the insulating medium 

other than at isolated points or along isolated lines. 

For convenience the medium of infinite resistivity may 

be called the "upper" medium, the second the "lower" 

medium.

In a "uniform geophysical space" the lower medium 

is uniform where the word "uniform" implies homogeneity and isotropy. A homogeneous medium has

the same resistivity in a selected direction at every 

point. An isotropic medium has the same resistivity 

at a selected point in every direction. Accordingly, 

in a uniform medium, the resistivity is the same for all 

points and all directions. Physical properties other 

than resistivity do not enter the present discussion.

The ideally uniform medium has the same resistivity 

for every volume regardless of size. In practice, this 

limiting condition is not necessary. A body may be 

considered as uniform if variations from ideal uniformity 

are not detectable by measurements. Thus, a mixture 

of clay, sand, and gravel is not ideally uniform, but the 

distribution may be such that the medium may be 

considered as uniform for a selected accuracy of surveying and a selected spacing of electrodes.

CONFIGURATIONS

The configurations and concepts of homogeneous 

space apply to a uniform geophysical space, with all 

current poles and potential points in the lower medium. 

The form factor is calculated by the same equation 

1.54, but the potential function of equation 1.2 must 

be determined for each specific arrangement of current 

poles. If the configuration has all of its points on the 

separating surface, the value of k in equations 1 becomes 

p/(2r) instead of p/(4ir), which is equivalent to replacing 

the current I by 27. Thus, for a surface configuration 

the potential drop is

(2.1)

For a four-electrode surface configuration, the potential 

drop is

_

rl~2TF

and the resistivity is

(2.21)

(2.22)

If the four-electrode surface configuration has the 

source at P (fig. 1), the sink at JV, the potential reference 

point at L, and the potential testpoint at H, the form 

factor is F, where

(2.3)

SINGLE-PROBE CONFIGURATION

If the point L is remote from the other three, the 

distances r12 and r22 are large and the form factor, F, 

is determined by

1-i Lj- A ~ -"1" '

where

A= 1 l

(2.41)

(2.42)
APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

is small and may be assumed to be zero as a first approximation. This configuration has only one moving 

point and the potential of H is such that the potential 

vanishes at infinity.

If the points L, N, and P are kept fixed the value of A 

is constant, but not necessarily negligible. There is 

only one moving point and the potential at infinity is

COLLINEAR CONFIGURATION

If N has the coordinates ( I, 0, 0) P the coordinates 

(I, 0, 0), H the coordinates (x, 0, 0), and L the coordinates (X, 0, 0), the system is collinear as well as 

surficial. (fig. 2).

N O H l

 Q 0 ^ 6

U 7 ......._... -.

D z. B ^> 

»-|i

FIGUBE 2. Collinear configuration. 0= (0, 0, 0) = origin of abscissae; N= ( 1, 0, 0) 

=negative current pole; P= (1, 0, 0) = positive current pole; H= (x, 0, 0) =field potential test-point; £=(X, 0, 0)= potential reference point.

For this configuration, the form factor has the value F, 

where

^ }

_

F~V x2 X2 P

WENNER CONFIGURATION

If the linear surface configuration has the points in 

the order NLHP with common separation I, it is the 

usual Wenner configuration of figure 3, for which the

Bed (£>

FIQUEE 3. Wenner and Lee partitioning configurations. Notation as in figures land 

2, with X= - and a: = -; A, thickness of overburden; pv resistivity of overburden; 

P4 , resistivity of bed.

form factor is

F=L (2.61)

uniform geophysical space, the resistivity is

(2.62)

LEE PARTITIONING CONFIGURATION

If a third potential point 0 is located at the center of 

the Wenner configuration and the potential drop is 

measured from 0 to L for the "left" resistivity and 

from H to 0 for the "right" resistivity, the form factor 

for each half has the value

F=2l (2.71) 

and the resistivity for a uniform geophysical space is

(2.72)

BASIC CONFIGURATIONS

For a selected set of points in a four-electrode system, there are only three distinct groups. Interchanging the two current stakes or the two potential 

electrodes leaves the magnitude of the potential drop 

unchanged, but changes the sign. Interchanging the 

pair of current stakes with the pair of potential electrodes leaves the magnitude of the potential drop 

unchanged in uniform geophysical space. Ignoring 

the sign of the potential drop, for four selected locations 

considered in a specific sequence, there are the three 

groups :

1. PNHL, NPHL, PNLH, NPLH, HLPN, 

HLNP, LHPN, LHNP.

2. PHNL, NHPL, PLNH, NLPH, HPLN, 

HNLP, LPHN, LNHP.

3. PHLN, PLHN, NHLP, NLHP, HPNL,

HNPL, LPNH, LNPH.

In the first group, the two current poles are consecutive 

and the potential electrodes are consecutive. In the 

second group, the current and potential poles alternate. 

In the third group, one type of electrode is interior, the 

other exterior. Experimentally, the order of PN determines the order of HL and when P is interchanged 

with N, H is automatically interchanged with L, if 

the sign of the measured potential drop is to remain 

unchanged.

NONUNIFOBM GEOPHYSICAL SPACE 

INTERPRETATION

If the medium of finite resistivity in a geophysical 

space is not uniform, the simple formulas of the uniform 

case do not apply. However, the departure of the 

measured resistivity from that calculated for the 

uniform case furnishes a basis of exploration. If the 

measured resistivity is the same for every station point 

and every direction, the medium of finite resistivity 

may be considered as uniform.

NORMAL RESISTIVITY

If the formulas for uniform geophysical space are 

applied to a potential drop as measured in a nonuniform 

medium, the resistivity will depend on the type and size
PHYSICAL PROBLEM

of the measuring configuration as well as on its location 

and direction. If a selected type of configuration is 

made smaller, the limiting value of the resistivity so 

determined, if independent of direction, is a value that 

may be called the "normal" resistivity at that point. 

The normal resistivity of a uniform medium is independent of the location, direction, or dimensions of the 

measuring configuration.

RIGID CONFIGURATIONS

If a selected configuration of current and potential 

poults is considered as a unit that may be translated 

or rotated in space without altering the respective distances between its member points, it may be considered 

as a "rigid configuration". The "station" may be a 

point of the configuration determined by a selected 

method applied to the elements of the configuration 

and the "direction" of the configuration may be a line 

and sense determined by a second method. For example, the station may be the midpoint of the current 

poles, or the midpoint of the potential points, or a point 

determined by a more complicated method, such as the 

point the sum of whose distances from all points of the 

configuration is a minimum. Likewise, the direction 

of the configuration may be the direction from the current sink to the current source, the reverse direction, 

the direction from the current source to the reference 

potential point, the direction from the station to the 

current source, or some other direction, associated with 

the configuration, but changing as the rigid configuration is translated or rotated or both.

In the Wenner configuration, the station usually is 

selected as the midpoint between the two current poles 

and the direction as that from the current sink to the 

current source. In the Lee partitioning configuration, 

the station usually is taken as that of the corresponding 

Wenner configuration, and each direction as that from 

the station to the second potential point of that potential drop measurement. These choices are not 

critical, but some specific choices must be made for 

clarity and definiteness in referring the configuration 

to the earth, in field surveys.

The "size" of a configuration is the distance between 

two selected points of it. In a Wenner system, the 

size may be taken as the "separation", which is the 

distance between adjacent elements, or as the "spread", 

which is the distance between the outer points and is 

three times the separation. In the Lee partitioning 

system, the size may be selected as that of the corresponding Wenner configuration, disregarding the 

central electrode.

For the single-probe arrangement, the station may be 

taken as the local current pole, the direction, that from 

the station to the second current pole or to the potential

647123 60 2

reference point and the size as the distance between 

current poles. As this system is not necessarily 

coUinear, a fourth quantity is involved; this quantity 

may be taken as the angle at the local current pole 

from the direction of the second current pole to the 

direction of the potential test point. Unless the 

potential reference point is very remote, the angle 

at the local pole from the direction of the second current 

pole to the direction of the potential reference point 

is a fifth constant.

Other configurations are possible. In each, there are 

constants of the configuration and parameters which 

determine the position and direction of the specific 

configuration. In a selected rigid configuration the 

size is fixed.

APPARENT RESISTIVITY

If the potential drop for a selected current in a 

selected configuration is substituted in the formula for a 

uniform geophysical space, the computed resistivity is 

an "apparent resistivity" for that configuration. By 

equation 2.22, if F is the form factor for the configuration used, V is the measured potential drop, and / is 

the current strength, the apparent resistivity has the 

value

(3.11)

If p0 is the normal resistivity of the medium near the 

station, a uniform medium having the resistivity p0 

would cause a "normal" potential Vn where

Po =27rFFn//. (3.12)

Any variation of pa from p0 indicates a nonuniformity 

of the medium.

DISTURBING FACTOR

Because of the diagnostic value of the apparent 

resistivity for a selected configuration, it is convenient 

to define a disturbing factor as

PO

(3.2)

Thus M is the ratio of the apparent resistivity to the 

normal resistivity of the lower medium near the station. 

It is also the ratio of the measured potential drop to 

the potential drop which would have been measured if 

the entire lower medium had been uniform, with its 

resistivity that of the overburden.

If the disturbing factor is unity for all configurations 

a uniform lower medium is indicated. If the disturbing 

factor exceeds unity for a selected configuration, there 

must be a region of resistivity higher than that near the 

station somewhere in the lower medium but close enough 

to the station to influence the value of the potential 

drop. This indicates the presence of a relative insulator
6 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

within the lower medium. Similarly, a disturbing 

factor below unity indicates the presence of a relative 

conductor within the lower medium.

KESISTrVTTY SURVEYING METHODS

By varying the parameters of a system, the size of 

the system, or by using various systems, the resistivity 

properties of the earth below, but near its surface, 

often can be detected and the variations in the apparent 

resistivity can be used for purposes of interpreting 

geological variations.

In well-logging, a set of electrodes with a selected 

configuration is lowered into a drill hole. In some 

applications, part of a configuration is located underground and part at the surface. The theoretical 

solutions of such problems can be obtained from the 

formulas of this paper, but in this paper numerical 

methods are applied only to surface configurations in a 

geophysical space, and the expression "surveying" is 

restricted to such applications.

HORIZONTAL SURVEYING

In horizontal surveying, the direction and size of a 

system are kept fixed and the station position is changed. 

A variation in the apparent resistivity indicates the 

presence of a discontinuity or change whose position 

horizontally can be determined approximately with 

respect to the station positions. Such a change may 

be a finite body, a change in thickness of the overburden, 

a horizontal change in material of the earth, or a change 

in water content. However, a horizontal surveying cannot detect changes associated only with varying depth.

The simplest method of horizontal surveying is horizontal profiling, in which the configuration of electrodes 

is collinear and all stations are located on the common 

line of the electrodes. The configurations most frequently used for horizontal profiling are the Wenner and 

the Lee partitioning, with fixed separations.

VERTICAL SURVEYING

In vertical surveying, the station of a system is kept 

fixed and the size or the size and direction are varied. 

As the size is increased, the depth of penetration of the 

current lines usually is increased so that vertical surveying usually can detect changes in the earth as the 

depth changes. Vertical surveying is affected by horizontal variations in the earth, so that depth effects can 

be masked by lateral effects.

The Wenner and Lee partitioning systems are the 

systems most commonly used for vertical surveying, 

and usually the direction of the systems is kept fixed. 

In vertical surveying, the direction of the system is that 

of the line along which the stations are placed and a set 

of apparent resistivities is determined for each station. 

However, these resistivities can be rearranged later to

furnish a set of horizontal profiles for as many separations as desired. Horizontal profiling is not used 

ordinarily to furnish vertical profiles, as it involves too 

much moving of equipment.

AZIMUTHAL SURVEYING

In azimuthal surveying, the direction of a system is 

varied, for a selected station. The size may be kept 

constant or a set of sizes may be used for each azimuth. 

Although usually slower in field operation than vertical 

or horizontal surveying, azimuthal surveying is helpful 

in some types of exploration, such as locating or identifying faults, inclined beds, or relatively small bodies.

SINGLE-OVERBURDEN EARTH 

NOTATION

In the single-overburden earth, the separating plane 

of a geophysical space coincides with the surface of the 

earth, the upper medium is air of resistivity p= <x>, and 

a plane at depth h (fig. 4) separates a uniform medium

Air

z-o m

//////Y/ /////////////////////

r

Overburden

Y\\\\\\\\\\\\\\\\\\\YV ('.'frttff (((({(«. 

Bed

FIGURE 4. Single-overburden earth. O, origin of coordinates; m, current pole; 

P, test point; B\, upper boundary, between air and overburden; Bs, lower boundary 

between overburden and bed; h, thickness of overburden; z, depth of P below 0; 

I, axial distance of Pfrom O (horizontally); r, distance of P from O; p= », resistivity 

of air; pp. resistivity of overburden; and pt,, resistivity of bed.

of resistivity p0 from a second uniform medium of 

resistivity P&?^PO. A current pole of strength m is 

located at the surface of the earth. The origin is taken 

at the pole, the 0-axis is directed downward, and the 

z-axis and y-axis are taken as mutually perpendicular 

in the surface of the earth. Except as bounded by the 

planes 2=0 and z=h each medium extends indefinitely 

in every direction.

NORMAL POTENTIAL.

The normal potential at the testpoint P corresponding 

to a current source / at the origin is

tfn=f£> (4.11)

where r is the distance of the testpoint from the pole. 

For multiple poles, the normal potential drop between

two points is

r M

(4.12)
PHYSICAL PROBLEM

For the four-electrode system

,. pQI J 1 1

V n=7T~ o \ - r

27T [r J

(4.13)

MODIFIED POTENTIAL

For the single overburden earth, the potential drop 

for a surface configuration is shown in the development 

of the mathematical problem to have the value:

(4.21)

(4.22)

(4.23)

where Q is the reflection factor, whose value is

Q=Pb po

Pb+PO

I

and W(Q, a) is a "modified potential" which can be 

calculated, and values of which are shown in table 38. 

For the four-electrode surface configuration, the potential drop is

(4.24)

TABLES OF MODIFIED POTENTIAL

Table 38, page 59, contains numerical values of the 

modified potential for values of the reflection factor 

from 1 to +1 by steps of 0.1, omitting the trivial 

case Q=Q, and for values of the ratio of the distance 

between current pole and potential point to twice the 

thickness of the overburden by various steps from 0 to 

30. The tabulated values may be rounded before 

calculation if full accuracy is not needed. In applications, the forcing errors may result in an error of 

several units in the last retained decimal place, so that 

one extra decimal should be used for evaluations and 

rounded out in final results.

INTERPOLATION

For the reflection factor of each, table 38 shows 

selected values of the argument, the corresponding 

values of the modified potential, and three differences 

which are convenient for interpolating within the 

table. For interpolation within each table, the spacings 

have been selected to permit interpolation to orders 

not exceeding the fourth. Interpolation between 

tables for intermediate values of the reflection factor 

follows the usual rules, but the differences must be 

calculated as needed and there is no assurance that the 

spacings between tables will justify interpolation of the 

fourth order or lower.

The formula for interpolation to the fourth order 

may be accepted as adequate. However, it frequently

saves time in preparing calculations, to estimate the 

order needed for interpolation before starting the 

calculation. For convenience, use may be made of 

the working rule:

if |At 2 |>43, use quartic interpolation;

if |Al 2 |^43 and JAljOl.6, use cubic interpolation;

if |A12 |^43, |A3_i|^16, and |A2_i|>8, use quadratic 

interpolation;

if |Atj|£43, |A8_i|^16, and lAlJ^S, use linear interpolation; where the table has been rounded to the desired 

extent before the criteria are applied. For derivation 

of this rule see the discussion of Lagrangean interpolation (page 41).

In interpolation, the argument lies between two 

tabulated values. Let the desired argument be a and 

let a lie between a0 and a® + h; where <z0 is the largest 

tabulated argument below a and h is the tabular 

spacing from a0 to the next larger tabulated argument. 

Then #=(« a0)/A is the "phase" of the interpolated 

argument and lies between zero and one.

For linear interpolation, the interpolant is

which may be written 

/,=/(*) *)/(0).

(4.311)

(4.312)

As the expressions 4.31 are so simple, there is no need 

to tabulate the first difference, which is /(I) /(O). If 

linear interpolation is adequate, the difference /(I ) /(O) 

is small. For use with a computing machine, equation 

4.312 is direct and does not use the first difference 

explicitly.

For quadratic interpolation, the linear interpolant is 

increased by F-^-i where Fz= %x(l x) is tabulated 

in table 37 (p. 59) for two decimal places in x. As x lies 

between 0 and 1, the quadratic contribution has a sign 

opposite that of the second difference.

For cubic interpolation, the quadratic interpolant is 

increased by -F3A3_i where F9= %(l+x)x(l x) is tabulated in table 37. The cubic contribution has a sign 

opposite that of the third difference.

For quartic interpolation, the cubic interpolant is 

increased by F^i.z where F±=%\ (1 +#)#(! x)(2 x) is 

tabulated in table 37. The quartic contribution has 

the same sign as the fourth difference.

For direct calculation, quartic interpolation leads to

4=/(*) =* /(I) + (l-* 4-2. (4.32)

If the value of Fk is needed between those tabulated in 

table 37 (p. 59), it may be calculated directly or ob-
8 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

tained by interpolation. Linear interpolation is usually 

adequate. The values of Fk may also be found in more 

extensive published values of Lagrangean interpolation coefficients (see Mathematical Tables Project, 

1944).

ASYMPTOTIC EXPANSIONS

The arguments in table 38, page 59, extend to 30, 

corresponding to distances between the current pole 

and potential point of not exceeding 60 times the 

thickness of the overburden. For larger values of the 

argument, asymptotic expansions are available.

If the bed is a perfect conductor, the reflection 

factor is Q= 1 and for a ^ 30, the modified potential 

can be shown to have the value, to at least eight 

decimals,

TF(-l,a)=log.2-^, (4.411) 

where

log. 2=0.693 147 181. (4.412)

If the bed is a perfect insulator, Q=-\-l and for 

a ^30, to at least eight decimals, the modified potential 

can be shown to have the value

J^2a'

where

#=0.809 078 696.

(4.421)

(4.422)

If the bed is neither a perfect conductor nor a 

perfect insulator, the reflection factor has a numerical 

value less than unity. For these cases Q2 <Cl, and the 

modified potential for large values of a can be shown 

to have the value

TF=log.(i Q) 

n=l

(4.431)

where the values of Bn for the first five values of flare shown in table 1. In this expansion, Bn eventually 

increases as the value of n becomes larger so that for 

a selected value of a, the terms eventually increase 

in numerical value. Such a series, in which the terms 

initially decrease but pass through a minimum value 

and eventually increase, is an asymptotically divergent 

series, which can be of practical utility if the minimum 

term has a negligible value. In such series, the error 

usually does not exceed twice the first neglected 

term, so that the expansion should stop after the term 

immediately preceding the term having the minimum 

absolute value. For Q^0.6, asymptotic expansions 

check the tabulated values at a=30, and hence may 

be used for all larger values of a to the eighth decimal 

place, except for forcing errors. For Q=+0.7 the 

agreement is to six decimals, for Q=+0.8 to four 

decimals, and for Q=+0.9, to only two decimals. 

For values of a much larger than 30, the agreement 

is better, but for values slightly larger than 30 caution 

is advisable.

For two points close to each other, it may be more 

accurate to calculate the difference directly, in the form

- TF(a2 ) = (4.432)

n=l

as the minimum term in this difference may be negligible. In this form, the logarithmic term disappears.

TABLE 1. Coefficients in asymptotic expansions

0

-0.9 

- .8 

- .7

- . 6 

- .5 

- .4

- .3 

- .2 

- . 1

+ .1 

+ .2 

+ .3

+ .4 

+ .5 

+ .6

+ .7 

+ .8 

+ .9

Bi

+ 0. 473 684 210 

+ . 444 444 444 

+ . 411 764 706

+ . 375 000 000 

+ . 333 333 333

+ . 285 714 286

+ . 230 769 231 

+ . 166 666 667 

+ . 090 909 091

- . Ill 111 111 

- . 250 000 000 

- . 428 571 429

- . 666 666 667 

- 1. 000 000 000 

- 1. 500 000 000

- 2. 333 333 333 

- 4. 000 000 000 

 9. 000 000 000

£2

- 0. 006 560 723 

- . 013 717 421 

- .021 371 871

- . 029 296 875 

- . 037 037 037 

- . 043 731 778

- . 047 792 444 

- . 046 296 296 

- . 033 809 166

+ .075 445 816 

+ . 234 375 000 

+ . 568 513 120

+ 1. 296 296 296 

+ 3. 000 000 000 

+ 7. 500 000 000

+ 22. 037 037 04 

+ 90. 000 000 00 

+ 855. 000 000 0

B3

 0. 009 800 194 

- . 020 195 092 

- . 030 560 296

- . 039 825 439 

- . 046 296 296 

- . 047 524 841

 . 040 510 459 

- . 023 148 148 

+ . 000 209 561

- . 140 413 047 

- . 834 960 938 

- 3. 559 008 151

- 13. 935 186 19 

- 56. 250 000 00 

- 258. 750 000 0

- 1 559. 120 370 

- 16 267. 500 00 

- 693 191. 250 0

Bt

- 0. 034 513 792 

- . 069 727 872 

- . 101 350 283

- . 122 916 698 

- . 126 028 807 

- . 102 465 050

- . 050 577 938 

+ . Oil 252 572 

+ . 031 692 485

+ . 655 170 329 

+ 8. 042 907 715 

+ 61. 356 233 83

+ 414. 904 835 4 

+ 2 926. 875 000 

+ 24 789. 843 75

+ 306 389. 802 0 

+ 8 167 556. 250 

+ 1 561 123 659.

Bi

- 0. 218 709 985 

- . 430 814 191 

- . 594 086 090

- . 653 030 165 

- . 553 876 600 

- . 283 957 451

+ . 060 482 081 

+ . 233 178 298 

+ . 059 492 146

- 6. 059 494 754 

- 152. 164 363 9 

- 2 074. 095 551

- 24 214. 693 53 

- 298 503. 515 6 

- 4 655 051. 953

- 118 Oil 731. 7 

- 8 037 473 112. 

- 6. 890 926 X 1012
PHYSICAL PROBLEM 9

APPARENT RESISTIVITY

For a single overburden earth, the apparent resistivity 

of a surface configuration has the value

__r s\iwn

Pa~L1+Trm~Jpo<

DISTURBING FACTOR

(4.5)

For a single overburden earth, the disturbing factor 

of a surface configuration has the value

(4.61)

which reduces for a four-electrode surface configuration to

M= S{W]

WENNER OR LEE CONFIGURATION

(4.62)

For the Wenner Configuration of separation /, the 

distances are ru=r22=l and ri2=r2i=2I. By equation 

4.23, the corresponding arguments of the modified 

potential are an=a22 =//(2A.) and 0,12=021= I/ h. Accordingly, the disturbing factor for a Wenner configuration 

has the value

For each half of the Lee partitioning configuration, 

the disturbing factor is given by equation (4.63). Although the applicable distances and arguments of W 

are not those of the Wenner configuration, the final 

form of the disturbing factor is the same.

For large values of I, the asymptotic expansions of W 

lead to approximations for these two configurations:

For 

For

For

(4.641)

(4.642)

(4.643)

The errors in these expressions must be determined by 

comparison with the calculated values if I is not large 

with respect to h. Thus, for Q= 1, M< 0.01, by direct 

calculation, if />4.4A. If the asymptotic expansions 

are sufficiently close to the direct values for a tested 

value of l/h, they will also be sufficiently close for every 

larger value of l/h.

SUPERPOSITION

If the apparent resistivities as measured match the 

values predicted by theory, the characteristics of the 

theoretical medium may be assumed as approximating 

those of the earth. If the observations are made by 

the Wenner or partitioning configurations, as in vertical 

surveying, the modified potential makes it possible to 

determine the disturbing factor, M, as a function of l/h 

and the observations determine the apparent resistivity, 

pa, as a function of I.

As the disturbing factor is the ratio of the apparent 

resistivity to the resistivity of the overburden,

Directly,

log M=\0g pa log po.

log (l/K)=log I log h.

(4.711)

(4.712)

Accordingly, if log M is plotted against log (l/h) on a 

reference chart, and log pa is plotted against log I on an 

observation chart, either may be interpreted as a map 

of the other, if the proper value of Q has been chosen 

for the theoretical curve and if the earth approximates 

the assumption of a single overburden. The notation 

u = log (l/h), v log M, x = log l,y = log pa leads to 

the curves v = j(u) and y = g(x) and to the mapping 

functions u = x log h, v = y log p0 where log h 

and log po are constants. Accordingly, the iw-plane 

and the a^/-plane are identical except for translation of 

the uv origin to the point x = log h, y = log p0 . Corresponding curves on the two planes are superposed 

exactly after the translation.

On the reference sheet, cross-sectional lines should 

not appear, except for the resistivity index, for which 

M = 1 or log M = 0 and the depth index, for which I 

= h or log (l/h) = 0. The index point, which is the 

intersection of the two index lines, should be designated 

distinctly. If the reference curves are prepared on 

transparent sheeting, a small hole may be cut at the 

index point to permit marking through the sheet directly 

on the observation graph sheet. On the observation 

sheets, the cross-sectional lines should be retained, and 

the observations should be shown isolated without attempting to pass a curve through them.

In interpretation, the superposition and the observation curves are shifted, maintaining the disturbing 

factor axis parallel to the apparent (measured) resistivity axis. If a satisfactory agreement can be found 

between the observed curve and one of the reference 

curves, the earth may be considered as having a single 

overburden. The reflection factor, Q, is determined by 

the curve of fit. The resistivity of the overburden is 

determined as the observed resistivity corresponding to 

the resistivity index, and the depth is determined as 

the observed depth corresponding to the depth index.
10 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

If the superposition sheet is laid over the observation 

sheet, the intersection of the two index lines can be 

shown directly on the observation sheet, or read and 

recorded. Finally the resistivity of the bed is determined by

(4.72)

Values to be used in preparing a superposition sheet 

are shown as the disturbing factors in tables 2 and 3 

with the arguments Q and l/h. The Wenner or partitioning configuration is assumed. Superposition 

curves, plotted from tables 2 and 3, are shown in figures 

5 and 6, in which each curve is labeled with both the 

reflection factor and the ratio of bed resistivity to overburden resistivity for which it is plotted.

TABLE 2. Disturbing factor, Wenner or Lee configuration, buried conductor 

[Items marked with asterisk (*) not needed for figure 5]

Disturbing factor for indicated values of Q

Z/ft

0.2 _________________ .

0.4.___.-... _______ ..._______._

0.6.___..._. _..__.._.__.__.._..

0.8... .._____-._.-.-.___.__._-.

1.0._..._ __ ...... __ _-_-....

1.2 __ . _ ....._..._._.._._-__

1.4____.____. .._._...-_-._. -._-

1.6.._. ..__._. -_._-._-__-__..__

1.8... _-_.----.__-----------.--

2.0

2.2-_.____- ._.._..___._-..__--.

2.4 ___ ._....__. _ __________

2.6... ....... __ ._._._._._.._-

3.0__._.__. ______...___.._.....

3.2... ....._-..._-_._.._._.....

3.6__.__. _..._.._._._.._.___-. _

4.0-_.________. ________________

4.4...__-_. _....-..-._._._... _.

5...... _______________________

6.

7.. ._._._._._..__.___....__...

8..... _...._._._.._._.___-..._

9 __ _________________________

10_______________. ____________

12_. __________________________

14____________________________

16 _ _ _ __ _ ___ __ _ _ __._

18______-___-____-_-_________.

20____________________________

22____________________________

24__________________ _ _______

26____-___-___________________

30.____________.___________.__

50-__._____________________.__

100. __ ________ __ __________

CO

-0.1

(*)

(*)

(*)

0. 9768

. 9633

. 9490

. 9350

.9220

.9101

.8996

.8904

.8823

.8753

.8639

.8593

.8519

.8461

.8416

.8366

.8311

.8278

.8255

.8240

. 8229

.8215

.8206

.8200

.8196

. 8194

.8192

. 8190

.8189

.8187

.8184

. 8182

.8182

-0.2

(*)

(*)

(*)

0. 9543

. 9279

. 9002

. 8732

.8482

.8257

.8059

.7887

.7738

.7610

.7406

.7324

.7195

.7097

.7022

.6941

. 6856

.6804

.6771

.6749

.6733

.6712

.6700

.6692

.6687

. 6683

. 6680

.6678

.6676

. 6674

.6669

.6667

.6667

-0.3

(*)

(*)

0. 9654

. 9326

.8938

. 8534

.8142

.7783

.7462

.7182

. 6940

. 6734

. 6558

.6283

.6175

. 6005

.5882

.5790

. 5691

. 5590

. 5532

.5496

.5471

.5454

.5432

^41 Q

. 5411

.5406

.5402

. 5399

.5396

. 5395

. 5392

.5387

.5385

.5385

-0.4

(*)

(*)

(*)

0. 9114

.8609

.8084

. 7579

. 7118

.6711

.6358

. 6056

.5801

.5587

. 5256

.5130

.4934

.4794

.4693

.4587

.4483

.4425

.4390

.4366

.4350

.4330

.4318

.4310

. 4305

.4301

.4298

.4296

.4295

.4293

.4288

.4286

.4286

-0.5

(*)

0. 9804

. 9439

.8909

.8290

.7650

.7039

.6485

.5999

.5582

.5229

.4934

.4689

.4316

.4176

.3965

.3817

.3712

.3607

.3508

.3455

.3423

.3402

.3388

.3371

.3361

. 3354

.3350

.3346

.3344

.3342

.3341

.3339

.3335

.3334

.3333

-0.6

(*)

(*)

(*)

0. 8710

.7981

.7233

.6522

.5881

.5324

.4850

.4452

. 4124

.3854

.3452

.3304

.3083

.2935

.2833

.2733

.2644

.2598

.2572

.2555

.2544

.2530

. 2522

.2516

.2513

.2510

.2509

.2507

.2506

.2505

.2502

.2500

.2500

-0.7

(*)

(*)

(*)

0. 8516

. 7682

. 6829

. 6024

.5304

.4682

.4157

. 3723

.3367

.3078

.2655

.2503

.2282

.2136

.2041

. 1950

. 1874

. 1838

. 1818

. 1805

.1797

. 1787

. 1781

. 1777

. 1774

. 1772

. 1771

. 1770

. 1769

. 1768

. 1766

. 1765

.1765

-0.8

(*) (*)

0. 9134

.8326

.7391

. 6439

.5546

.4751

.4071

.3502

.3036

.2657

.2354

. 1918

. 1766

. 1548

. 1412

. 1323

. 1245

. 1186

. 1159

. 1146

.1137

. 1132

. 1125

. 1121

. 1119

. 1117

. 1116

. 1115

. 1114

. 1114

. 1113

. 1112

. 1111

. 1111

-0.9

(*)

(*)

(*)

0. 8141

.7108

.6061

. 5085

.4222

.3488

.2880

.2387

. 1991

.1677

.1236

.1085

.0876

.0751

.0674

.0610

.0567

.0550

.0543

.0539

.0536

.0533

.0531

.0530

.0529

.0529

.0528

.0528

.0528

. 0527

.0527

.0526

.0526

-1.0

0. 9948

.9629

.8942

.7960

.6833

.5696

.4640

.3714

.2932

.2289

.1773

.1364

. 1044

. 0604

.0457

. 0259

.0146

.0082

.0034

.0008

.0002

.0000

.0000

.0000

.0000

.0000

.0000

.0000

. 0000

.0000

. 0000

.0000

.0000

.0000

.0000

.0000
PHYSICAL PROBLEM 11

TABLE 3. Disturbing factor, Wenner or Lee configuration, buried insulator 

[Items marked with an asterisk (*) not needed for figure 6]

I

0.2 --_----_----__-________-_

0.4_. -------________-________ .

0.6.-- ------_---______________

0.8.__ -_ _______

1.0. ________ ____ ... __ _ _

1.2.__ ________________________

1.4. -__----.__. ____ ________

1.6.... _ ...- _ . _ .__._....

1.8. -_--_.-_._-.__._._.._.....

2.0. __________________________

2.2. -_--..__-.- _ _____ _ ____

2.4__. -._.___________.___._.__

2.6._ _--__-_--.-_-____________

3.0.__ --_-_------_____-_-____

3.2. --_--_-_---_._____________

3.6. __________________________

4.0. -_-.-___-.__--___. ________

4.4_. _-_-..-__ _ _____________

5-___--------- - --._--.- _ __.

6 ______ 

7___-_ _-____-__ ----_-_-__-_

8 __ _ __ _ _ ----__. _ ______

9 _____ -.-_-_________ ____ _

10__---_--_---_-_-___.________

12__________. _________________

14___------ ________ _______

16_ _ _ _ -_-_ _ _ ___ _______

18_-_----_------_-_____.______

20__-_-_-_-___-_._____________

22 ___ __________ _ _____ __ _

24 _ _____ __ _ _ _ _ ____ _ _

26__--_--_--________-________-

30_.__ _._ _ _ ___ __________

50____________________________

100. ____ -----__--_-___-____.

CO

+0.1

(*) (*) (*)

1. 0241

1. 0383

1. 0534

1. 0685

1. 0828

1. 0959

1. 1078

1. 1186

1. 1281

1. 1367

1. 1508

1. 1567

1. 1668

1. 1748

1. 1813

1. 1888

1. 1976

1. 2034

1. 2074

1. 2103

1. 2124

1. 2153

1. 2170

1. 2182

1. 2190

1. 2196

1. 2201

1. 2204

1. 2207

1. 2211

1. 2218

1. 2221

1. 2222

+0.2

(*)

(*)

(*)

1 04QO

1. 0782

1. 1095

1. 1408

1. 1708

1. 1987

1. 2243

1. 2475

1. 2683

1. 2871

1. 3190

1. 3325

1. 3558

1. 3747

1. 3903

1. 4091

1. 4313

1. 4465

1. 4573

1. 4652

1. 4711

1. 4792

1. 4844

1. 4879

1. 4903

1. 4921

1 4Q34.

1. 4944

1. 4952

1 4QR4

1. 4987

1. 4997

1. 5000

+0.3

(*)

(*)

1. 0380

1. 0750

1. 1200

1. 1686

1. 2176

1. 2649

1. 3094

1. 3505

1. 3882

1. 4225

1. 4536

1. 5077

1. 5309

1. 5715

1. 6053

1. 6336

1. 6683

1. 7110

1. 7411

1. 7630

1. 7794

1. 7919

1. 8095

1. 8210

1. 8288

1. 8343

1. 8384

1. 8415

1. 8439

1. 8458

1. 8486

1. 8540

1. 8563

1. 8571

Disturbii

+0.4

(*)

(*)

(*)

1. 1022

1. 1639

1 2312

1. 2995

1. 3660

1. 4292

1. 4883

1. 5429

1. 5932

1. 6395

1. 7210

1. 7567

1. 8200

1. 8737

1 Q1QQ

1. 9775

2. 0506

2. 1039

2. 1439

2. 1745

2. 1985

2. 2329

2. 2559

2. 2720

2. 2836

2. 2923

2. 2989

2. 3040

2. 3081

2. 3141

2. 3262

2. 3315

2. 3333

ig factor for i

+0.5

(*)

1. 0226

1. 0658

1. 1307

1. 2104

1. 2977

1. 3872

1. 4753

1 ^^Q7

1. 6395

1. 7143

1. 7838

1. 8486

1. 9646

2. 0164

2. 1096

2. 1905

2. 2613

2. 3517

2. 4702

2. 5599

2. 6292

2. 6838

2. 7275

2. 7922

2. 8369

2. 8688

2. 8924

2. 9103

2. 9242

2. 9351

2. 9439

2. 9569

2. 9837

2. 9958

3. 0000

ndicated val

+0.6

(*)

*)

(*)

1. 1607

1. 2596

1. 3689

1. 4819

1. 5942

1. 7031

1. 8072

1. 9059

1. 9989

2. 0865

2. 2462

2. 3189

2. 4518

2. 5698

2. 6751

2. 8130

3. 0003

3. 1475

3. 2653

3. 3609

3. 4396

3. 5599

3. 6463

3. 7101

3. 7586

3. 7961

3. 8257

3. 8495

3. 8688

3. 8980

3. 9602

3. 9897

4. 0000

ues of Q

+0.7

(*)

(*)

(*)

1. 1926

1. 3123

1. 4458

1. 5851

1. 7251

1. 8624

1. 9954

2. 1230

2. 2450

2. 3613

2. 5772

2. 6776

2. 8640

3. 0338

3. 1886

3. 3965

3. 6898

3. 9306

4. 1308

4. 2990

4. 4418

4. 6691

4.8402

4. 9720

5. 0754

5. 1580

5. 2250

5. 2800

5. 3254

5. 3959

5* 5554

5. 6367

5. 6667

+0.8

(*)

(*)

1. 1131

1. 2268

1. 3694

1. 5298

1. 6991

1. 8713

2. 0424

2. 2103

2. 3737

2. 5321

2. 6852

2. 9757

3. 1133

3. 3746

3. 6184

3. 8464

4. 1614

4. 6252

5. 0251

5. 3728

5. 6771

5. 9452

6. 3941

6. 7527

7. 0439

7. 2834

7. 4827

7. 6502

7. 7923

7. 9138

8. 1091

8. 5961

8. 8825

9. 0000

+0.9

(*)

(*)

(*)

1. 2640

1. 4322

1. 6235

1. 8278

2. 0385

2. 2510

2. 4627

2. 6721

2. 8782

3. 0808

3. 4743

3. 6653

4. 0360

4.3922

4.7348

5. 2250

5. 9839

6. 6787

7. 3171

7. 9055

8. 4495

9. 4421

10. 265

11. 002

11. 650

12. 223

12. 732

13. 187

13. 596

14. 296

16. 387

18. 060

19. 000

+1.0

1. 0070

1. 0511

1. 1512

1. 3062

1. 5044

1. 7329

1. 9810

2. 2410

2. 5083

2. 7799

3. 0540

3. 3294

3. 6056

4. 1593

4. 4364

4. 9907

5. 5452

6. 0997

6. 9315

8. 3177

9. 7041

11. 090

12. 477

13. 863

16. 636

19. 408

22. 181

24. 953

27. 726

30. 498

33. 271

36. 044

41. 589

69. 315

138.63

CO

If the logarithm of the disturbing factor is plotted 

against the natural value of the reflection factor for 

a selected set of values of the relative separation, 

l/h, a set of smooth curves results and graphical interpolation permits the plotting of superposition curves 

for reflection factors intermediate between those shown 

in figures 5 and 6. The sets of such curves shown in 

figures 7 and 8 may be used for this purpose, or new

curves may be drawn on cross-section paper to permit 

direct reading with better accuracy.

ILLUSTRATIVE EXAMPLES 

SUPERPOSITION

To illustrate the method of superposition, consider 

the values shown in figure 9, in which the observations 

are indicated by circles and the curve for $=-1-1.0
12 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

Resistivity index 

0.3 5 6 7 8 9 10 20 30 40 50 60 70 8090 100 0.4 05 0.6 0.7 OS 0,9 1.0

O.OI 0.01

50 60 70 8090 100

SEPARATION/DEPTH, l/h 

FIGUKE 5. Superposition curves, buried conductor, Wenner or Lee configuration.

has been copied from a family of superposition curves. 

The circles show the values of pa plotted against I 

on double logarithmic paper. As the apparent resistivity increases with increasing spread, a buried insulator 

is indicated. The family of curves for a buried insulator (fig. 6) was laid over the graph and translated 

without rotation to obtain the curve of best fit. In 

this example, the curve Q= + 1.0 fits best. The index 

was indicated through the reference sheet, showing 

a depth of 6.0 feet and an overburden resistivity of 

2,440 ohm-cm. As Q= + l, the bed resistivity is 

infinite.

FOUR-EIJSCTRODE SURFACE CONFIGURATION

As an example of the four-electrode surface configuration, let the current source lie at the origin, the 

current sink at (2000,0,0), the potential reference point 

at (1500, 500, 0), and the potential test point at 

(400, 300, 0). Then Zn=P#=500, 112=PL=500^10, 

l21 =NH=WQ^/265, and laa=NL=5QQ^2. The normal 

potential for a pole of strength 1000 units is

2.167 462 913. For the approximation in which N 

and L are assumed to lie at infinite distances, the potential is 2 units so that the approximation error 

amounts to about 8.4 percent.

If a perfect conductor is taken at a depth of 100, 

Q= 1 and the calculations are as follows:

On = 2.5,

=7.905 694 150, 

a2i=(l/2)265 =8.139 410 300,

=3.535 533 905,

Wu = 0.4934 8923 

JFi2 = 0.6299 0162 

Wsl = 0.6317 1767 

JF22 = 0.5517 3698.

Thus S{W}= -0.2163 9308 and V= +0.003 5321. 

Assuming, N and L at infinity, Z12 =Z21 =Z22 = » and by 

equations (4.41), W12 =TF2 i==IF22=log. 2=0.6931 4718 

so that V= 0.003 4205, the difference being about 3.2 

percent. It may be noted that the use of eight significant figures in the values of the modified potential 

leads to only five significant figures in the calculated 

value of the potential.

If the bed has a resistivity 19 times that of the 

overburden, Q=+0.9, the calculated potential is
PHYSICAL PROBLEM 13

30 40 50 60 708090100

100

1

02

- 2

03 0.4 05 0.6 0.7 Q8 09 10 2 3 4 56789 10 

SEPARATION/DEPTH, l/h

20 30 

Resistivity i

50 60 70 80 90 100

FIGUEE 6. Superposition curves, buried insulator, Wenner or Lee configuration.

13.736 310 and the limiting value as N and L recede 

to infinity is 16.893 981. Accordingly, neglecting of 

the contributions of N and L introduces an error of 

about 25 percent.

INTERPOLATED VALUES OF THE MODIFIED POTENTIAL

The use of interpolation in table 38 (p. 59) may be 

illustrated by specific examples. Let Q= 0.8 and let 

the argument lie between 13.5 and 14.0, which appear 

in the table. Then a0=13.5, ^=14.0 and the tabular 

spacing is &= 0.5 so that the phase is x=(a 13.5)/0.5 

=2(a-13.5).

The table shows that TF(a0)== 0.5548 7048 and that 

IF(ai) =0.5560 4567. The second, third, and fourth 

order differences, with the unit position in the eighth 

decimal place of W, are A2_!= 9035, Al^+934, A12 

= 149. Let L be the result of interpolating linearly 

between the tabulated values. Then

=0.5560 4567z+0.5548 7048(l-z)

The correction to L is, in the proper decimal position,

or a reduced form.

Example 1

Let a= 13.71 and W be to 4 decimals. As the values 

of the differences to 4 decimals are each less than 1, 

linear interpolation is adequate. The phase is 

0.21/0.5=0.42 so that

TF(13.71)=0.42(0.5560)+0.58(0.5549)=0.5554.

Example 2

Let a= 13.71 and W be to 6 decimals. Here the 

differences are listed as

Ai!= 90, Al^+9 and A12=-1,

and the phase is x=0.42. As the fourth difference is 

less than 43, the third less than 16 and the second
14 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

0.8- 

0.8-

0.6- 

O.SZ 0.08-

g 0.08-

3

tCO

Q 0.06-

-0.1 0.2 0.3 0.4 0.8 Q.« 0.7 0.8 0.9 -1.0

4.0

0.2 as 0.4 as 0.6 0.7 0.8 o.s 

REFLECTION FACTOR, Q

FIGURE 7. Interpolation chart, burled conductor.

greater than 8, quadratic interpolation should be used 

so that C=Ftb*_ l . By table 37, page 59, the value of 

F2= 0.12180. The linear interpolant is L=0.555 364, 

and the quadratic correction is +11 so that TP(13.71) 

= 0.555 375.

Examples

Let a =13.71, and W be to 8 decimals. As A12>43 

quartic interpolation is required. The phase is x 0.42. 

Table 37, page 59, shows that F2= 0.12180, F3 

= -0.05765, and F*= +0.0228. 

Accordingly,

L= (0.42) (0.5560 4567)+ (0.58) (0.5548 7048) = 

0.5553 6406

and

(7= (-0.12180) (-9035) + ( 0.05765) (+934) 

+ (0.0228) (-149) = +1043

+0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.8

0.2 0.3 0.4 as O.S 0.7 0.8 0.8 + 1.0

REFLECTION FACTOR, Q 

FIGURE 8. Interpolation chart, burled Insulator.

in the eighth decimal place, so that

TF(13.71) =0.5553 7449 

with the last digit subject to forcing errors of rounding.

Example 4

Let a=13.713 962 and W be to 8 decimals. 

The phase is x= 0.427 924. 

By direct calculation

0253

6050

9748

L=0.5553 7337 

(7= + 1048 

W= 0.5553 8385

As the differences are to 4, 3, and 3 significant digits, 

the tabulated values of Fk may be interpolated linearly
MATHEMATICAL PROBLEM 15

100,000

10 50 100

50,000

CO

cc. 

uu

UJ

z

Ul

o

10,000

CC.

5000

1000

Index

h = 6.0 feet 

P=2440 ohm-cm

100,000

50,000

10,000

5000

5 10

SEPARATION, IN FEET 

FIGURE 9. An Illustrative example of superposition.

50

1000

100

to the same accuracy, using the phase x== 0.7924 and 

leading to

F2= -0.1224, F3= -0.0583, ^=+0.0229, 

C= + 1048 and TF=0.5553 8385. The difference is 

within the forcing error of rounding. If the values for 

Fk are taken directly from the table for a=0.43, they

are

^2=-0.1226, ^3=-0.0584, and ^4=+ 0.0229 so that 

(7=1050 and TF=0.5553 8387, also within the forcing 

error of the refined value.

As this example illustrates, the values of table 37, 

page 58, may be used without interpolation for most 

purposes and with linear interpolation for better

accuracy. Only in rare calculations is it necessary to 

calculate the coefficients by formula. The magnitudes 

of the differences and their rates of change determine 

the validity of the simpler approximations.

MATHEMATICAL PROBLEM 

NOTATION

Mathematically, the present problem is that of 

finding a function that determines the value of the 

difference in electrical potential for each pair of fieldpoints when the locations of the current sources and 

sinks are known. As the potential function for a 

selected field is the algebraic sum of the potential 

functions due to the separate current poles, the funda-
16 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

mental problem is that of determining the potential 

function corresponding to a single current pole. The 

extension to more complicated fields is direct.

As actual measurements of electrical potential 

always involve the measurement of the difference in 

potential between two points, the potential function 

lacks uniqueness to the extent of an additive constant, 

which may be selected as convenient for each measurement of potential difference and may be altered between pairs of points when such an alteration is convenient. As a constant multiplier does not affect the 

fundamental properties of a potential function, the 

solution may be made for a unit current pole and the 

function may be multiplied by a factor appropriate to a 

specific current strength.

The basic problem is that of an infinite space separated into three sections by two parallel planes. Although rotation of the entire space may be needed in a 

geological application, such as exploration over a 

sloping plane, the separating planes may be considered 

as horizontal. The section between the two planes may 

be considered as "the overburden", one external section 

as "the air", and the second external section as "the 

bed". These terms have no mathematical significance 

but are convenient in discussing the regions, and the 

terminology transfers conveniently into geological 

significance. The air, in this use, may have an arbitrary value of resistivity.

The Z-axis may be taken perpendicularly to the 

boundary planes with its positive sense from the air 

toward the bed. The x- and y-axes may be taken 

arbitrarily, but mutually perpendicularly, in the plane 

separating the air from the overburden. For the fundamental problem, there is a single current source so that 

the field has cylindrical symmetry about the Z-axis. 

Let the air, the overburden, and the bed be designated 

as Gi, G2, and G2 , respectively (fig. 10), and let the

z

/

Axis

Air 

p

Po!e

:B\

Overburden

Test point -Hf

'mmmmmme*

FIGTTKE 10. Two-boundary space, Internal pole. G\ is the air; Ot, Is the overburden; 

Ot, Is the bed; m is the current pole of unit strength; pt, is the resistivity of Ok; 

Bi, is the upper boundary; B8, is the lower boundary; 6, is the distance of the pole 

below BI; c, is the distance of the pole above Bj; h=b+c, is the thickness of the 

overburden; z, is the depth of the test-point below BI; I, is the axial distance of the 

test point (horizontally from the axis) ..........

corresponding resistivities be pi, p2, and p3 , respectively. 

Let the plane BI separate the air from the overburden 

and the plane B2 separate the overburden from the bed. 

In the geophysical case, the resistivity of the air is 

infinite, but this restriction may be removed for the 

mathematical problem. If the distance between the 

separating planes is h, the plane BI is specified by 2=0 

and the plane B2 by 2=h. There are two distinct 

cases, according to the location of the current source. 

For an "internal pole", the current source lies between 

the two planes; for an "external pole", the source lies 

in one of the external regions. Let the source lie at 

P= (0,0,6) and have a strength m=l. Let c=h b. 

Let the test point lie at H=(x,y,z). Because of the 

cylindrical symmetry, the values of x and y enter only

through l=-Jx2 -{-y'2 , the distance of H from the "axis" 

of the space.

As the term "electric" or "electrical" is implied 

throughout this discussion, it will be omitted except 

where need for clarity or emphasis.

Unless otherwise specified, each counter will be assumed to have the range of positive integers, excluding 

zero and including infinity. A summation sign without 

limits will mean that each counter covers the range of

positive integers. For example, XX means

____ CO ____

bk=*>~\a 1k means bk=y~\a,k, and bk T~*

j=\ 

means

. Specifically Xja^-i is equivalent to

POTENTIAL FUNCTION

The potential function due to a single pole has several 

properties, not mutually exclusive, each generally 

accepted:

1. The potential function is continuous at every point 

of space, except at the pole.

2. At the pole, the potential function becomes infinite 

in value in the order of l/r, where r is the distance 

from the pole to the field point at which the 

potential is measured.

3. At points infinitely remote from the pole, the value 

of the potential approaches a fixed value Um , 

independently of the path of recession, in such a 

manner that the difference U U« approaches 

zero in the order of Ifr, where U is the potential 

value at the test point.

4. At a boundary separating two media, the product of 

the conductivity by the derivative, in the direction 

perpendicular to the boundary, of the potential 

function is continuous. At a pole, located on the 

boundary, this condition is meaningless but is not 

needed, as it can be treated as the limiting condition as the pole approaches the boundary.
MATHEMATICAL PKOBLEM 17

The derivative, in a direction parallel to a boundary,

of the potential function is continuous. 

Except at the pole and at a boundary, the derivative,

in an arbitrary direction, of the potential is

continuous. 

Except at the pole, the potential is harmonic, so

that the potential function satisfies Laplace's

equation

(5)

8. A special potential function is 1/r. Multiplication 

by a constant and addition of a constant are 

permissible.

For multiple sources, the potential function is the 

sum of the functions that represent the potentials of 

the separate sources, each multiplied by an arbitrary 

constant. If the number of sources is infinite, the 

sum becomes an infinite series, which satisfies the 

conditions if it converges or if it consists of two parts, 

of which one converges and the other diverges in a 

manner independent of the distance from each pole.

UNIT POLE

In order to use the unit pole for problems involving 

electric current, a relationship must be established 

between a pole of strength TO and a current of strength 

7. If the current / enters a medium of uniform 

resistivity p, the differential form of Ohm's law is

, IF""'7' (6.1)

where U is the potential, r is the distance from the 

source to the test point and IT is the current density, 

at the testpoint, measured perpendicularly to the 

equipotential surface. As the total current 7 crosses 

each closed surface enclosing the source and as the 

equipotential surface for a single source in a uniform 

medium is a sphere with center at the source, the 

current density has the value

Ir=

47TT2

and equation 6.1 becomes

(6.2)

(6.3)

(6.4)

where the integration is taken along the path of the 

testpoint. The drop in potential from a point at the 

distance TI from the source to a point at the distance

so that the potential is

ip IpCdr

rz is independent of the path followed by the testpoint 

and has the value

TT TT Ipr*dr Ip /I 1\ ==U i L/2= -r l r=T~ I I* 

4-a-J ri r2 4x \ri r2 /

The drop in potential from the distance 7*1 to the distance 

r2 due to a pole of strength TO is

/I 1 = m [ 

\ri r

so that

l-^, l~47T

(6.52)

(6.61)

and a unit pole corresponds to a current of strength

7=1*. (6.62)

Accordingly, the solution for a unit pole must be multiplied by 7p/(4ir) to obtain the solution for a current of 

strength 7.

BOUNDARY CONDITIONS

At the boundary BI, the conditions on the potential 

function are:

2 = 0

P2

and at the boundary B2 , they are:

Ut- U3 =0

C)2 52

(7.11)

(7.12)

(7.13)

(7.21)

(7.22)

(7.23)

As usual, in evaluating a function for specified conditions, the conditions are imposed after all indicated 

operations have been performed. Specifically, in equations 7.13 and 7.23, the differentiation is performed before the values of z are substituted.

SIMPLE-IMAGE METHOD

The general aspects of the Kelvin image method have 

been known for about a century (Thomson, 1884, 1897, 

especially article 109, p. 174-176), and the method has 

been applied to the solution of a variety of problems. 

(See, for example: Jeans, 1925, p. 185-299, 341-363; 

Maxwell 1892, especially article 317, p. 443). The 

method is based on the principle that if a surface 

separates two media, the potential at every point of 

space is the same as if the medium of one side occupied
18 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

all space and the surface were replaced by properly 

chosen fictitious poles, or images, determined by the 

original pole and the surface.

Applied to a space consisting of two media separated 

by a single plane, the images are determinable without 

difficulty. This fundamental case may be used to 

determine the image sets in spaces involving plane 

boundaries, provided no contradictions of the fundamental conditions are involved (see Keller, 1953). 

The method has been applied to nonplanar boundaries, 

but such boundaries are not considered in the present 

study.

In figure 11, let the boundary plane between the 

media Oi and G2 be defined by 2=0, the resistivities of

Supplement

B

Complement

Pole

 z-o

FIOUEE 11. Simple images. B is the boundary plane between media Oi and GI. 

P, is the test point; pt, is the resistivity of <?*; m, is the unit pole; 6>0, is the distance 

of m below B; 6s>0, is the distance of complement below B; mi, is the supplementary 

image strength; ms, is the complementary image strength; 6i>0, is the distance of 

supplement above B; r, is the distance from pole to test point, P; rt, is the distance 

from m* to test point, P.

GI and G2 be PI and p2 , respectively, and a unit pole be 

located at (0, 0,6) in G2 where 6^0.

The method of images replaces the medium GI by one 

of resistivity p2 and two images: a supplementary 

image in the space occupied by GI and a complement in 

the original medium (?2 . Although the potential is 

mathematically continuous and a single function, the 

analytical expression for the potential function takes a 

different form in the spaces occupied by GI and by G2 . 

For the space originally occupied by GI, the actual pole, 

located at (0,0,6) is replaced by a complementary pole of 

strength m2 located in the region of G2 at (x2) y2) 62) 

where 62 ^0; for the space originally occupied by G2, 

the actual pole is augmented by a supplementary pole 

of strength Wi located in the region of GI at (xi, yi} &t) 

where 6j^0. The distances of the test point P from 

the pole, supplement, and complement are r, rh and r2 , 

respectively, where

(8.12)

(8.13)

The potential in the space occupied by GI has the value

i7!= 2(8.21) fa

and that in the space originally occupied by <72 has the 

value

But

5r_g 

6z r

so that

52 ~ n 

6r2 _z bz

62 r|

_b-z mi (61+g) ^T ~~ ^1 ' 

i

(8.311)

(8.312)

(8.313)

(8.321)

(8.322)

At the boundary, the testpoint has z Q, so that

Ui= (8.411)

S2

t72=I.L.El (8.412)

S Si

where

(8.421)

(8.422)

(8.431)

(8.432)

(8.433)

By equations 8.4, the boundary conditions of equations 7.1 become

(8.511) OT2 _ 1_m\ __

& si

& _j_ m,'&' n fo *19\ jj-t 8 " (O.014)

Solution of equations (8.51) for mi and m2 leads to

(8.521)

where

» gLii_

(8.522)

(8.523)

To have significance, the image strengths and positions must be independent of the position of the test
MATHEMATICAL PROBLEM 19

point. Accordingly, mt must be independent of x and

y where the test point is P= (x,y,$).

Specifically,

*i3 (x-x2 )} 

:-xl)+28ihi 1

(8.531)

As the right member of equation 8.531 is a polynomial 

in pi and p2 , the coefficients must vanish. The vanishing of the coefficient of px2 reduces to

0 = 3&1 6sis2«s[(s2 -si2 )a;-s2a;i]. (8.532)

Hence, if xyb^Q, either 61 =0 or Zi 0 and «i=s. 

The coefficient of p22 reduces to

(8.533)

Hence, if xyb^O, either 62=0 or ^=0 and Si=s. 

If SIT^S, bi=bz Q, which is impossible for b^Q by 

equation 8.512. Accordingly Xi=Q and Si=s. Similarly, yi=0, so that bi=b.

Setting xi~yl =Q, and bi=b, Si=s, equation 8.531 

reduces to

 2s*z2}\ 

f

and the coefficient of pip2 reduces to:

0=2&&2s3[2(«2 -sf)z-2s2a;2].

(8-534)

(8.535)

Hence 62 =0 or z2=0 and s2=s, with y2 =0 similarly.

If 62=0, equation 8.512 shows that mi=j-^ I, so

2s that by equation 8.511, m2= - Hence m|s2 =4si

s

which leads to m2=±2, z2=y2 =0 and b2 =b. Accordingly, in either choice, b2 =b and s2=s.

If Xi X2=yi=yz Q and bi=bi=b, it follows from 

equations 8.43 that s=Si=s2 , so that, by equations 

8.52,

m2 mi=l (8.5411)

 + =- (8.5412) 

pi Pt Pt

Equations 8.541 have the solutions:

_Pl~P2 

P1 + P2

= 2P1 

P1 + P2

(8.5421)

(8.5422)

The notation

P2

P1 + P2

P1

(8.551)

(8.552)

reduces the solutions of equations 8.542 to

mi=Q (8.561) 

m2 =r. (8.562)

The quantity Q may be called the "reflection factor" or 

"supplement factor," in B, of the medium Ga , and the 

quantity Tmay be called the "reflection complement," 

or "complement factor," in B, of medium (?2 .

Substitution of equations 8.56 in equations 8.2 

furnishes the two potential expressions:

Ui=T/r over the space occupied by GI (8.61) 

U2=l/r+Q/ri over the space opcupied by 0?2 (8.62) 

where r and rt are defined by equations 8.1.

Stated in words, the image method applied to a 

single horizontal boundary plane with a pole located 

in the lower medium leads to the conclusion that the 

upper medium may be considered as a continuation of 

the lower medium by the introduction of two additional 

poles obtained by "reflection" in the boundary. Below 

the boundary, the potential is that due to the original 

pole added to that due to the supplementary image 

located symmetrically above the boundary. The 

strength of the supplementary image is the product 

of the strength of the original pole and the reflection 

factor, in the boundary, of the lower medium. Above 

the boundary, the potential is that due to the complementary image located at the position of the original 

pole. The strength of the complementary image is 

the product of the strength of the original pole and the 

reflection complement, in the boundary, of the lower 

medium. As this method of analysis does not change 

the value of the potential at any point of space, the 

conclusions apply to the original space without change. 

It must be remembered that the solution implies the 

extension of medium G2 to all space so that, by equation 

6.61, a current of strength / corresponds to a pole of

strength m=-r?- to 4?r

As a matter of notation, if a medium Gj is reflected 

in the boundary that separates it from a medium Gk, 

the reflection factor has the value

"P»+P, (8.71)
20 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

and the reflection complement has the value

2P*

(8.72)

The images contributing to the potential over a 

selected region may be said to be "in" that potential, 

but each such image is external to the region over which 

the function applies. Specifically, an image contributing to the potential Uk is " in Uk" but lies externally 

to the region Qk over which Uk expresses the potential. 

In the limiting case for which the pole is on the boundary 

plane, both images coincide with the pole in position. 

However, this causes no difficulty if considered as a 

problem in limits.

POLE INTERNAL TO A PARALLEL PLATE

The two-boundary problem with plane parallel 

boundaries includes two distinct cases, according as the 

current source lies between the two boundaries or 

external to the medium between them. An internal 

pole lies in G2 (fig. 10, p. 16), and external pole lies in 

GI or G3 . The problem with the pole in G3 is omitted 

from this discussion as it is obtainable by a simple 

change in notation from the problem with the pole in 

ft.

For the problem of an internal pole, figure 10, the 

original pole must be reflected in the boundary BI 

and in the boundary B2 . Each of these reflections introduces into the external potential a complementary 

image located at the position of the original pole, and 

also introduces into the internal potential a supplementary image located in an external region. The 

new space has the uniform resistivity p2 . At each 

boundary, the original pole and the two images satisfy 

the potential conditions at that boundary. The complementary image belongs to an external potential and 

its region does not touch the other external region. 

However, the supplementary image belongs to the 

internal potential, whose region does touch the other 

external region. Accordingly, this new image must be 

reflected in the boundary which did not introduce it, 

and the solution of the problem involves two sequences 

of images, each involving reflections alternately in the 

two boundary planes.

If a pole or image of strength m is located at the 

depth £, reflection in a plane z u leads to two images. 

The complement has the strength T m and is located 

on the axis through the pole at the depth £. The 

supplement has the strength Q m and is located on the 

same axis at the depth (2it £). For the present 

problem, reflection is either upward in Bl or downward 

in B2 . All images are located externally to the regions 

to whose potential function they contribute. For

some plane boundaries, the application of the image 

theory leads to contradictions and can not be used 

unless modified (Keller, 1953).

For the present problem, reflection is from G2 into 

GI or Gs. The two reflection factors are

$23 =

P1 + P2

P3 ~P2 

P3 + P2

and the two complements are

2pi rn _ I 21 

P1 + P2

_2p3_ 

 T 

P3 -TP2

It is convenient to write

(9.111)

(9.112)

(9.121)

(9.122)

(9.13)

The sequences of images are shown in table 4.

Inspection of table 4 shows that the images fall into 

four groups whose depths are

z &-!,!= |2(fc-l)/i+b] (9.21)

,,=2(fc-l)A+& (9.22)

)h-b (9.23)

l)A-Bl. (9.24)

The images of the three potentials are mjku where j 

indicates the region or potential expression and k,u 

indicate the indices of the image. With this notation, 

the image strengths are

mi,*-i,2= T21Q*-i at depth z k-w

mlkz= T2lQ23Q>r-i at depth z fc3 

mz, k-l} i=Q2iQ k-1 at depth z k-1}1 

m2k2=Qk at depth z fc2 

 QisQ k~1 at depth z &3

= Q k at depth z M 

mZ)k-i,i=Q!i iT23Q k-1 at depth z k-i} i 

w3, s-i,4 = TWQ k~1 at depth

In addition,

The notation

(9.311)

(9.312)

(9.321)

(9.322)

(9.323)

(9.324)

(9.331)

(9.332)

(9.341)

(9.342)

(9.343)

(9.41)
MATHEMATICAL PROBLEM 21

TABLE 4. Two-boundary image sequences for an internal pole

Original pole is in Uj, has unit strength, and has depth b.

Q=Qai QJS.

Each line represents a single reflection from the preceding line.

Sequence with first reflection upward

Reflection

from precedingPole

Supplement. _______ _ __________ _____

Supplement___________________ _ _______ _

Supplement. ____ __ ___ ____ ___ _ _ _ ___

Supplement___________ ___ _____ _____

Supplement _____ _ _ _ _ _____ __________

And so on, with the general terms: 

Supplement ______ ______ ______ _ __ ___

Supplement______ _ _ ____ _ _ _ _ _____

in 

CQCQCQCQCQCQ OQO

introduces

supplement in Uj

strength

Qn

Q

<?2lQ Q2

QnQ2

Q*

On Q*-1 

Q*

depth

-b 

2h+b 

-(2A+6) 

4/i+ & 

-(4/1+&) 

6/1+ &

-[2(fc-l)fc+b] 

2fc/i+&

complement

in 

u,

Us 

Ui 

Us

u,

U3

u,

Us

strength

?«

^23 $21

T21 Q 

Qu TW Q T21 Q2 

Qn T2s Q2

Tn Q*-1 

Qn Tu Q*-i

depth

6 -6

2/i+ & 

-(2/i+&) 

4/i+ & 

-(4/1+&)

2(k-l)h+b 

-[2(k-l)h+b]

Sequence with first reflection downward

Reflection

from preceding 

Pole __

Supplement... ____ _____ ____ ____ __ _ _

Supplement.. ________ ___ _ _ ____ __

Supplement. ___ _ ____ _ _ ___ ___ _______

Supplement ____ _.__ ____ __ __ _ ____

And so on, with the general terms: 

Supplement. __ __ __ __ _____ _____

Supplement- _____ _ __ _ _____ _ ____ ____

in 

DQCQ CQDQCQCQCQCQ 

introduces

supplement in Uj

strength

<?23

Q

Q2

<?23<?2 

Q*

<?23 Q*-1

depth

2h- b 

(2h 6)

4/1-6

-(4/1-6) 

6/1-6 

-(6/1-6)

-(2kh-b) 

2kh-b

complement

in 

Us

u,

Us 

Ui 

Us

u, 

u,

strength

ff f\ 1 21 V23

T2sQ rrt s\ s\

j_ gi SJ23 *c 

/TT ^"j2

T^QxQ2 

T2S Q k~l

depth

6 

2/1-6 

-(2/1-6) 

4/1-6 

-(4/1-6) 

6/1-6

2kh-b

leads to the potential functions:

.

(9.421)

(9.423)

where S represents the sum in k over all positive 

integers, excluding zero.

For the limiting case of b = 0, in which the pole is 

located on the upper boundary, corresponding to the 

surface of the earth, the depths of equations 9.2 reduce 

to

*-ifl= -2(fc-l) A

**-!,!= -2(fc-

(9.511)

(9.512)

(9.513)

(9.514)

so that

and

(9.521)

(9.522)

(9.523)

(9.531)

(9.532)

(9.533)

For this limiting case, the potential may be written

Tn 

' roi

(9.62)

(9.63)
22 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

If the pole lies on the lower boundary, a corresponding 

change of notation reduces it to the present problem.

POLE EXTERNAL TO PARALLEL PLATE

In the problem of a pole external to a parallel plate, 

the pole may be taken at z= b, using the notation of 

figure 10. The first reflection is downward in BI, 

leading to the supplement Q12 in U\ at depth 6 and to the 

complement TV in U2 at depth ( 6). The supplement 

needs no further reflection, but the complement is in 

U2 and must be reflected in B2 . This introduces a 

complement in Us and a supplement in U2 . Accordingly, the £72 images are reflected alternately in the 

two planes to form a single sequence of images, each 

with its associated image in an external potential. 

The new space has a uniform resistivity pi. The 

details are indicated in table 5, with the same notation 

as for an internal pole, except that the reflection 

factors are

Ql2 =

p2 PI

P2 + P1

Q*=^

P1 + P2

f\ P3 P2 ^23 = T 

P3TP2

and the reflection complements are

(10.111)

(10.112)

(10.113)

Inspection of table 5 shows that the images fall into 

two groups, for which the depths are

z k-i.i=-[2(k-l)h+b] 

z k-i,z=2(k-l)h+b

and the corresponding image strengths are 

mioi=l at depth ZQI 

at depth 202

1 at depth «u 

m2>k-1} i= TuQ*-1 at depth z k~i,i

m2 jfc2=T1?Q23Q fc-1 at depth Z M 

mZ ki= Tu TnQ*-* at depth

(10.21)

(10.22)

(10.311)

(10.312)

(10.313)

(10.321)

(10.322) 

(10.33) 

Accordingly, the potentials for the external pole case

are

r*2

 

(10.41)

(10.42)

(10.43)

where r*_i,« is defined by equation 9.41.

For the limiting case 6=0, in which the original 

unit pole lies on the upper boundary, the depths of 

equations 10.2 reduce to those of equations 9.5, so 

that the potentials are

-t 1 

Tz

T<

2 , {iv.iti}

P1 + P2 

Pi / -I i\ -t oo\

P1 + P2

C/i= 12+Q*T12 TZ1 ^ (10.51)

C*2 " * 12 x i' "" ~| *K 23 - 12 X i \J.\J»v&J Tk-i,i Tja

Q k-1 

ft A C0\

P2+P3 "* 'ffc-I.l

TABLE 5. Two-boundary image sequences for an external pole

Original pole is in Ui, has unit strength, and has depth ( 6). Q

Each line represents a single reflection of an image from the preceding line.

Reflection introduces

from preceding 

Pole... _______________________

Complement. ________ _ _ _____

Supplement.. __ ____________

Supplement.. ______ _ __ _ ___

Supplement.. _______ _____ __

Supplement _ _ ______ ________

Supplement...... ____ ______ _

And so on, with the general terms: 

Supplement.. ______ __________

Supplement____________ ____ __

in 

Bi

Bz

Bi

Bz

Bi

B2

Bi

B2

Bi

in 

Ui

U2

U2

U2

U2

uz

U2

U2

U2

supplemen

strength

Qa r\ T1

QTn

Ti2Q*

TuQ23 Q*

rp fn

QnTnQ*-i

TuQ*

t

depth

&

2/1+&

-(2/1+&)

4/1+&

__ fAl _L J,\

&h+b

-(6/i+&)

2kh+b

-(2fc/i+&)

in 

ut

us

Ui

us

Ui

us

Ui

ut

Ui

complemeE

strength

Tit

T12 T23 Q2

f\ FT* fT1 /"i2

TttTnQ*-i

QaTuTuQ*-*

t

depth

-&

-&

2/i+ &

-(2M-&)

4fe+&

-(4/1+&)

6/1+ &

-[2(k-l)h+b]

2kh+b
MATHEMATICAL PROBLEM 23

If equations 9.6 are compared with equations 10.5 

an apparent inconsistency is detected. However, the 

unit pole corresponding to the current / is /p2/(47r) 

for equations (9.6) and 7p1 /(4ir) for equations 10.5. 

Furthermore,

rr _ m _ 2pip2 p2 J 21 Pl-t 12 j ' 

P1+P2

(10.6)

so that the potentials of both limiting cases reduce 

to

Lr02

Q*-i"J

r k2 J

-A Q fe~n

-T^J

(10.73)

SOME SPECIAL CASES

By choices of special values of the parameters 

involved, the general problem reduces to forms of 

interest for special purposes. The special case of 

6=0 has been considered. Except for the geophysical 

case, for which 2=6=0 and PI = °°, special cases will 

be indicated without detailed consideration.

IMBEDDED PLATE

If the parallel plate G2 is imbedded in a uniform 

medium of resistivity p1; p3=p1 

so that

n V21 

 

n V23 

 

Pi~P2 i 

P1 + P2

T21= TVS 

P1 + P2

Q=Qli

Qu=-Qii

ru=i-QM .

INSUtiATED PLATE

(11.12)

(11.13)

(11.14)

If the imbedding medium is a perfect insulator, 

Pi Pa °°> so that

Q21 = Q23= 1 (11.21)

The pole must be internal.

COVER PLATE

If the upper medium is a perfect insulator, the 

problem reduces to a conducting cover plate in contact 

with a second medium. For this case, Pi=<», so that

T2l =2

(11.31)

(11.32)

Q Qzs P3 P2 

P3+P2

Qu=-

(11.33)

(11.34)

(11.35)

(11.36)

(11.37)

(11.38)

GEOPHYSICAL CASE

In the geophysical case, the upper medium is air 

and both current pole and test point lie on the upper 

boundary, BI. Accordingly, for this case, PI=°° and 

6=2=0, so that

£21=1 (12.11)

P3 P2

(12.12)

(12.13)

(12.14)

(12.15)

(12.16)

By equations 10.7, the potential at the surface of 

the earth has the value

z=l

The introduction of the "relative displacement"

changes equation 12.21 to

Ip2l, 1 Q k

(12'22)

For Q=l, as k increases, the summand approaches 

l/k so that U becomes asymptotic to a harmonic series, 

which diverges. Accordingly, for every value of the 

reflection factor, Q, it is convenient to introduce the 

auxiliary variable

(12.31)

so that the potential function of equation 12.23 becomes

As the potential function lacks uniqueness to the 

extent of an additive constant and as 2Qk/k is independent of a and hence of /, the potential function may 

be written as
24 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

If k=l}2, then a=l and 2*i=2M=&, so that 

' x represents the excess of the

reciprocal of the depth of an image over the reciprocal 

of the distance from the testpoint to the image. Accordingly, it may be called the "reduced reciprocal 

distance". It depends only on the horizontal and vertical distances between the testpoint and an image. 

Accordingly, the summand is the product of Qk, which 

depends only on the ratio of resistivities p2 and p3 , 

and on ^fc(^)> which depends only on the distances involved. Each term of the summation is the product 

of a "resistivity factor" and a "distance factor." For 

many purposes, it is convenient to take the unit of 

length as twice the thickness of the parallel plate.

If P3=P2, the earth is homogeneous and Q=Q so that 

the potential reduces to the normal potential

Un==^l (12<41)

Accordingly, the surface potential for a two-layer 

earth is

U=Un SQV*- (12.42)

The medium, G2 has been called the overburden and 

the medium 6r3 the bed (see p. 16). Then BI will 

be the surface of the earth, assumed plane, and B2 

will be the contact of the overburden and the bed, 

also assumed plane. In applications, it is convenient 

to replace p2 by p0 and p3 by p6 , as a matter of notation. 

Then

Q=Pi pot

Pi + PO

(12.5)

Interchanging p0 and p6 reverses the sign of Q(see 

equation 12.5) but leaves its numerical value unchanged. 

This suggests the separation of even and odd terms in 

the potential, permitting simultaneous analysis of the 

buried conductor and buried insulator, having the two 

materials interchanged in the overburden and the bed. 

Such an interchange corresponds to an interchange 

of resistivities.

If the bed is more conducting than the overburden, 

Pb'Oo and Q<0. For a homogeneous earth, Q=0.

For purposes of analysis, it is convenient to write

2=M (12.61)

Wi=Zg2k~1^2k-1 (12.621)

W2 =2g2V2fc. (12.622)

As ^fc and g are both positive, W] and W2 will each be 

positive except for the trivial case in which Q=q=Q. 

If the bed is relatively an insulator,

Q=q

where

If the bed is relatively a conductor,

where

(12.71)

(12.721)

(12.722) 

(12.73)

(12.741)

(12.742)

(12.81)

(12.821)

(12.822) 

(12.83)

(12.841)

(12.842)

Inspection of equations 12.7 and 12.8 indicates that 

W{ is negative and We is positive, except for the trivial 

case Q=0. Accordingly, for a buried insulator, the 

actual potential is less than for a uniform earth 

having the resistivity of the overburden; whereas for a 

buried conductor, the actual potential exceeds that 

for a uniform earth. This paradox of a buried insulator 

corresponding to a decrease of potential arises from the 

fact that the introduction of the reduced reciprocal 

distance reverses the sign of the sum in the potential as 

given by equation 12.23 when expressed by equation 

12.33. In the original notation, the potential becomes 

infinite for 1=0 and approaches the value zero for Z <». 

In the modified notation, the potential vanishes for Z=0

Io 0* and approaches the value ^-T 2D TJT f°r £=°°. The

discrepancy disappears in the difference in potential 

between two points of observation. Specifically, consider two points at distances Zi and 4 from the surface

pole, so that 

tion 12.23,

l\ -i LZ i=~ and a2=-zT.

£JIV £lu

so that

"=^ Ff +i 

2,r Lfe fc

In the original equa-

(12.911)

(12.912)

(12.913)
FORMULAS FOR THE EVALUATION OF THE MODIFIED POTENTIAL 25

In the modifiied notation of equation 12.32,

U u V" u U'=~- -4-- u

Inspection shows that equations 12.913 and 12.923 are 

identical.

As W\ and W2 depend only on q and a, tables may 

be prepared using one table for each value of Q and the 

value of a as the argument in each table.

For some purposes, it is convenient to write

U=Un+^j-W (12.931)

without reference to whether the bed is a better or a 

poorer conductor than the overburden. For this use

JT=-SQV*. (12.932)

For a buried insulator W=TFf and for a buried conductor W We. For this reason it is convenient to call 

W the modified potential function.

FORMULAS FOR THE EVALUATION OF THE MODIFIED 

POTENTIAL

CALCULATION OF THE REDUCED RECIPROCAL 

DISTANCE

DIRECT CALCULATION

The basic calculation in evaluating the modified potential is that of evaluating the reduced reciprocal 

distance

(12.31)

If

and the terms of

are respectively smaller than those of

(12.932)

that

occur in equation 12.23. Furthermore ^ft decreases 

more rapidly than (a2 +&2 )~^ as k increases.

If an adequate table of square roots is available, the 

calculation is simplified. If not, the method of iteration may be used for finding the square root. If D 

is an approximate value of -J~N, better approximations 

are

V¥=(l/2) (13.11)

(13.12)

If a calculating machine with adequate capacity is 

available, the first form is simplest. However, the 

second expression leads to a somewhat more accurate 

value, as it determines a correction to the approximation, to the same degree of accuracy as the first form 

determines the entire value. 

After VA:2 a2 has been determined, determined the value of is

calculated by some convenient routine, such as

(12.31)

(13.21)

(13.22)

(13.23)

(13.24)

depending on the available computing machine and the 

preferences of the operator. Of this group equation 

12.31 has the advantage that the values of l/k may be 

preprinted on the calculation form. However, it is 

less occurate than the others because of forcing errors 

of rounding. The values should be calculated by two 

independent methods, as a check.

RESCAUQSTQ

As $t is homogeneous of order minus one in k and a 

jointly, it follows that fat (no) = so that if fa (a)

IV

is known ^wft(7ia) is available by division. In preparing 

tables of the reduced reciprocal distance, this device 

of "re-scaling" avoids the need for calculating a considerable number of values of ^*. As an example, 

direct calculation shows that

so that

and so forth.

= 0.055 983 235, 

= 0.027 991 618 

-0.0 18 661 078,

CONTBOL VALUES

The modified potential is given by equation 12.932, 

in which the terms decrease as the counter of the term 

increases. As each term has a resistivity factor which 

is a power of g, the largest value of g needed in the 

calculations will determine the largest value of k needed
26 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

to attain a selected accuracy. For g=l, the series 

converges very slowly. For the range g= [0(0.1)0.9], 

the value [#=0.9] determines the control on the value 

of k needed.

By the definition in equation 12.31, the value of the 

reduced reciprocal distance, $%, decreases as k increases. 

Hence

-i (13.31)

(13.32)

as fa<i^]c+i- The values of the controls are shown in 

table 7, for nine decimal calculations, with the unit in 

the ninth decimal place. For <£<i0.9, the series converges more rapidly than for # =0.9 and the fa usually 

vanish to the ninth decimal place before the control 

value is reached. If the $k do not become negligible, the 

same control values may be used as for g=0.9.

TABLE 6. Error factors in direct evaluation of W for varying 

factors of q

Equation 12.62 may be written

U co

J^F. ^ ' g24 l,fc . _L 'X A 02«+2ife \J, , , - (13411)

fc=l fc-1

U co

"W ^ ' O2^ 1 "X ' g2u+2kJ, (13 412}

so that

U co

fc=l fc=l

U co

0<TF2 -XJ fl'*to*<fl*"fc« S 22 *- (13.422)

K _ p i r. i

"D.-J. 

JtSUt

2"} g2fc=- » (13.43)

fc^l 1~~?2

so that

U / 2 \

''1^^* >/MJ 5 lr2&*"~l~T~l -| 2 / ^i Ir2tt I/ ^lO.T:OJ.y

fc=l ^ 2 '

W / 2 \

TT^ "C" ^ ' 0* fc li t-t-I 1 (09' tl \L ^ (1° 4^2^

~

If TFj or TF2 is calculated term by term for the first u

terms, the last term is <£u~lfau -i or #2 "^2 «, respectively,

so that the remainder, in either evaluation, can not

exceed the product of #2/(l #2 ) by the last retained

term. The remainder is always positive, so that if the

remainder is taken as the product of the last retained

term by q2/[2(l q2)], the error can not exceed this

remainder, except for the forcing errors of rounding.

This method fails for #=1, but is not needed, as convergence is too slow for direct calculation.

The values of the error coefficient g*/[2 (1 g2 )] are

shown in table 6. For q= 0.9, the error coefficient is

81/38 or about 2.13. Accordingly, for #=0.9, the error 

in Wi or W2 can not exceed 81/38 times the last retained

term. For an error not in excess of one in the ninth

decimal place, the last term must not exceed the product

of 1X10-9 by 38/81 or about 5X10~10. This furnishes

a control, ^k, such that the last term needed in the direct

calculation is the last one for which fa^^k . As

fa+i^tkrfk+i need not be calculated if fa<^)c+iAccordingly, direct calculation may be stopped as soon

Error factors for indicated values of Q

0 0.1 0.2

0 -L 1

198 48

0.3 0.4

9 2

182 21

0.5 0.6

1 9

6 32

0.7 0.8

49 8

102 9

0.9 1

81

38 "

TABLE 7. Control values for q=0.9

[Unit: 10-»]

k

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

QQoo

39

40

41

42

43

44

45

46 

47

48 

49

50

51 

52

53

54

55

56 

57

58 

59

60

61 

62 

63 

64

65

**

5

6

6

7

7

8

9

10

11

12

14

15

17

18

20

23

25

no£o

31

34

38 

42

47 

52

58

64 

71

79 

88

98

108 

120

134 

148

165

183 

203

226 

251

279

310 

344 

382 

425

J.79 tl£

k

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

QO

oo

84

85

86 

87

88 

89

90

91 

92

93 

94

95

96 

97

98 

99

100

101 

102

103 

104

105

106 

107 

108 

109

110

**

524

582

649

719

798

887

986

1 095

1 217

1 352

1 502

1 669

1 854

2060

2289

2 543

2 826

3 140

3489

3 876

4 307 

4 785

5 317 

5908

6564

7293 

8 104

9 004 

10005

11 116

12 351 

13723

15 248 

16 942

18825

20916 

23240

25 823 

28692

31 880

35 422 

39 357 

43730 

48589

53988

k

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

IQfi liO

129

130

131 

132

133 

134

135

136 

137

138 

139

140

141 

142

143

144

145

146 

147

148 

149

150

151 

152 

153

154

155

**

59 987

66 652

74057

82 286

91 429

101 588

112 875

125 417

139 352

154 835

172 039

191 155

212 394

235 993

262 215

291 350

323 721

359 691

399 656

444063

493 403 

548 225

609 139

676 821

752 024

835 582 

928424

1 031 582 

1 146 203

1 273 558

1 415 065 

1 572 294

1 746 993 

1 941 104

2 156 782

2 396 424 

2 662 694

2 958 548 

3 287 276

3 652 529

4 058 365 

4 509 295 

5 010 327 

5 567030

6 185 589

As fa<^ l/k for every value of a, no value of k in excess

of 155 will be needed for nine decimal calculations,

because (0.9) 156/156<5X10-10. As fa is large for small

values of k, controls for &<C21 are seldom needed and

do not appear in table 7.
FORMULAS FOR THE EVALUATION OF THE MODIFIED POTENTIAL 27

ASYMPTOTIC LIMITS OF W

As the value of a increases indefinitely, the value of 

W defined by equations 12.932 and 12.31 approaches 

the limit

^«=-]Cir (14JL)

From the customary series expansions of the logarithm,

locr ("1 r} = - ^S N * _ N * , ("14212 1)

 "- &« V x */ / I i. ,Z_J 07.__1 Z t 07, \J-~t-4mj

it follows that

Accordingly,

(14.221)

(14.222)

(14.31)

Also, by equations 12.62, as a becomes infinite W\ and 

W2 approach, respectively, the limits

_ (14.321)

Wam = 5-log. (1 g2 ). (14.322) £

As 2 lies between zero and unity, Wi m and W2a> are 

positive, as are Wi and W2, for all positive values of a. 

For 2= 1, equations 14.3 hold in a limiting sense but 

are of no use in computation. The numerical values 

of the limits are shown in table 8.

TABLE 8. Asymptotic limits of W

Q

0.1 

0.2 . ____ ...

0.3 

0.4 

0.5. .. ..

0.6 

0.7 

0.8 

0.9 _ . _ .....

1.0 

Win

0. 100 335 348

. 205 732 554

. 309 519 604

. 423 648 930

. 549 306 144

. 693 147 181

. 867 300 528

1. 098 612 289

1. 472 219 490

00

wta>

0. 005 025 168

. 020 410 997

. 047 155 340

.087 176 694

. 143 841 036

. 223 143 551

. 336 672 277

. 830 365 603

00

-wia>

0. 105 360 516

. 223 143 551

. 356 674 944

.510 825 624

. 693 147 181

. 916 290 732

1. 203 972 804

1 fiftO 437 Q19

2. 302 585 093

+wem

0. 095 310 180

. 182 321 557

. 262 364 264

. 336 472 237

. 470 003 629

. 530 628 251

. 641 853 886

. 693 147 181

EULER-MACLAURIN SUMMATION FORMULAS

For reflection factors numerically less than unity, 

the series Wi and W2 converge with sufficient rapidity 

that direct evaluation is at least feasible. It is true 

that as many as 78 terms may be needed in each of 

these series to assure an accuracy of a few units in the 

ninth decimal place, but the process is definite and the 

error can be limited arbitrarily. However, if the 

numerical value of the reflection factor is unity, the

terms of each series approach those of a harmonic 

series, which diverges, and direct evaluation is not 

practicable.

For this case, a group of formulas developed independently by Euler and by Maclaurin almost simultaneously (Charlier, 1907, p. 1) are applicable and 

permit the evaluation of the remainder after a number 

of initial terms have been calculated. For many 

series, the Euler-Maclaurin formulas permit the transformation of a slowly convergent series into an equivalent series that converges more rapidly. For this 

reason, one of the Euler-Maclaurin formulas is derived 

in a form convenient for the present problem (Roman, 

1931, 1936).

In the series

k=a

(15.1)

let the limit a be finite and the limit b be finite or 

infinite. If a function j(x) can be found, such that 

for every positive integer k from a, to 6,

/(*)=«*, (15.2)

the series S may be replaced by an integral, as a first 

approximation. All needed derivatives of j(x) must 

exist over the range (a,b) and for values somewhat outside this range. Usually, the function j(x) can be selected directly from the general term of the series. 

The translation

(15.31)

(15.32)

(15.41)

permits writing Taylor's theorem in the form

f(x) =

kl

Integration of /(#) from (n h) to (n+h) in x is equivalent to integration of <t>(h) from ( h) to (+h) in h, so 

that term by term integration of equation 15.41 leads to

(15.42)

For odd values of k, the corresponding members of the 

summation have the same value at each limit so that

rn+h _ 

Jn-h

which may be written

(15.431)
28 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

Application of this equation iof{zld (n) leads to

The notation

(2*+ 1)1

1 fn+h

(15.51)

(15.521)

(15.522)

(15.61)

(15.62)

Successive applications of equation 15.62 to 15.61 

lead to

converts equations 15.43 to

oo

/(») =

(n

(15.71)

If &2« represents the sum of all possible products of the 

a2jt for which the sum of the indices is 2u, with each 

term having a coefficient that is the number of possible 

distinct permutations of the indices, equation 15.71 

may be written:

(15.72) 1=1

Inspection of equations 15.7 shows the structure of 

62M . For u=l, the only term of equation 15.71 which 

can contribute to the term involving h2 in equation 

15.72 is the one for which k is 1 in the second term. 

The corresponding term is

so that

2=-1/6.

(15.811)

(15.812)

The term involving h* in equation 15.71 can come from 

the second term for k =2 and from the third term for 

j==k=l. Accordingly,

=. (15.82)

In a similar manner, other values of b2u are available. 

For example,

&«= (15.83)

(15.84)

Arbitrary selection of

leads to the relation

&o=l (15.91)

(15.92)

The common fractional values of b2u for values of u 

from zero to eight and the decimal values of 62« to 18 

decimals for values of u from 0 to 18 are shown in 

table 9. It can be shown that the ratio of 620+2 to 

&2« approaches 1/Tr2 as u increases indefinitely.

TABLE 9 Euler-Maclaurin coefficients 

[Values of M

2u

0

2

4

6

8

10

12

14

16

18

20

22

24

26

28

30

32

34

36

sign

+

+

+

+

+

+

+

+

+

+

Decimal

1. 000 000 000 000 000 000

. 166 666 666 666 666 667

. 019 444 444 444 444 444

.002060264 550264550

.000209 986 772486 772

. 000 021 336 045 641 601

. 000 002 163 347 442 779

. 000 000 219 232 713 446

. 000 000 022 213 930 854

. 000 000 002 250 767 479

. 000 000 000 228 021 077

.000000000023 101 422

. 000 000 000 002 340 586

.000000000 000 237 149

.000 000 000 000 024 028

.000 000 000000 002435

.000 000 000 000 000 248

.000000000000000 025

. 000 000 000 000 000 002

Common fraction

Numerator

1

1

7

31

127

73

1 414 477

8 191

16 931 177

Demoninator

1

6

360

15 120

604 800

3 421 440

653 837 184 000

37 362 124 800

762 187 345 920 000

By means of the definitions in equations 15.5 equation 

15.72 becomes

(16.11)

(16.12)

(16.2)

Introduction of the notation

transforms equation 16.11 into

f(n)=Z(n+h) Z(n h).

Successive applications of equation 16.2 results in the 

relations

sh]=Z[k+(2s+l)h]-Z[k+(2s-l)h] (16.311) 

Z[Jb+(28+3)ft]-Zflb+(2«+l)fc] (16.312) 

and so on to

f[k+2(t-l)h]=*Z[k+(2t-l')h]-Z[k+(2t-S)h] (16.321) 

f[k+2th]=Z[k+(2t+l)h]-Z[k+(2t l')h]. (16.322)
FORMULAS FOE THE EVALUATION OF THE MODIFIED POTENTIAL 29

Addition of equations 16.3 and cancellation of balancing 

terms leads to

(16.33)

Several special cases of equation 16.33 are important. 

The basic equations may be rearranged to furnish 

additional results (Roman, 1936). 

For A=l, equation 16.33 becomes

where

For k=Q, equation 16.411 takes the form

t

]£/(2«)=Z(2*+l)-Z(2«-l) i=s

and for k=l, it takes the form

(16.411)

(16.412)

(16.421)

(16.422)

The utility of the Euler-Maclaurin summation 

formulas is determined by the function Z(x). To be 

useful, the integral and odd order derivatives of j(x) 

must exist over the interval (s K) to (t+K) and be 

obtainable by direct calculation or by available tables. 

The derivatives must decrease as the order increases or 

at least increase so slowly that b2uf(2u~1) (x) converges 

with convenient rapidity as u increases.

The error in applying the Euler-Maclaurin formula 

in the original form usually does not exceed twice the 

first neglected term (Charlier, 1907} p. 14; Scarborough, 

1950, p. 184). Accordingly, if the terms begin to increase as the counter increases, the calculations should 

stop after the term preceding the minimum. Without 

proof, the present form of the Euler-Maclaurin formula 

may be assumed to have the same error criterion. The 

Euler-Maclaurin formula permits the evaluation of a 

divergent series if a term sufficiently small can be 

reached. For convergent series, the use of the formulas 

often increases the speed of convergence, thus reducing 

the amount of numerical calculations needed.

In the present study, the Euler-Maclaurin formula 

has proved convenient for q=l, but for §<X it has not 

proved practicable because the integral has not been 

obtainable in closed form or in available tables. Fortunately, it is not needed.

As the Euler-Maclaurin formula uses only the 

difference between two values of the auxiliary function, 

a constant term may be dropped from that function.

547123 60 3

The formula is especially convenient if Z(t) approaches 

a determinable constant as t increases indefinitely.

LOCAL EXPANSIONS 

LOCAL EXPANSIONS FOB

If the first n terms of the modified potentials are 

calculated directly, equations 12.62 may be written

(17.11)

where

k=n+l

k=n+l

(17.12)

(17.21)

(17.22)

The quantity Enk is the remainder, or residue, of order 

n, for the series WK -

For Q2=l, 2=1 and each of the series Wi, W2 converges very slowly. The Euler-Maclaurin summation 

formula expedites the numerical evaluation unless the 

argument a is so small and the degree of accuracy 

needed is so low that direct calculation is adequate. 

By equation 12.31 the simplest choice of equivalent 

function is

Direct integration transforms equations 18.1 to

//(*)<&= -log [1+Vi+3 

where

/o a=[ -

Differentiation shows that

where 

and

15

21 35

a02

315

55 . 495 , 1,155 , . 5,775 , 693 

- ~~-~

(18.1)

(18.2)

(18.3)

(18.41)

(18.42)

(18.51)

(18.52)

(18.53)

(18.54)

(18.55) 

«5 (18.56)
30 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

- i 1- 2,145 , 2,145 . , 45,045 

-i - * - --

9,009 . , 3,003 '-,,_ __,. (18-57)

j =1 105 a 4'095 -f 25,025 ^ 225,225

- 

1.785 of . 7,735 , , 425,425 , !- aH a*

153,153 , . 357,357 . 36,465 _ , 109,395 . ,, 0 _m ^- a5 -)- ' a8 jj a7 -| - a8 * (18.59)

It can be shown by mathematical induction that

(-1) *-? *(!)* (18.61)

(t*-l)!

where

and A is the largest integer not exceeding (l/2)(u 1). 

In particular, _4=0 for u=2 and odd values of u are not 

needed in the present study. 

With the notation

the Euler-Maclaurin function of equation 16.42 becomes, for 2=1,

 [tog.

By equations 15.8 and 19.11, the early values of 

02* are

1

60:c*

06 =

31

126a;6

127

120s8 

511

014=~

66s10

1 414 477 

: 16 SSOa;12

8191

016= 118 518 239 

4080a:16

For a selected value of a, as k increases indefinitely, 

the values of a and 02* approach zero and the values of 

DQ, _92*, and E_* approach unity. Accordingly, equations 17.2 and 19 show that, for g=l,

# != Z(2ri) (19.41) 

JBn2= Z(2»+l) (19.42)

In each remainder, the argument of the Euler-Maclaurin function Z(x) is one unit greater than the counter 

of the last term of the direct calculation.

In an isolated application, the value of Z(x) must be 

computed term by term until a term of negligible value 

is reached. In group calculations, it usually is preferable 

to predetermine the number of terms needed. Direct 

analysis leads to the values of (1 DajJEa) shown in 

table 10. On each interval between the values shown for 

a, the expression is monotonic. Numerical values are 

shown for a [1(1)25 (5)50] and k ^5, in table 11.

(18.62)

(*-l). 

are not

(19.11) 

(19.12)

42 be-

(19.2) 

lues of 

(19.31) 

(19.32) 

(19.33) 

(19.34) 

(19.35) 

(19.36) 

(19.37) 

(19.38)

TABLE 10. Variations of (1 DzkEzk) 

[Monotonic between values shown]

2k

2_.__ __ ______ _____

4 __ __________ _ __

6 __ _ __. ____ _ _ .__

8-.-. ____- _ _

10_______ ____ _ ______

a

0

00

0

2/3 

4/3

oo

0 

0. 217 786 631 

0. 450 806 662 

2. 448 880 035 

3. 549 193 338

oo

0 

0. 110 117 150 

0. 235 232 105 

0. 818 616 636 

1. 180 638 620 

5. 071 266 214 

6. 584 129 275

oo

0 

0. 066 855 287 

0. 146 317 486 

0. 430 721 573 

0. 624 921 564 

1. 657 987 349 

2. 129 978 868 

8. 511 102 456 

10. 432 115 413

oo

l-D»J2j*

0 

1

0 

1

1. 051 532 505 

1

0 

1 

1. 112 759 589 

1 

0. 998 344 737 

1

0 

1 

1. 158 338 770 

1 

0. 990 473 786 

1 

1. 000 031 129 

1

0 

1 

1. 192 181 694 

1 

0. 978 416 470 

1 

1. 000 509 415 

1 

0. 999 999 619 

1

In calculating the residue, the number of terms 

needed depends in a complicated manner on the value 

of the argument and the number of terms initially calculated. A reduction in the number of initial terms 

usually requires an increase in the number of terms 

needed in the residue to obtain a selected accuracy. No
FORMULAS FOR THE EVALUATION OF THE MODIFIED POTENTIAL 31

TABLE 11. Values of (l-DSkESk)

a

1 

2. 

3 

4.. _

5 ~

6 

7 

8 

9 

10 

11 

12.. . 

13 

14 

15 

16 . 

17 

18 . 

19 

20 

21 

22 

23 

24 

25 

30 

3fi 

40. _ - 

45 

50 

1-DtEi

0. 646 446 610

.807 649 911

. 875 000 000

. 9iO 557 281

QQ1 QKfi ft 1ft

.946005075

.955 805827

. 962 962 963

. 968 377 223

.972 589878

. 975 943 739 

. 978 665 377 

. 980 909 911 

. 982 786 741 

. 984 375 000

.985 733 198 

.986 905 430 

.987 925 488 

. 988 8x9 660 

. 989 608 672

.990309 058 

. 990 934 156 

.991 494827 

.992000 000 

. 992 457 072

.994 206281 

. 995 370 370 

.996 190 884 

.996 794 740 

.997254353

1-D4£4

1.044 194 174

1.042 766 687

1. 027 343 750

1.017888 544

1.012 285249

1.008 815 498

1.006 560 072

1.005029 921

1. 003 952 847

1.003 171 419

1.002 589389 

1.002 146086 

1. 001 801 871 

1. 001 530 067 

1.001 312256

1.001 135420 

1. 000 990 176 

1. 090 869 632 

1. 000 768 648 

1.000683330

1. 000 610 689 

1.000 548406 

1.000 494 658 

1. 000 448 000 

1. 000 407 273

1.000 265269 

1.000 183 971 

1. 000 133 693 

1.000 100 732 

1.000 078 115

1-JDtEt

1.046956310

1. 003 563 891

. 998 596 192

.998425808

QQfl 7QQ fl^fl

. 999 134 192

. 999 375 552

. 999 542 753

.999 658 869

. 999 740 708

.999 799 444 

.999 842 388 

. 999 874 339 

. 999 898 506 

. 999 9i7 060

. 999 931 502 

. 999 942 885 

. 999 951 960 

. 999 959 270 

. 999 965 216

. 999 970 095 

. 999 974 131 

. 999 977 495 

. 999 980 319 

.999 982 704

. 999 990 349 

. 999 994 149 

. 999 996 224 

. 999 997 442 

.999 998 199

1-DtEt

0.992058859

.996040 i22

.999 128 342

.999 833 994

.999995351

1.000028454

1.000 030293

1.000 025298

1.000019 766

1.000015 147

1.000 Oil 585 

1. 000 008 906 

1.000006905 

i. 000 005 404 

1. 000 004 271

1. 000 003 409 

1.000 002 746 

1. 000 002 231 

1. 000 001 828 

i. 000 OOi 510

1. 000 001 256 

1.000 001 052 

1.000 000 886 

1. 000 000 751 

1. 000 000 641

1.000000 311 

1. 000 000 166 

1. 000 000 096 

1. 000 000 059 

1. 000 000 038

1-DioSio

0. 991 077 006

1.000 487 652

1.000 261 620

1.000 083 575

1. 000 026 369

1.000 008 304

1.000 002383

1.000 000 412

. 999 999 787

. 999 999 629

.999999 630 

. 999 999 679 

. 999 999 737 

. 999 999 788 

.999 999 831

.999 999 866 

.999 999 893 

. 999 999 915 

.999 999 931 

.999 999 944

. 999 999 955 

. 999 999 963 

. 999 999 970 

. 999 999 975 

. 999 999 979

.999 999 991 

.999 999 996 

.999999 998 

. 999 999 999 

.999 999999

simple working rule has been found. Usually the terms 

of the residue are more difficult to compute than are 

those of the original series.

If the value of a is small, it usually is simpler to calculate log DO by series than to interpolate in a table of 

natural logarithms. Specifically, by equation 19.12 and 

the usual series expansions,

so that by

10g e DO J0t

which may

where

21 . 33 7

2048 ' 4096 *

equation 14.211

3 , , 5 35 63

32 ' 96 1024 ' 2560 

77 429 6 435

4096 ' 28 672 524 288 a '

be written

loge DQ= / , Uk i

a

0 4

u$ 8

MS 5a

u3 __21a

u* 18a 

ui~ 25

Ml~~"72~ 7/5 55a 

. . . (20.11)

. . . (20.12)

(20.21) 

(20.221)

(20.222)

(20.223)

(20.224)

(20.225)

(20.226)

u9 _ 39« 

us~ 49

MB 128

(20.227)

(20.228)

Except for rounding errors, the errors do not exceed 

five units in the 10th decimal place for the approximations

lOge D0=-Ct

log. D0=«.-

log. Z)o=aMore generally,

for «<0.000 073 030 (20.31)

for «<0.002 125 317 (20.32)

for a<0.010 997 665- (20.33)

1 \ u 1 \ *** ) * m* f nf\ A \

!) o->../..i\o, « ' (20.4)

For &>0, equations 18.42 and 18.61 may be written 

,, (2fe + 2u-3)l(fc-l)l

M=l

(20.51)

(20.52)

where A is the largest integer not exceeding (& 1)/2. 

If equation 20.51 is written

jD*==1+S(~ 1 ) tljD *«a"» (20.53)

the values of Dku may be calculated and tabulated as 

in table 12, which shows the values of Dk» for u ^ 8 

and even values of k not exceeding 10.

TABLE 12. Values of D ku

k

4__..__

. ___-_

10-. 

u=l

3

2

7

2

11

2 

15

2

19

2

M=2

15

8 

63

8 

143

8 

255

8 

399

8

M=3

35

16 

231

16

715

16 

1615

16 

3059

16

M=4

315

128 

3 003

128 

12 155

128 

33 915

128 

76475

128

M = 5

693

256 

9009

256 

46 189

256 

156009

256 

412 965

256

M=6

3003

1 024 

51 051

1 024 

323323

1 024 

1 300 075

1 024 

3 991 995

1 024

M=7

6435

2048 

138 567

2 048 

1 062347

2 048 

5 014 575

2 048 

17 678 835

2048

M=8

109 395

32 768 

2 909 907

32 768 

26 558 675

32 768 

145 422 675.

32 768

583 401 555.

32 768

The Euler-Maclaurin function may be written

where

(20.61)

(20.62)

Values of Ak, 2v are shown in table 13 for k ^ 8 and 

values of 2u not exceeding 10.
32 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

TABLE 13. Values of Ak,«u,

k

1

2

Q

4________.__ _____._-

5____.-__--________.

6

7 ___ _______.__._.

0

+i^8 

3

64

4- 5

1 192 

35

2048 

. 63

1 5 120

77

8 192 

429

1 57 344 

6 435

1 048 576

2

1

8

+ 5

^32 

35

192 

105

1 512 

231

1 024 

. 1001

1 4 096 

2145

8192 

. 36 465

1 131 072

4

,7

+ 24 

49

64

49

^32 

2695

1024 

21 021

1 5 120 

49 049

8 192 

17 017

1 2 048 

2 909 907

262 144

2u equals 

6

31

24 

,93

+ 16 

1705

96 

22 165

1 512 

93 093

1 024 

. 527 527

1 3 072

4 295 577

14 336 

64 433 655

1 131 072

8

, 381

+ 40 

4191

64 

L 18 161

^ 64 

1 906 905

2 048 

,6 483 477

h 2 560 

123 186 063

20 480 

52 794 027

1 4 096 

13 356 888 831

524 288

10

2555

24 

. 33 215

1 32 

1 162 525

192 

19 762 925

1 768 

45 059 469

512 

,525 693 805

^ 2 048 

2 714 296 585

4 096 

203 572 243 875

1 131 072

Noting that a^ is not to be confused with a having 

no indices, the notation

ate=--^L

«*.«= x2

leads to the relation

(20.71)

(20.721)

(20.722)

(20.73)

Early values of ak ,2j are shown in table 14 with akQ =Ako. 

If a good table of natural logarithms is available, it 

may be preferable to write

where

/=! X2 '

(20.81)

(20.82)

(20.83) 

LOCAL EXPANSIONS FOR QK1

Forjpoints near the current pole, the axial displacement, /, of the test point, is small and the relative 

displacement, a l/(2h), likewise is small. For such

so that, by equations 20.62 and 20.82,

pouits, the reduced reciprocal distance of equation 

12.31 may be written

(21.11)

where

and the definition of equation 21.121 is extended to 

include

H0=l. (21.122)

The value of Hv depends only on the value of u, and is 

shown as a common fraction for u ^ 24 and as a decimal 

fraction for u ^32, hi table 15.

If the first (n 1) terms of the modified potential are 

calculated directly and if w is an arbitrary constant, 

the notation

(21.21)

transforms equation 12.932 into

which is reducible to

n-l

(21-2221)
FORMULAS FOB THE EVALUATION OF THE MODIFIED POTENTIAL 

TABLE 14. Values of ctkji

33

k

1 ____ . ____ .__

2 _______ _____

3 _____ ________

4

5-_-_-_--_-_---_

6 ____- .._._.__

7_-___--__.__-__

8-_-___.____._-__

2; equals 

0

r -I. 1 + 8 

1 + 0. 125

f 3

< 64 

1-0. 046 875

f + 5

< + 192 

L + 0. 026 041 667

r 35

{ 2048 

L-0. 017 089 844

( 1 63

< +5120 

L + 0. 012 304 688

r 77

< 8192 

L-0. 009 399 414

f 1 429

< +57 344 

L + 0. 007 481 166

r 6435

I 1 048 576 

1-0.006 136 894

2

1 

1

10

3 

3. 333 333 333

7 

7 

12 

12 

55

3 

18. 333 333 333

26 

26 

35 

35 

136

3 

45. 333 333 333

4

7

3 

2. 333 333 333

49

10 

4.9

42

5

8. 4

77

6 

12. 833 333 333

91

5 

18.2

49

2 

24.5

476

15 

31. 733 333 333

399

10 

39.9

6

31

7 

4. 428 571 428

372

49 

7. 591 836 735

1705

147 

11. 598 639 456

806

49 

16. 448 979 59

155

7 

22. 142 857 14

4216

147 

28. 680 272 11

1767

49 

36. 061 224 49

310

7 

44. 285 714 29

8

1143

155 

7. 374 193 548

1397

124 

11. 266 129 03

4953

310 

15. 977 419 355

2667

124 

21. 508 064 52

4318

155 

27. 858 064 42

21 717

620 

35. 027 419 35

2667

62 

43. 016 129 03

32 131

620 

51. 824 193 55

10

12 775

1143 

11. 176 727 91

66 430

4191 

15. 850 632 30

89 425

4191 

21. 337 389 64

347 480

12 573 

27. 636 999 92

48 545

1397 

34. 7491463 14

178 850

4191 

42. 674 779 29

58 765

1143 

51. 412 948138

255 500

4191 

60. 963 970 41

TABLE 15. Values of Hu

u

i _ . ___ . ___ .....

2 __ . __ . _____ ...

3 . 

4 . _ _ __

5 - _____ - __ -

6 

7.. ......................

8.. ......................

9...- _ ___ . _ ._

10.. _____

11 ____ - ____

12. _____

13 

14. _________ . _ 

15.. ______ _ 

16 _ .... __ - __ ..

17. 

18- 

19 _ ___

20 

21 ___________ . __ .

22 ______ .. ______ .

23- 

24 

25 . __ -

26 _ . ____ . _ - _ .

27. 

28 

29 

30 

31 

32. ____ . .....

Decimal

0. 500 000 0000 

. 375 000 0000 

. 312 500 0000 

. 273 437 5000 

. 246 093 7500

. 225 585 9375 

. 209 472 6562 

. 196 380 6152 

. 185 470 5811 

. 176 197 0520

. 168 188 0951 

. 161 180 2578 

. 154 981 0171 

. 149 445 9808 

. 144 464 4481

. 139 949 9341 

. 135 833 7596 

. 132 060 5996 

. 128 585 3206 

. 125 370 6876

. 122 385 6712 

. 119 604 1787 

. 117 004 0879 

. 114 566 5027 

. 112 275 1727

. 110 116 0347 

. 108 076 8489 

. 106 146 9052 

. 104 316 7861 

. 102 578 1730

. 100 923 6863 

. 099 346 7537

Numerator

1 

3 

5 

35 

63

231 

429 

6 435 

12 155 

46 189

88 179 

676 039 

1 300 075 

5 014 575 

9 694 845

300 540 195 

583 401 555 

2 268 783 825 

4 418 157 975 

34 461 632 205

67 282 234 305 

263 012 370 465 

514 589 420 475 

8 061 900 920 775

Denominator

2 

8 

16 

128 

256

1 024 

2 048 

32 768 

65 536 

262 144

524 288 

4 194 304 

8 388 608 

33 554 432 

67 108 864

2 147 483 648 

4 294 967 296 

17 179 869 184 

34 359 738 368 

274 877 906 944

549 755 813 888 

2 199 023 255 552 

4 398 046 511 104 

70 368 744 177 664

Upon isolating the term .7=0, equation 21.2221 becomes

£-{ ^ Jpl uW |_n2u+1 ' -f={ (

which may be written

«-i

Jk=l

AJ 1 A \ ati-rl 1

(21.2222)

-=i n* u+i L n £1 ( n+wu+i j x * '

As u increases indefinitely, the quantity in the square 

brackets approaches unity so that the summand in the 

second sum approaches

<-'> (%)( ) <"-

which suggests that w should be selected as having the 

value n, for which choice

W= 23 QVfc+X. ( !)"( )<h8" l+na«+1 _5;7i (n+j)2«+1J 

(21.2224)
34 APPARENT RESISTIVITY OF A

and equation 21.21 becomes

2 (21.23)

SINGLE UNIFORM OVERBURDEN

and the ratio Xn« approaches the limit

. _2u+l An 

For systematic calculation, it is convenient to write

Q=eq (21.31)

where e= 1 for a buried insulator and e= 1 for a buried 

conductor. The notation

leads to

g

so that equation 21.2221 becomes, for w=n,

n \ OD 

fc=l w=l

With the notation

equation 21.331 may be written

Q V*+ jb

(21-322)

(21.331)

(21.332)

(21.333)

For selected values of g and n, the values of (7B« may be 

tabulated for values of u. If powers of /3 are available, 

equation 21.333 is convenient for small values of 0, 

as the remainder series converges rapidly, especially if 

n is selected as greater than a.

If powers of /3 are not available, a ratio form is more 

convenient. The notation

'n.u+1

leads to

a.,

which starts the recursion sequence with

(21.411)

(21.412)

(21.413)

Because of equations 21.12 and 21.332, the ratio of 

equation 21.411 may be written

f «

As the value of u increases indefinitely the value 

of <7nu approaches the limit

C»=(-l)»-1^^ (21.511)

(21.512)

The series for $£ usually converges rapidly so that it 

frequently is simpler to calculate

(21.521)

(21.522)

in terms of which

and

As u increases indefinitely, junu approaches zero, leading 

again to the limits of equations 21.51.

Omitting the constant subscript n, for a selected 

value of v and large value of u, \«« approaches the 

value

Xu== 2jHil:=1 _0tt+i_ M (21.611)

where

1 (21.612)

Dropping the subscript n, equation 21.411 may be 

written

C U+1=-\UCU .

In particular,

(21.613)

(21.621)

(21.622)

(21.623)

(21.624)

If the first (n 1) terms of the modified potential are 

calculated directly, the remainder has the value, by 

equation 21.333, upon dropping the subscript n,

C,+3 = - (1-&) (1-&) (l-<h

u=i u e+l

which may be written

where

(21.631)

(21.632)

-*i)i8'. (21.633)
FORMULAS FOB THE EVALUATION OF THE MODIFIED POTENTIAL 35

The value of S is expressible as a power series in #, and 

the various fa. Specifically, the term free of fa has the 

value

For <?2<1, and n^>a, iS<l, so that

So= i+T

(21.641)

(21.642)

The terms of S which are linear in fa have the sum

(21.651)

i E Z)

i=i

Reversal of the order of summation transforms this 

relation to

(21.652)

(21.653)

which may be written, by equation 21.612, in the form

(21.654)

Summation in i reduces Si to

g _^(-l)W

v+j

Separating the even and odd terms, this equation may

be written

(21.655)

in which all of the summands are positive, and converge 

conveniently to zero as j increases, for small values of 8.

comes

_ (-!)'

which may be written

Si _ (-D

But

(-1)'--'ft'

=log,

so that

The notation

(21.656)

(21.6562)

(21.6563)

(21.6564)

(21.6565)

reduces equation 21.6564 to

8lSS2(l+ft)ft'' (21.6566) 

The terms of S which are quadratic in fa have the sum

 i *

Q ,__ ^y^ ( ,_, 1 \ f l/3t "^ A ft\ ^V ^ A\ ('91 fifil ^

Interchanging the j and k summations, this becomes

(21.662)

Interchanging the i and k summations, this becomes

CO 00 fc 1

fc-2 i-fe }=l

Since the limits of i and j are mutually independent, 

this may be written

(21.664)

.=Z3 S Z) (-i)*-1 */*^'.

i=2 fc=2 J=l

j-fc

and the summation in i may be performed, leading to

ft-i

By equation 21.612 this is equivalent to

1 _____ 

"4(1 + 18)

(21.665)

(21.666)

For small values of 18, equation 21.666 is convenient. 

For values of 18 near unity, a procedure similar to that mvemently to zero as ? increases, tor small values ol 8. , . «. I T\ u *u- j / ^ i . rt .. At j.11 used for Si, is convenient. Interchanging the j and k For values of |8 near unity, another form is preferable. ,. ,. 01 aaa ,

TT , ,. ,, /' ,. «- _,, summations, equation 21.666 becomes Upon retardmg the counter by v, equation 21.654 be-

° 2~4(l + 0)^lft^.

If the k counter is retarded by v, this becomes

1 " " ( \\Tt-v-\ak~v ~ i v » v v v 1 1 a

(21.6671)

(21.6672)

which may be written

k

(21.6673)

By equations 21.6563 and 21.6565 this becomes

(21 ^Lbb74) 6674 
36 APPAEENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

Retarding the counter by v, this becomes 

- (-!)'

which may be written

«.. (-1)" te»fr -

(21.6675)

(21.6676)

Summarising, the first (n 1) terms of W are calculated by the products Qkfa, the next v terms are 

calculated by the product Cnu&u, and the residue is 

calculated as Cn^ v (SQ +Si+S2). Additional values 

of Sk are not needed as the method is used only when the 

term C7» r|8* is numerically small. The values of S% need 

not be calculated with great accuracy. The value of SQ 

may be calculated directly. The residue factors are 

independent of Q and are needed only for values of j8 

near unity. However, Q enters the residue through the 

coefficient Cn^v. Table 16 shows the values of S0 , Si, 

S2 , and S to 6 decimals, for a/n on the range [0.80(0.01) 

1.00] for #==30. If intermediate values are needed, S0 

should be calculated directly, Si may be obtained by

In preparing table 16, the values of SQ were computed 

for each entry. The values of Si and S2 were calculated 

on the range a/ft=[0.80(0.05)1.00]. The intermediate 

values were interpolated from a fourth degree polynomial passing through the five points, with adjustments 

when needed.

ASYMPTOTIC EXPANSIONS 

ASYMPTOTIC EXPANSIONS FOR

The modified potential function has been defined as

where

 PO

P6+PO

a 2h

(12.932)

(12.5) 

(12.31) 

(12.22)

The limiting value of the modified potential as I increases indefinitely is

so that

(14.1)

(22.1)

from the table. The value of S is the sum of SQ, Si, and

S2 . 

TABLE 16. Residue factors for w=30 

[Last figure in columns Si and Ss are in sixth decimal place]

a/n

0. 80______-_-___.

82

83

84

85

86

87

88

89

. 90

92

93

94

95

96

97

98

.99 __ _________

1. 00_____ ________

+So

0 390 244 

396 172 

402 057 

407 899 

413 696

419 448 

425 155 

430 816 

436 429 

441 995

447 514 

452 984 

458 406 

463 778 

469 102

474 376 

479 600 

484 775 

489 900 

494 975

. 500 000

-s,

3886 

3907 

3928 

3947 

3964

3980 

3996 

4010 

4023 

4034

4045 

4055 

4064 

4071 

4078

4083 

4088 

4091 

4094 

4096

4097

-ft

24 

25 

25 

26 

26

27 

27 

28 

28 

29

29 

29 

30 

30 

31

31 

31 

32 

32 

33

33

Factor

0 386 334 

392 240 

398 104 

403 926 

409 706

415 441 

421 132 

426 778 

432 378 

437 932

443 440 

448 900 

454 312 

459 677 

464 993

470 262 

475 481 

480 652

485 774 

490 846

. 495 870

Fora> 

l

VJfcH^2 

so that 

where

16

Wa> W= (22.31)

(22.321)

(22.322)

(22.323)

(22.324)

(22.325)

with similar expressions for higher order terms.

Each of the coefficients Bn is of the form 

with a constant factor. The ratio of the (&+l)-th 

term to the k-th term in Bn is Q[(k+l)/k]*H-\ Although the coefficient of Q in this ratio exceeds unity, 

it approaches unity as k increases indefinitely so that 

for Q?<i\, BH converges. However, only the early
FORMULAS FOR THE EVALUATION OF THE MODIFIED POTENTIAL 37

values of n lead to values of Bn that are convenient for 

calculation.

Excluding the cases Q2=l, the values of Bn may be 

evaluated directly. Specifically, Bl is a geometric 

series of ratio Q and first term Q. Accordingly

Q

1-Q (22.41)

As B2 involves the factor k2, whose third difference 

vanishes, it is convenient to isolate the initial terms 

and write

(22.511)

(22.512)

(22.513)

(22.514)

The third difference of k* is (&+3) 2 -3(&+2)2 + 

3(Ar-f-l) 2 kz, and the exponents of the summations in 

equations 22.5 may be made Ar+3 by multiplying by 

1> Q) Q?) and &, respectively, leading to

*+3 (22.521) 

*+3 (22.522)

3Q2 (252) = 3Q»+Z}3(*+ 1) 2 Q*+3 (22.523)

(22.524)

(22.531)

(22.532)

Addition of equations 22.52 leads to

so that

A procedure similar to the determination of B2 leads

to

128(1

The notation -'0-0

u=Q+Q

transforms equations 22.53 and 22.54 to

(22.551)

(22.552)

(22.561)

(22.562)

(22.563)

(22.564)

B _ 140(1 + Q)(M8 +246M2 +4,044M+11,080)

(22.565)

Each quantity Bn depends only on Q so that numerical 

values may be calculated. For Q2<1, the first five 

values of Bn are shown in table 1, as noted on page 8.

As Q, approaches unity, v approaches zero and u 

approaches 2, so that each value of Bn becomes infinite 

and the usefulness of these asymptotic expansions 

disappears. As Q approaches minus one, v approaches 

 4 and u approaches 2 so that B\ approaches one-half 

and all other Bn vanish. This property is convenient 

for large displacements but its validity may need 

verification for small displacements.

The use of asymptotic expansions may lead to 

divergent series as the value of n increases. Usually 

an asymptotically divergent series has a remainder 

less than twice the first neglected term, so that the 

evaluation should stop after the term ahead of the 

smallest has been calculated. If the first neglected 

term is sufficiently small, the series may be used for 

calculation.

ASYMPTOTIC EXPANSIONS FOR Q=+l

If the underlying bed is a perfect insulator, p& <» 

and Q=l, so that the modified potential function of 

equation 12.932 may be written by equation 12.31,

w_W!__L_A ^\k VfcH^y (23.1)

In calculating this function, the Euler-Maclaurin 

summation formula uses the integral and odd order 

derivatives of the function

Direct differentiation shows that

f"(x) = (2z2 -a2 )

/"'(a) = - (6z3 -9a2z) (z2 +a2 )-7/2 .

These derivatives suggest the relation

k 

u=0

(23.21)

(23.22)

(23.23)

(23.24)

(23.31)

547123 60 4
38 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

where AkV must be determined for £1=1 and Q=* 

By differentiation, equation 23.31 leads to

r fe

fe-1

fe-1

(23.321)

Replacing fc by (fc+1), equation 23.31 becomes

H-l

/(*+D(x ) == (_i)t+i(a;2 +02)-*-3/2^ (23.322)

Equating the right members of equations 23.32 and 

dividing by the common factors leads to

fc-1

fc-i 

X)

W=0

-«. (23.331)

Upon retarding the counter of the final summation by 2, 

this becomes

w=0

k-l

fe+1

(23.332)

Upon equating coefficients of corresponding terms, 

this equation determines the conditions:

(23.341)

(23.342)

(2<M<fc) (23.343) 

(23.344)

For positive integral values of k, equations 23.34 are 

satisfied by the values

fe.2»-i=0 where 1^2w 1^ 

W

(23.351) 

where 0^2w^fc. (23.352)

Specifically, equations 23.35 take the special forms

(23.361)

(23.362)

(23.363)

By equations 23.35 and 23.36 the derivative of equation 

23.31 may be written

(23.37)

M=0

The Eulerian constant is defined as

where

(23.411)

(23.412)

To 16 decimals (Jolley, 1925, p. 19), the value of the

Eulerian constant is

*=0.5772 1566 4901 5325. 

Equation 23.412 may be written

so that

8-1

(23.413)

(23.421)

(23.422)

For A=Kj Ar=0, the Euler-Maclaurin summation 

formula of equation 16.33 reduces to

(23-431)

where equation 16.12 reduces to

u=l

(23.432)

Accordingly, by equations 23.21, 23.37, and 23.43,

£>-* H> (23-441)

where 

=log e

i. (23.442) «=i

Combination of equations 23.42 and 23.44 leads to

S s

(23.451)

(23- 452)

For large values of t, $4 approaches $ and 

^+a2) approaches Iog4 (2x) . For specific values

of a, Uj and «?, the summand in equation 23.442 approaches a finite constant multiplied by ar2a~2f and
FORMULAS FOR THE EVALUATION OF THE MODIFIED POTENTIAL 39

accordingly approaches zero as x becomes infinite. 

Hence Z(x) approaches loge (2x), and equations 23.1 

and 23.45 lead, for the limiting values, to

fc=l

fc=«

(23.461)

- (23.462)

For small values of x, corresponding to large values 

of a, Ma'claurin's expansion may be used for Z(x). As

/i . .__.. i

(23.51)

the notation

leads to the relations

(23.521)

'(*)=/(«) (23.5221)

(23.5222)

where j(x) is the function defined by equation 23.21. 

Accordingly, by equations 23.35, 23.36, 23.37, 23.521 

and 23.522,

<KO)=log ea (23.531)

(23.532)

(23.533)

(23.534)

(23.541)

If the definition of Aik is extended to include 

Aoo=l,

equations 23.532 and 23.534 may be combined to

0<2«-» = 42u-2,2 U-2 0-2U+1. (23.542) 

Accordingly, Maclaurin's series leads to

Combining of equations 23.442 and 23.55 results in the 

relation

U-l

By equations 18.42 and 20.51,

u-u,a2 ^^^. (23.56)

I«', (23.571)

so that

(23.572)

The notation

reduces equation 23.572 to

so that

(23.581)

(23.582)

T;[yra

co oo u 1 ^ \ -^r-v ^r~\

"-^-T ^-» 2-1 

14=1^=05=0

5T> &2« A r, /X\ So2l^ A2«-i.2 «£«»( a) 

=n * x \a/

4«4-2te-2»-l

(23.583)

If the term u 1 is isolated and the counter advanced 

by unity in the first summation, equation 23.583 

becomes

co co U-l t

_^r-\ ^r-\ ^r-\ t?2tt

Z i 2 1 2-1 o~\2<

For x=1/2, this determines

4u+2u>-2»

(23.584)

ce co u 1 4tt+2«r-2»-l

(23.591)

If the counters v and w in the triple summation are 

retarded by unity, this equation becomes

U co 

M=lU=l«)

x i \4o-f2to 2 » l

^G^-A ) (23.592)

\^a /

If the order of summation in v is reversed, this equation 

becomes

00 " *° /I \22tt4-2»4-2» 8

~Z) Z) Z) &2«^2«-i,2«-2 ,G!«,»-i( ^ ) (23.593) «=l p=iu?=i \*a/

In equation 23.593, the coefficient of l/(2a)2*+1 is

The value selected for k restricts u-\-v to the range 

2r£tt+0:gfc+l> so that u is restricted to the range 

l^u^k and t> can not exceed either u or (fc «+!). 

Two cases exist, according as & is even or odd. If k 

is even, it may be written as k=2j. For u£j, k-\-l 
40

 / ' / i

APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

where

For a positive reflection factor,

u^j+1, so that it<&+l u and 1 ^v£u. For u^j 

1, k+l u^j, so that k+l u^u and 1 ^v^k+l u. 

Accordingly the coefficient of l/(2a) 4J+1 in equation 

23.593 is

2j 2J+1-M

X^i / i &2»A2«-l,2w-2iX?i<,2j-*-t>+l' 

M=J-H 0 = 1

A j+i w fl.4>+2. 4t+2 Xr-\ -sr-v r ., r

r A , q\ I s t / i O2*-'i2« 1,2« 2i"w, 2j u »+2

2j+l 2J+2-M 

/i

If k is odd, and greater than 3, it may be written as 

&=2j+l, where j>l. Forit^j+l, k-\-l u^j+l, so 

that ii^^+1 ii and l^v^u. For w^'+2, fc+1 

w^j, so that k+l u^u and 1 ^0^2^+2 u. Accordingly, the coefficient of l/(2a) 4^+3 in equation 23.593 

is

» 1, 2w 2»tr», 2j u 1+2-

For the coefficient of l/(2a) 3 in equation 23.593, all

counters must be taken as unity so that the coefficient

^ of l/(2a) 3 is -f M-io<V But by equation 23.363,

o!

A22= 1; by equation 15.812, 62 = 1/6; by equation 

23.361, A10=l; and by equation 23.581, #10=1; hence 

the coefficient reduces to zero. Direct calculation for 

&^6 has shown that the coefficient of l/(2a) 2ft+1 

vanishes, so that the modified potential of equation 

23.461 reduces to

W=loge 4-*-log. (2o) i (23.61)

at least for powers not above the 14th, in l/(2a), and 

except for errors in the application of the Euler-Maclaurin summation formula. To 16 decimals, Ioge4= 

1.3862 9436 1119 8906 so that for large values of a, 

the modified potential is given approximately by the 

asymptotic expansion

where

= 0.8090 7869 6218 358.

(23.621)

(23.622)

Comparison with the direct series calculations shows 

that the asympotic expansion is valid to eight decimals 

if a exceeds about three.

REDUCTION FORMULA FROM POSITIVE TO NEGATIVE 

REFLECTION FACTOR

Displaying both arguments, the modified potential is 

given by

(24.11) fc=i

and for a negative reflection factor,

where «=IQ|.

(24.12)

(24.21)

(24.22)

(24.23)

If the terms for odd and even values of k are 

separated, equations 24.2 become

(24.31) 

«). (24.32)

(24.41)

(24.42)

But, by equation 24.12

so that

1 . /« 

.2

Accordingly, by equations 24.3 and 24.42,

W(-q,a) =

~|Z) rt (f) 

-M ~|Z) «2 ¥/b (f ) (24.52)

But

so that

which leads to the reduction formula

W(-q,a) =

(24.71)

(24.72)

Hence if W is available for Q=g_, and for Q,=q*, a 

simple subtraction furnishes W for Q= g;. In the 

present study, the spacings in g; are by one-tenth of a 

unit, so that the only useful case is that for g=l, for 

which

TF(-l,a)=Tr(l,-|)-TF(l,a). (24.81)

However if tables of W are available for reflection 

factors on the range Q=[ 1(0.1)1], equation 24.71 

furnishes W(g2,a/2).

Combination of equations 23.621 and 24.81 furnishes 

the result, for large values of a,

TT(-l,a)=bg.2~, (24.91)
FORMULAS FOR THE EVALUATION OF THE MODIFIED POTENTIAL 41

where

log e 2=0.6931 4718 0560. (24.92)

As for the case Q=-\-l, equation 24.91 must be considered as an approximation which is valid only for 

values of the relative displacements that are sufficiently 

large. To eight decimal places, the approximation of 

equation 23.621 is valid for values of a above 3, so that 

equation 24.91 is valid for values of a above 6. For 

fewer decimal places, the lower limit of a may be 

decreased.

TABLE SMOOTHING AND ERROR DETECTION

Two important uses for differences in tables are detection of gross errors of calculation and smoothing for 

small errors due to forcing errors of rounding. (Scarborough, 1950, p. 57; Steffensen, 1927, p. 46; Willers, 

1948, p. 101). Specifically, the descending differences 

of f(x) at x=a are defined as

A.=/(a+fc)-/(a) (25.11) 

AS = A^ 1- AIT 1 for n^2, (25.12)

where h is the constant spacing in the argument x. If 

a unit error occurs at some position in a table of exact 

values, the successive differences have the values 

shown in table 17, where correct values have been

TABLE 17. Error patterns

V

0

0

0

0

0

0

0

1

0

A

0

0

0

0

0

0

+1-1

0

A2

0

0

0

0

0

+1

_ 2

+1

0

A3

0

0

0

0

+1

-3

+ 3

-1

0

A*

0

0

0

+1

-4

+ 6

-4

+ 1

0

A«

0

0

+1

~~ O

+ 10

-10

+ 5

_ i

0

A«

0

+1

-6

+ 15

-20

+ 15

-6

+ 1

0

subtracted from the functional values y, leaving only 

the error residuals, and the error shown is assumed to 

be the only error in this portion of the table. This 

example illustrates the general rule that the errors in 

the n-ih descending differences begin on the ascending 

diagonal through the error in the function, end on the 

line of the error, and are the value of the error multiplied 

by the coefficients of x in the expansion of (1 x) n. 

The errors on the ascending diagonal through the error 

are the same as the error in the function, and the errors 

on the line of the error are the error with alternating 

signs, beginning with the error itself. If the function 

is a polynomial of degree k in the variable x, the k-ih 

difference is constant and the (k-\-l)-th difference 

vanishes. Hence, the presence of the error pattern in 

the (&+l)-th difference detects the error, locates it, 

and determines its value.

If the functional values are approximations, they 

are subject to forcing errors of half a unit in the last 

digit, if the table contains no gross errors. Hence the 

various differences will not approach zero, as for a 

polynomial, but will eventually alternate and the sum 

of (£+2) successive values of the (&+l)-th difference 

will approximate zero. To be of much value for 

interpolating, the successive order of differences must 

decrease in numerical value except where they change 

signs for two successive lines. In the early order of 

differences, an error may not be recognizable; for large 

order of differences, the forcing error causes a pattern 

of large numerical value. For some intermediate 

order of differences, the error pattern may be recognizable, and the error recognized and determined. 

The term in error can then be verified and corrected.

If the tabulated function is subject to normal forcing 

errors of rounding, but contains no gross errors, the 

table may be "smoothed" by making arbitrary adjustments by small amounts to improve the error pattern. 

Although such smoothing does not necessarily improve 

the accuracy of the tables, it does tend to do so by 

producing a table that is more regular in behavior. To 

insure greater accuracy, the only safe method is to 

retain a sufficient number of decimal places in each 

calculation to prevent an accumulation of forcing errors 

from causing a tabular error in excess of half a unit in 

the last tabulated digit. For data obtained from 

observations, no such improvement is possible.

Although the forcing errors due to rounding increase 

the numerical errors of successive differences, the coefficients used in interpolation usually decrease with the 

order so that the net effects of such increased differences 

usually tend to decrease, and smoothing a table usually 

does not lead to errors which exceed the permissible 

tolerance associated with the table.

LAGRANGEAN INTERPOLATION

Numerous formulas are available in the literature 

(for example, Scarborough, 1950, chap. 3; Steffensen, 

1927; Willers, 1948, chap. 2) for interpolation but the 

simplicity and directness of the Lagrangean formulas 

favor their use except for specialized problems.

The Lagrangean interpolation formulas are based on 

the assumption that a polynomial of a selected degree, 

not inconveniently high, can be used with sufficient 

accuracy to determine the value of a function for arguments between those in a table of that function, or only 

slightly outside the range of the table. As a polynomial 

of degree n in a single variable has (n-\-l~) terms, its 

determination requires (ft+1) tabulated values of the 

function. Accordingly, the degree of the polynomial 

must be less than the available number of functional 

values. Usually, a polynomial of small degree is suf-
42 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

ficient. The arguments need not be equally spaced, 

but the formulas and their use are simplified if they are. 

Interpolation by a polynomial of degree (n l) is called 

"interpolation of order (n i)" or "n-point interpolation".

To illustrate the method, consider a table of two 

entries. For such a table, the polynomial must be 

linear and the usual formulas of interpolation result. 

Let the two arguments be x0=a and Xi=b, and let the 

corresponding functional values be 2/o=/(a) and yi=j(b) 

The desired function is y=j(x). Then

y=a?/o+j8?/i, (26.11)

where a and /3 are functions of x to be determined. For 

x=a, 2/=2/o so that /3=0 for x=a and /3 may be taken 

as the product of (x a) and a constant 7. For x=b, 

y=yi, so that a=0 for x=b and a may be taken as the 

product of (x b) and a constant 5. Hence,

(26.12)

(26.131)

(26.132)

But, for x=a, yQ=8(a b)y0, so that

1 o~6

and for x=b, yi=y(b a)yi, so that

7 =

Accordingly,

6-0

a; 6 , x a (26.14)

Introducing the phase of x on the interval (a,b) as

x a

equation 26.14 may be written

y=> (1 0) 

or

(26.151)

(26.152)

 yo). (26.153)

Equation 26.153 is the customary formula for linear 

interpolation without a computing machine or where 

the first difference A=i/1 2/0 is tabulated or small 

enough to be calculated quickly. Equation 26.152 is 

preferable if a computing machine is available, as 

differencing is not needed.

For three tabulated entries, linear interpolation 

may be used between the first two and between the 

second two, if the results are sufficiently accurate. 

Usually quadratic interpolation on the whole interval 

is more reliable. Let the arguments be xQ a, 2*1=6,

and x^ c, and let the corresponding functional values 

be 2/o=/(a), 2/1 =/(&)> an<* 2/2=/(<0, respectively. Let 

the function be y=j!(x). As /(a) =2/0, the coefficients 

of 2/1 and y2 must vanish for x a and hence, each must 

have the factor (x a}. Similarly the coefficients of 

2/o and 2/2 must have the factor (x b) and the coefficients of 2/0 and 2/1 must have the factor (x c) . Hence, 

the function may be written y=a(x b) (x c)yQ -\- 

P(x a)(x c)y1+y(x a)(x b')y2. For x=a, y=y0 so 

that (a b)(a c)a=l, with similar relations for 

j8 and 7. Accordingly,

(x-a)(x-c) 9 .22)

Similar formulas may be derived for higher order of 

interpolation.

The formulas derived by the method just discussed 

are general and take simpler forms if the arguments 

are equally spaced. For most functions, the results 

are most accurate when the interpolating argument 

is as near the center of the range as possible. As a 

matter of notation, it is convenient to select the arguments so that the range of x is (0, 1), whence 0<a;<l, 

and so that where possible there are the same number 

of arguments below x as above for odd order interpolation and the same number of positive as of negative 

for even order interpolation. If the available functional values do not permit this choice, the restriction 

0<j#<O niay be removed or the notation may be 

changed. The results may be somewhat less accurate 

for such conditions, as near the beginning or end of a 

table.

For the choices of the preceding paragraph, linear 

interpolation has x0=Q, #1= !,$=#, so that equations 

26.152 and 26.153 may be written

y'=(l x')y0 -i-xyi (26.311)

(26.312)

For quadratic interpolation, x0 1, #1=0, x2 =l 

so that equation 26.22 may be written

y"= (1/2) (l-a0a;y-,+ (1-aO (l+o02/o+ U/2Xl+a0y, (26.32) 

For cubic interpolation, similar analyses show that

+ (1/2) (2-3Ml+a0yi- (1/6) (l-*)ap(l+a0ya . (26.33) 

The corresponding quartic interpolation formula is :

y"= (1/24) (1+zMl-aO (2-a0y_,- (1/6) (2+a0s(l-a0 (2-oOy-i 

+ (1/4) (2+a0 (l+a0(l-s)(2-*)y0+ U/6) (2+a0 (l+a03 

(2-3) yt- (1/24) (2+3) (1+3)3(1 -3) y,. (26.34)
FORMULAS FOR THE EVALUATION OF THE MODIFIED POTENTIAL 43

The corresponding quintic interpolation formula is

y*= (1/120) (l+aO*(l-*) (2-*) (3-*)y_,

- (1/24) (2+»)*(l-») (2-*) (3-z)2M 

+ (1/12) (2+s) (1+a;) (1-ar) (2-*) (3-a?) ft 

+ (1/12) (2+s) (l+aO»(2 a?) (3 3)2/1

- (1/24) (2+*) (1 +»)*(! -*) (3-ar)»«

+ (1/120) (2+o?) (1+zMl-z) (2-z)?/3. (26.35)

The value of the function as determined by a selected 

order of interpolation may be considered as the value 

as determined by interpolation of order one lower plus 

the "contribution" of the selected order. Thus, the 

contribution of order n is

LB =2/<»)-2/<»-«. (26.41)

Direct calculation shows that the quadratic contribution 

is

1/2= - (l/2)z(l-as) A8-! (26.421)

where

A8_i = 2/i 22/0+2/-1- (26.422) 

The cubic contribution is

L3= - (1/6) (1 -*)*(!+*) A8-i (26.431) 

where

A3_j = y2 3?/i+3?/o 2/-i. (26.432) 

The quartic contribution is

L4 = (1/24) (l+x)x(l-x) (2-*)A4-j (26.441)

where

A4-2=2/3 4?/i+6?/o 4y_i+ y~2. (26.442)

The quintic contribution is

1/5= (1/120) (2+a)(l+»)*(!-*) (2-*) A»-j (26.451) 

where

A5 2=2/3 5j/2~l~ lOyi 10yo~l~ 5y i y-2. (26.452)

The notation

Fr,= -(1/2) (!-»)* (26.51)

^1=-(1/8) (1-*)*(!+*) (26.52)

F4 = (1/24) (1+*)*(!-») (2-*) (26.53)

F8 = (1/120) (2+*) (l+»)*(l-») (2-*) (26.54)

permits equation 26.35 to be written as the quintic 

interpolation formula

i. (26.55)

The coefficients of the differences hi equation 26.55 

may be written

(26.61)

(26.62)

(26.63)

F4=-(l/4)(2-z)F3

with F2 given by equation 26.51.

Table 37 (p. 59) shows the numerical values of F2, F3, 

and F^ for the range x= [0(0.0 1)0.99]. The values of 

F2 are exact, but the values of Fa are rounded to five 

decimal places and the values of F^ are rounded to four 

decimal places. If values not in the table are needed, 

they may be calculated directly by equations 26.5 and 

26.6 or obtained by interpolation in the table. Values 

of F6 are not needed in the use of table 38 (p. 59) but 

have been used in the selection of arguments.

Excellent tables are available for interpolating without differences (Mathematical Tables Project, 1944), 

but the present method usually is preferable if the 

necessary differences are tabulated.

Analysis of equations 26.5 is of assistance in selecting 

the order of interpolation to be used. The coefficient 

F2 has a minimum value of % for x Y^ so that its 

absolute value on the interval (0,1) cannot exceed %. 

The coefficient Fa has a minimum of v/3/27 for 

a;=V3/3 and a maximum value of V3/27 for #= V3/3, 

so that its absolute value on the interval (0,1) cannot 

exceed V3/27. The coefficient F^ has a minimum value 

of 1/24 at 2;= 1/2 (1 - \/5), a maximum value of 

3/128 at 2;= 1/2, and a minimum value of 1/124 at 

x=l/2 (l+-\/5), so that its absolute value cannot 

exceed 3/128 on the interval (0,1). The coefficient Fs 

has extrema at a;=±loVl50±loVl45 where the signs 

are mutually independent. These four extrema are 

located at approximately #=±0.55 and approximately a; =±1.64. On the interval (0,1), F5 has a 

maximum value ________ of (l/6000)Vl50- 10V145V1+V145 at 

x= (1/10) V150 lO-v/145. Approximately, the maximum 

is 0.011 822 at a;=0.543 912.

The assumption that the successive differences decrease monotonically, or at least do not increase too 

rapidly, as the order of the differences increases, leads 

to the working rule:

IfJA12 |>43, use quartic interpolation. 

If|Al2j^43 and |A3_!|>16, use cubic interpolation.

If|Al 2 |^43, lAl^ld and |A2_!J>8,use quadratic interpolation.

If|Al 2 |^43, |A3_!|^16 and |A2_!|^ 8, use linear interpolation. Table 38 (p. 59) has been arranged so that 

quartic interpolation is always sufficient.
44 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

NUMERICAL EVALUATION OF THE MODIFIED 

POTENTIAL

PLAN OF CALCULATION

In the numerical evaluation of the modified potential 

for selected values of the reflection factor Q and the 

relative displacement a, various methods were tested; 

some of them proved to be relatively inferior to others 

previously developed and some proved to be superior 

but the ranges of superiority were so short that they 

were discarded. In the present discussion only those 

methods are considered that could be recommended for 

recalculation or for extensions of the present tables. 

The accepted values of the modified potential, W(Q,a) 

are shown in table 38 (p. 59).

As a framework for determining the accuracy of approximation formulas, the modified potential was calculated directly for each reflection factor on the range 

Q=[ 1(0.1)1], omitting the trivial case Q=Q, and for 

each relative displacement on the basic range a [0(0.1) 

10(0.5) 15(1) 30]. In addition, the values were calculated for a=50, a=100, and a=°°, but these values 

were used only for checking and are omitted from the 

tables.

In making the direct calculations, the control values 

^ of table 7 (p. 26) were preprinted on a standardized 

calculation form and the values of ^&(a) for each value 

of a were calculated for each value of k for which 

^k<^^k+i- The values of ^fc were entered in a double 

entry table, for the selected value of a, with the arguments k and g_, under the heading 2=1. The values of 

2V*were calculated for each pair of values (k,q). If 

the product did not vanish before the control value of k 

was reached, the remainder was calculated by the error 

coefficients of table 6 (p. 26) for 2<1 and by the EulerMaclaurin formula for 2=1. In these calculations, the 

odd and even values of k were considered separately, 

so that there were two sets of calculations for each value 

of a, one for odd values of k, the other for even values. 

Finally, the modified potentials were calculated by 

algebraic addition of the contributions for the odd and 

for the even values of k.

In most of the computations, 10-bank desk calculators were used. The computations usually were made 

to nine decimal places or to ten significant figures. For 

purposes of checking, a few computations were made to 

ten decimals and occasionally more significant figures 

were retained, by double precision methods, to test the 

effect of accumulated forcing errors. For some of the 

auxiliary functions, as many as thirty significant figures 

were used.

For small values of the relative displacement, the 

modified potentials were calculated by the local expansion formulas; for large values, by the asymptotic formulas.

For both local and asymptotic expansions, the limits 

of validity were determined by comparison with the direct evaluations for relative displacements on the basic 

range. Between the limits of validity of the local and 

the asymptotic expansions, no satisfactory approximations were developed so that interpolation or special 

devices were used.

For each value of Q, and for each spacing in the value 

of a, the calculated values of W were differenced to detect gross errors and to determine the spacing of the 

argument needed to make fourth order interpolation 

valid, if that spacing did not exceed the spacing on the 

basic range. The criterion used was that the fifth difference A6_2 must not exceed 425 in the ninth decimal 

for any line of the table. This value of A5_2 is not included in the table, as it is not needed for the use of the 

table.

If the fifth difference exceeded 425, intermediate 

values were selected for the relative displacement and 

the corresponding values of the modified potential were 

determined by one of the approximate formulas, or "by 

means of interpolation of orders as high as the ninth. 

As a guide for selecting intermediate values of the argument, a simple approximation was used: if the equal 

intervals of a selected table are divided into n equal 

parts, the fifth difference is divided by n5 , neglecting 

higher order differences. As the new tables were differenced, the resulting fifth differences were calculated 

to check the choice of n, which was usually 2 or 5. If 

the fifth difference exceeded 425, a larger value of n was 

selected.

As a guide to consistency, the tabular spacing was 

selected as 0.02, 0.05, 0.1, 0.2, or 0.5. The break-points 

are subject to personal choices and in the tables for a 

few values of the reflection factor, some of these spacings 

are omitted. The choices of breakpoints were affected 

by the ease of calculation, by the number of lines in the 

resulting tables, and by calculations previously made.

In the tables of the modified potential, the functional values and the differences are listed to eight decimals. In most of the entries, the functional values are 

correct to within half a unit in the eighth decimal place, 

as they have been rounded from values calculated to 

at least nine decimals. In a few cases, forcing errors 

may cause the values of the function to be in error by 

more than half a unit in the eighth decimal place, but 

no functional value should be in error by more than a 

few units. The differences may have larger errors so 

that interpolated values of the modified potential may 

have errors of several units in the eighth decimal place. 

The results of calculations with these tables should be 

reduced to fewer significant figures, depending on the 

use made of the table, but no general rule can be given 

beyond that of observing the customary precautions.
NUMERICAL EVALUATION OF THE MODIFIED POTENTIAL 45

Because of forcing errors, the differences as rounded 

may not agree exactly with the corresponding differences calculated from the rounded values of the function. This discrepancy does not decrease the reliability 

of the interpolated value.

In some intervals of the tables, empirical or semiempirical methods were used. One of these deserves 

specific mention. If the calculated values of the 

function have differences that are too large for interpolation, the remainders after subtracting the first few terms 

of the series used in calculating the functional values 

may have differences that are small enough to permit 

interpolation. If this condition exists, the remainders 

are calculated for the intermediate arguments by 

interpolation and added to the sums of the initial terms 

that are calculated directly. This device is useful 

because the terms of the various series used decrease 

rapidly at first and more slowly after the first few terms. 

In some cases, a single isolated term is sufficient to 

justify interpolation in the remainder, in other cases two 

isolated terms are sufficient. Rarely is there a gain in 

isolating more than two terms. This method applies 

to direct series, to local expansions, and to asymptotic 

expansions. The restriction to fourth order interpolation does not apply. In some parts of the tables, interpolations of orders as high as the ninth were used for 

remainders. Judgment as to relative difficulties and 

accuracy of possible methods is the only guide for details 

in selecting procedures.

FORMULAS FOB CALCULATIONS

In an earlier section, various results were derived 

for use in calculating the value of the modified potential 

function for selected values of the reflection factor, 

Q, and of the relative displacement, a. To avoid the 

necessity of searching for these results, they are summarized here.

The calculations fall into three groups: direct series, 

local expansions, and asymptotic expansions, with 

auxiliary devices applicable to one or more of these 

groups.

For the direct series form the modified potential is, 

by equation 12.932:

/c=l

where

and

Rn=

fc=n+l

(27.11)

(12.31)

(27.12)

The remainder, Rn , after isolating the initial n terms, 

is available by the Euler-Maclaurin formula for g=l

and by comparison with a geometric series for 

For purposes of calculation, it is convenient to separate 

the terms into two groups, one for odd, the other for 

even, values of the counter k. For this purpose, the 

modified potential is given, because of equation 

12.932, by

-W=eW1 + W2 (27.13) 

where

k=l

k=l

By equation 12.61,

For 2=1, equations 12.62 may be written

(12.621)

(12.622) 

(27.14)

(27.211)

(27.212)

(19.41)

(19.42)

(19.2)

The values of g2k are obtainable from equations 19.1 

or 19.3 The value of D0 is given by equation 19.12. 

The values of D^ are given by equation 18.42 and the 

values of E21e are given by equations 18.5 or 18.6. A 

few values of (l-D2 icE2 ^) are shown in tables 10 and 11, 

pages 30 and 31.

where

Rm=-Z(2n) 

R n3=-Z(2n+l)

For 2<Cl, equations 12.62 may be written

k=l

/c=l

(27.311)

(27.312)

By equations 13.45, the maximum absolute values of the 

remainder are

and

respectively.

If the remainders are written

(27.321)

(27.322)
46 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

the error in equations 27.31 cannot exceed the applicable 

remainder. The second factor in each of the equations 

27.32 is the last retained term of Wv, where u=l or 2. 

The values of the first factor are shown in table 6, 

page 26.

For small relative displacements, local expansions are 

available. For 2=1, after isolating n terms, the 

remainders are given by equations 19.4 where

(20.61)

«=o

(20.62)

<*=- (18.3) x2 v '

The values of AK, 2u are shown in table 13, page 32.

For 2*0? after isolating (n 1) terms, the modified 

potential is

n 1 e»

(21.333)

where

w-i

1 n2 

_ (2u)l

(21.332)

(21.522)

(21.411)

(21.23)

(21.121) 

(21.321) 

(21.521)

Values of Hu are shown in table, 15 page 33. After 

isolating (n 1) terms, the remainder of equation 21.333 

has the value, by equations 21.632, 21.633,

w=l

(27.4)

where the values of SQ , Si} S2) and the factor

are shown in table 16, page 36, for 0=30, as discussed 

in the summary following equations 21.667.

The local expansion for 2<O is also valid as a limiting 

case for 2=1.

The asymptotic expansions are available for large 

values of the relative displacement. For <2= + l, the 

modified potential for large displacements is given by

,a) =loge (2o o) -K, (23.621)

wnere £=0.8090 7869 6218 358. (23.622) 

For Q= l, the modified potential for large relative 

displacements is given by

where

TF(-l,o)=loge2-l/(2o),

Ioge2=0.6931 4718 0560.

(24.91)

(24.92)

For 2*0) and large relative displacements, the modified potential has the value

where

Wa =loSe (l-Q).

(22.31) 

(14.31)

The values of Wm are shown in table 8, page 27, and 

the values of Bn are given by equations 22.32, and 

22.5 and are shown in table 1, page 28.

CALCULATIONS FOB. SEPARATE VALUES OF Q 

CALCULATIONS FOR Q=+l

For Q=-\-l, the local expansion may use the remainder calculated by the Euler-Maclaurin formula 

19.4 and 19.2 or by one of the forms used in evaluating 

the remainder of equation 21.333, setting e=l. For 

7i=l, 2, 3, and 4, the early values of 8%, given by 

equation 21.321, are shown in table 18 and the early 

values of Cnu, given by equations 21.332 and 21.522, 

are shown in table 19. The values of XnM, given in 

equations 21.42 and 21.523, are also shown in table 19.

For isolated calculations, n may be taken as 4. 

Increasing the value of n reduces the value of j8 for a 

selected value of a and causes the remainder series to 

converge more rapidly. In the calculations for the 

present study n was selected as the smallest integer 

not less than the value of a. Local expansions were 

used to evaluate the modified potential for relative 

displacements from 0 to 3.2, with various spacings.

For relative displacements of three or greater the 

asymptotic expansion of equation 23.621 is sufficiently 

accurate for eight decimal values. There is no gap 

between the regions of applicability of local and asymptotic expansions.
NUMERICAL EVALUATION OF THE MODIFIED POTENTIAL 47

TABLE 18. Values of S£ for q= +1 

[D, The number in the preceding column is to be divided by this power of 10]

a

3__ _____ _

5__ _________

7-_____ 

9.__ .__._

11 ______ _

13__. -..-.._

15 __________

17__ _____

19. _________

21. _________

23 __________

25. _ ______

27__________

29__________

31____ 

33__________

35 ____ .__

7B = 1

1. 051 799 790

1. 004 523 763

1. 000 471 549

1. 000 051 345

1. 000 005 666

1. 000 000 628

1. 000 000 070

1. 000 000 008

1. 000 000 001

1. 000 000 000

1. 000 000 000

1. 000. 000. 000

1. 000 000 000

1. 000 000 000

1. 000 000 000

1. 000 000 000

1. 000 000 000

7B=2

1. 502 571 129

3. 240 399 235

7. 877 728 730

1. 957 047 642

4. 885 225 530

1. 220 852 922

3.051 851 160

7. 629 452 798

1. 907 352 272

4 768 373 856

1. 192 093 038

2. 980 232 328

7. 450 580 652

1. 862 645 153

4 656 612 875

1. 164 153 218

2. 910 383 046

D

1

2

3

3

4

4

5

6

6

7

7

8

9

9

10

10

11

7B=3

K 170 070 no«

4. 523 762 795

4. 715 486 524

5. 134 518 384

5. 666 051 090

6. 280 554 218

6. 972 470 313

7. 744 839 456

8. 604 441 145

9. 560 116 531

1. 062 220 241

1. 180 238 743

1. 311 373 995

1. 457 081 262

1. 618 978 798

1. 798 865 178

1. 998 739 026

D

2

3

4

5

6

7

8

9

10

11

11

12

13

14

15

16

17

7B = 4

2. 525 711 289

1. 153 922 348

6. 522 872 955

3. 922 642 238

2. 413 030 294

1. 497 965 669

9. 334 788 912

5. 826 719 389

3. 639 626 916

2. 274 193 701

1. 421 212 267

8. 882 136 200

5. 551 212 869

3. 469 474 099

2. 168 411 885

1. 355 254 810

8. 470 335 290

D

2

3

5

6

7

8

10

11

12

13

14

16

17

18

19

20

22

7B=5

1. 476 275 323

4 085 364 577

1. 430 128 155

5. 399 204 185

2. 102 182 063

8. 299 474 160

3. 298 375 303

1. 315 080 730

5. 251 728 516

2. 098 951 642

8. 392 273 200

3. 356 190 276

1. 342 329 631

5. 369 019 901

2. 147 547 056

8. 590 063 973

3. 436 000 239

D

2

4

5

7

8

10

11

12

14

15

17

18

19

21

22

24

25

TABLE 19. Local coefficients Cnu and ratios Xnu for Q= +1

u

1 __________ . ______ . ___ ..

2 ___________ .. _____ .........

3... - 

4- ____ . _________________ .

5.. ________ . _______ . ___ ..

6.. _____________________ ..

7... 

8--... 

9... _____________________

10... __________________ . ....

11... __ . __ . __ . _________ . ....

12.._.._.__.._.__.._...._..____..__..___

13.......-- . _ 

14....... __ --. -. -. ..........

15.. 

16... ____ . __________ . ____ .

17... _ . ________________ . ...

18. ... ..................

19.. ________ ..... _ . .............

20.... . 

21 _____ . ___ . _______ ......

22... ____________ . __ . _ ....

23..... ..............................

24.... . 

25 . ... .. .. ..

26. ............. ...................

27 . _ ... __ ... ___ .........

28.... ..................................

29... ___ .. _____ .-..... .-..-...

30 __ . __ -. __ ........... _ ..

Coefficients (-l)«-iCB « where 

n=l

0. 601 028 452 

.388 847908 

. 315 109 149 

. 273 986 670 

. 246 215 367

. 225 613 620 

.209479064 

. 196 382 115 

. 185 470 935 

. 176 197 136

.168 188 115 

. 161 180 263 

. 154 981 018 

. 149 445 981 

. 144 464 448

. 139 949 934 

. 135 833 760 

. 132 060 600 

. 128 585 321 

. 125 370 688

. 122 385 671 

. 119 604 179 

. 117 004 088 

. 114 566 503 

. 112 275 173

. 110 116 035 

. 108 076 849 

. 106 146 905 

. 104 316 786 

. 102 578 173

n=2

0. 404 113 806 

. 221 566 531 

. 166 985 548 

. 140 587 498 

. 124 535 528

. 113 387 133 

. 104 978 827 

.098290 734 

. 092 777 302 

. 088 116 231

.084 101 551 

.080 593 323 

.077 491 873 

. 074 723 575 

.072 232 475

.069 975 075 

. 067 916 926 

. 066 030 320 

. 064 292 669 

.062 685348

. 061 192 837 

. 059 802 090 

. 058 502 044 

. 057 283 252 

. 056 137 586

.055 058 017 

. 054 038 424 

. 053 073 453 

.052 158 393 

.051 289086

n=3

0.346 756 064 

. 172 461 812 

.122284 597 

.099 151 775 

.085 843 247

. 077 090 657 

.070 792 519 

. 065 963 886 

. 062 088 826 

. 058 873 358

. 056 138 152 

.053 767 339 

.051 682 260 

.049 827207 

.048 161 272

. 046 653 495 

.045279839 

. 044 021 249 

.042 862 348 

.041 790 544

. 040 795 397 

.039 868 155 

. 039 001 415 

.038 188863 

.037 425 074

.036 705 354 

.036 025 621 

.035 382 299 

.034 772 260 

.034 192 724

W=4

0. 320 158 929 

. 150 002 765 

. 101 798 414 

. 079 969 123 

.067 692403

. 059 834 672 

. 054 344 086 

. 050 254 569 

. 047 058 007 

.044 464 747

. 042 299 086 

.040 448 926 

. 038 839 627 

.037419605 

.036 152 006

.035009 714 

. 033 972 239 

. 033 023 731 

.032 151 676 

.031 346 007

.030 598 501 

. 029 902 347 

. 029 251 838 

.028 642 138 

.028 069 114

. 027 529 210 

. 027 019 339 

.026 536806 

. 026 079 146 

. 025 644 512

Ratios A,,,, where 

n=l

0.646970883 

.810 366 065 

.869 497 666 

.898 639 948 

.916 326 315

.928 485 894 

.937 478 485 

. 944 439 034 

.949998 641 

.954 545 112

.958 333 248 

. 961 538 440 

.964285 708 

. 966 666 666 

.968 750 000

.970 588 235 

.972 222 222 

.973684210 

.975 000000 

.976 190 471

. 977 272 731 

.978 260868 

.979 166 668 

.980000 000 

. 980 769 230

.981 481 480 

.982 142855 

.982 758 621 

.983 333 334 

.983870968

n=2

0. 548 277 559 

. 753 658 719 

.841 914 164 

.885 822 209 

.910 480 205

. 925 844 266 

. 936 291 026 

.943906 906 

. 949 760 649 

. 954 438 813

.958 285 810 

.961 517286 

. 964 276 280 

.966 662 464 

.968 748 128

.970 587 401 

.972 221 852 

.973 684 046 

. 974 999 926 

.976 190 444

.977 272 712 

.978 260 863 

.979 166 664 

.979 999 999 

.980 769 230

.981 481 481 

. 982 142 857 

.982 758 621 

.983333 333 

.983 870 968

w=3

0. 497 357 740 

. 709 053 184 

. 810 827 999 

.865 776 198 

. 898 039 849

. 918 302 191 

.931 791 768 

.941 254 827 

. 948 211 799 

.953 540 852

.957 768 238 

.961 220 356 

. 964 106 571 

.966 565 761 

.968 693 159

.970 556216 

. 972 204 188 

.973 674 052 

. 974 994 279 

.976 187 255

.977 270 913 

.978 259 849 

.979 166 091 

.979 999 674 

.980 769049

.981 481 379 

.982 142 799 

. 982 758 589 

. 983 333 314 

.983 870 958

W=4

0. 468 525 923 

. 678 643 583 

. 785 563 542 

. 846 481 748 

.883 920 046

. 908 237 390 

.924747712 

. 936 392 380 

.944 892 488 

.951 294 873

. 956 259 248 

.960 214 870 

.963 438 824 

.966 124 740 

.968 403 081

. 970 366 082 

.972 079 909 

.973 593 081 

.974 941 617 

.976 153 071

.977 248 755 

.978 245 554 

.979 156 865 

.979 993 672 

.980 765 193

.981 478 909 

.982 141 199 

.982 757 562 

.983 332 655 

.983 870 534
48 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

CALCULATIONS FOR Q = -l

In preparing the tables for Q= l } shown in table 38, 

page 59, methods similar to those used for$= + l were 

used. In fact, the calculations for the direct series 

are the same for Q=([ and for Q= <L, except that the 

sum of the odd terms is subtracted from the sum of the 

even terms instead of being added as for Q=-\-l. 

Likewise, the values of 8%, Onu, and \nu are obtained 

from the same basic data in the two cases.

By equation 24.81, the modified potential for Q= l } 

and the argument a, is the excess of the modified potential for a/2 over that for a, each calculated for Q=l. 

Accordingly the values of 8%, Cnu, and \nu are omitted 

from the tables.

CALCULATIONS FOK Q=+0.1

For 2= +0.1, the early values of 8% are shown in 

table 20, the early values of Cnu in table 21, and the 

early values of \nu in table 22.

TABLE 20. Values of S£ for q=0.1 

[D, The value in the column preceding D is to be divided by this power of 10]

a

3 _ _ -

5_ _ __ -

7 

9 

11 

13 

15. _ -----

17.. ------

iq

21- _

23_- ___ .-

25 

27 

29 

31. - ___

33. _ _____

35 

37 

39 

41 ------

43 . _ --.

45 __ -----

47 ___ -_-_

49 ___ _ _

51 ___ ---_

53 

55 

57 

59 

7i=l

1. 000 371 173

1. 000 041 184

1. 000 004 574

1. 000 000 508

1. 000 000 056

1. 000 000 006

1. 000 000 001

C)

D

1

1

1

1

1

1

1

71=2

1. 251 567 149

3. 125 977 852

7. 813 110 709

1. 953 163 157

4. 882 836 345

1. 220 704 615

3. 051 758 744

7. 629 395 113

1. 907 348 669

4. 768 371 605

1. 192 092 897

2. 980 232 240

7. 450 580 597

1. 862 645 149

4. 656 612 873

1. 164 153 218

2. 910 383 046

7. 275 957 614

1. 818 989 404

4. 547 473 509

1. 136 868 377

2. 842 170 943

7. 105 427 358

1. 776 356 839

4. 440 892 099

1. 100 223 025

2. 775 557 562

6. 938 893 904

1. 734 723 476

D

3

4

5

5

6

6

7

8

8

a

q

10

11

11

12

12

13

14

14

15

15

16

17

17

18

18

1Q

30

20

7i=3

3. 711 732 996

4. 118 432 304

4. 573 754 925

5. 081 038 591

5. 645 234 120

6. 272 336 674

6. 969 204 708

7. 743 537 483

8. 603 921 215

9. 559 908 733

1. 062 211 932

1. 180 235 421

1. 311 372 666

1. 457 080 730

1. 618 978 585

1. 798 865 093

1. 998 738 992

2. 220 821 102

2. 467 579 002

2. 741 754 447

3. 046 393 830

3. 384 882 033

3. 760 980 037

4. 178 866 707

4. 643 185 230

5. 159 094 700

5. 732 327 445

6. 369 252 716

7. 076 947 463

D

5

6

7

8

9

10

11

12

13

14

14

15

16

17

18

19

20

21

22

23

24

25

26

?,7

28

29

30

31

32

71=4

1. 567 149 261

9. 778 515 700

6. 107 092 648

3. 815 690 302

2. 384 461 544

1. 490 192 703

9. 313 438 457

5. 820 825 174

3. 637 995 218

2. 273 741 313

1. 421 086 738

8. 881 787 714

5. 551 116 100

3. 469 447 223

2. 168 404 420

1. 355 252 737

8. 470 329 531

5. 293 955 936

3. 308 722 455

2. 067 951 533

1. 292 469 707

8. 077 935 670

5. 048 709 794

3. 155 443 621

1. 972 152 263

1. 232 595 164

7. 703 719 778

4. 814 824 861

3. 009 265 538

D

6

8

9

10

11

12

14

15

16

17

18

20

21

22

23

24

26

?,7

28

29

30

3?

33

34

35

36

38

39

40

n=5

8. 029 292 449

3. 205 966 899

1. 281 216 362

5. 122 480 679

2. 048 506 052

8. 193 032 502

3. 277 010 683

1. 310 762 993

5. 242 967 735

2. 097 169 905

8. 388 644 539

3. 355 450 657

1. 342 178 802

5. 368 712 226

2. 147 484 282

8. 589 935 885

3. 435 974 101

1. 374 389 589

5. 497 558 249

2. 199 023 278

8. 796 093 068

3. 518 437 218

1. 407 374 885

5. 629 499 538

2. 251 799 814

9. 007 199 256

3. 602 879 702

1. 441 151 881

5. 764 607 523

D

8

9

10

12

13

15

16

17

19

20

22

23

24

26

27

29

30

31

33

34

36

37

38

40

41

43

44

45

47

i For a 17, S"= 0.1 to 9 decimal places.

For n 7, the value of Cnu is

For n^8, |C» 

so that W= 

to nine decimal places.

(28.11)

(28.12)

As the number of isolated terms increases, the terms 

of the remainder converge more rapidly, so that less 

accuracy is needed in Cnu and especially in \nu.

TABLE 21. Local coefficientsjCnu for Q = -\-0.1

Terminal digit of Cn« is in ninth decimal place.

Sign is that of (-1)«-».

For u >15, CU = (-l)«-iJ?.Q»/7i

u

1 __ . _ . . . 

2. __ .. ___

3- - _ ... 

4 ______ ... .... ....

5- __ 

6 ___ .. .- __ .

7 _ 

8-- 

9_ 

10- _ 

12-. _

13- 

14 

15 ___ .. _ _ -

7i=l

50644342

3761 8769

3127 4559

2734 9105

2461 0578

2255 8869

2094 7330

1963 8077

1854 7062

1681 8810

1611 8026

1549 8102

1494 4598

1444 6445

71=2

257 7369

190 0297

157 1770

137 0771

104 7603

982003

92 7395

88 1003

R4 flCUR

80 5904

77 4906

74 7230

72 2322

71=3

17 4080

12 8068

10 5587

9 1840

82381

7 5375

6 9918

6 5510

6 1850

5 R7_tft

5 6070

5

eRqq

5 1663

4 9817

4 8155

71 = 4

13180

9695

7981

6930

6206

KRTt

5256

_tQ91

4644

4409

4207

4ftQO

3876

3737

3612

7i=5

1062

782

643

558

499

455

422

one

372

QEQ

337

322

310

299

289

71=6

89

65

54

47

At)

39

35

QQ

31

29

OQ

97

OR

ftK

24
NUMERICAL EVALUATION OF THE MODIFIED POTENTIAL 

TABLE 22. Local ratios XBU /or Q = + 0J

49

u

!__._____. ________________

2_ _

3____-____----.___________

4_________________________

5____-___--________-______

6--_--__-_-__--__-__-_____

7______-___--_-____-______

8__-_________-_____-_-__-_

9-___-____-_______________

10_______________.________

ll---_____--_-_______-_-__

12___. ____________________

13__-________-__-__-______

14________________________

15_-__-____----______---__

16_---___-___--______--_-_

17----__--_-_-__________-_

18__---___-__-_______.__-_

19

20_--_____-_---_._________

21________________________

22________________________

23___________--_._________

24

25_-____-__-_-____________

26________---____-_____-._

27__-_____________________

28_______---_-___-____-___

29__.______-___. __________

30__-____-_-_________--_._

Values of XB» where 

n=l

0. 742 802 985 

. 831 355 200 

. 874 484 103 

. 899 867 780 

. 916 633 054

. 928 562 923 

. 937 502 147 

. 944 443 904 

. 949 999 864 

. 954 545 421

. 958 333 324 

. 961 538 460 

. 964 285 713 

. 966 666 667 

. 968 750 000

. 970 588 235 

. 972 222 222 

. 973 684 211 

. 975 000 000 

. 976 190 476

. 977 272 727 

. 978 260 870 

. 979 166 667 

. 980 000 000 

. 980. 769 231

. 981 481 481 

. 982 142 857 

. 982 758 621 

. 983 333 333 

. 983 870 968

«=2

0. 737 301 229 

. 827 117 762 

. 872 119 485 

. 898 687 776 

. 916 076 902

. 928 305 637 

. 937 380 872 

. 944 391 144 

. 949 976 177 

. 954 534 818

. 958 328 589 

. 961 536 348 

. 964 284 773 

. 966 666 246 

. 968 749 813

. 970 588 152 

. 972 222 184 

. 973 684 192 

. 974 999 995 

. 976 190 470

. 977 272 727 

. 978 260 870 

. 979 166 667 

. 980 000 000 

. 980 769 231

. 981 481 481 

. 982 142 857 

. 982 758 621 

. 983 333 333 

. 983 870 968

«=3

0. 735 681 818 

. 824 463 968 

. 869 606 840 

. 897 205 037 

. 914 958 513

. 927 600 761 

. 936 949 785 

. 944 133 053 

. 949 823 964 

. 954 446 008

. 958 277 210 

. 961 506 786 

. 964 267 854 

. 966 661 004 

. 968 744 325

. 970 585 037 

. 972 220 418 

. 973 683 197 

. 974 999 429 

. 976 190 152

. 977 272 547 

. 978 260 786 

. 979 166 612 

. 979 999 967 

. 980 769 211

. 981 481 472 

. 982 142 848 

. 982 758 619 

. 983 333 332 

. 983 870 968

«=4

0. 747 685 

. 832 043 

. 874 153 

. 895 577 

. 913 789

. 926 725 

. 936 305 

. 943 675 

. 949 505 

. 954 227

. 958 129 

.961 407 

. 964 202 

. 966 613 

. 968 715

. 970 567 

. 972 207 

. 973 675 

. 974 994 

. 976 186

. 977 270 

. 978 260 

. 979 166 

. 979 999 

. 980 769

.981 480 

. 982 143 

. 982 759 

. 983 333 

. 983 871

n=5

0. 73613 

. 82273 

. 86734 

. 89456 

. 91284

. 92588 

. 93562 

. 94314 

. 94908 

. 95391

. 95789 

. 96123 

. 96406 

. 96652 

. 96864

. 97052 

. 97217 

. 97365 

. 97498 

. 97617

. 97726 

. 97825 

. 97916 

. 98000 

. 98077

. 98148 

. 98214 

. 98276 

. 98333 

. 98387

«=6

0. 7369 

.8226 

.8669 

.8966 

.9120

.9253 

.9350 

.9425 

.9487 

.9535

.9577 

.9610 

.9638 

.9664 

.9686

.9704 

.9721 

.9736 

. 9750 

.9762

.9773 

. 9783 

.9792 

. 9800 

.9808

.9815 

.9821 

.9828 

.9833 

.9839

n=7

0.737 

.823 

.867 

.893 

.910

.925 

.935 

.943 

.948 

.954

.957 

.961 

.963 

.967 

.969

.971 

.972 

.974 

.975 

.976

.977 

.978 

.979 

.980 

.981

.981 

.982 

.983 

.983 

.984

As seven initial terms are always sufficient and as 

the basic range furnishes values of the modified potential for which fourth order interpolation is adequate, 

local expansions were used for a<9, and asymptotic 

expansions were used for a ^9 on the basic range to 

check the direct calculations. For immediate values 

above a=10, interpolation was used.

CALCULATIONS FOR Q = -0.1

For Q= 0.1, the values of S^ are the same as for 

Q=+0.1 and are shown in table 20. However, the 

values of Cnu and of XKM are different for the two cases 

for small values of u. The values of Cnn are shown in 

table 23 and of XKM in table 24.

TABLE 23. Local coefficients Cnu for Q= 0.1 

[Terminal digit in ninth decimal place. Sign is that of (-1) »+«-i]

u

1- _ 

2 __. 

3. _ ... . . 

4- 

5- 

6_ - ... 

7- 

8 _ ._ . -

9 _ 

10- 

11- 

12- 

13-- 

14-- 

15- 

«=!

4939 2775

3738 4320

3122 5727

2733 8423

2460 8175

2255 8319

2094 7202

1963 8047

1854 7054

1761 9704

1681 8810

1611 8026

1549 8102

1494 4598

1444 6445

«=2

242 8900

185 0876

155 3475

136 3658

122 9052

112 7352

104 7124

98 1803

92 7311

88 0967

840932

80 5898

774904

74 7229

72 2322

«=3

15 9976

12 2127

10 2805

9 0470

8 1688

7 5018

6 9731

6 5411

6 1797

58718

5 6055

53723

5 1658

4 9814

4 8154

«=4

1 1895

9080

7653

6746

6100

5609

5219

4899

4630

4401

4202

4028

3874

3736

3611

«=5

946

721

608

537

486

447

416

391

370

351

336

322

310

299

289

w=6

78

59

50

45

40

37

35

33

31

29

28

27

26

25

24
50 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

TABLE 24. Local ratios X«»/or Q= 0.1

u

1__ _ _ _

2__. _ _ .

3 _

4 ___ .... ___ __._.__.__.

5 ___ .._-. _____ ________

6-__-____-___.____________

7--__-_-_---_____________-

8______ __ ______ _ ______

9-_______. _ __ ____ _ _ _

10

11. ___ _ __ _ _

12. _______ ____________

13________________________

14________ _ _____________

15__-__-_______________.__

16________________________

17. _ __ ___ _____________

18 __

19 __

20________________________

21 ___ __ _____ _____ __

22________________________

23. _ __ __ ______________

24_____ ___ ______________

25___. ____________________

26 __ ______ _ ___________

27________________________

28___. ____________________

29________________________

30_. ______________________

Values of Xn« where 

n=l

0. 756 878 312 

. 835 262 665 

. 875 509 583 

. 900 131 459 

. 916 700 192

. 928 579 925 

. 937 502 145 

. 944 444 985 

. 950 000 136 

. 954 545 489

. 958 333 342 

. 961 538 463 

. 964 285 714 

. 966 666 667 

. 968 750 000

. 970 588 235 

. 972 222 222 

. 973 684 211 

. 975 000 000 

. 976 190 476

. 977 272 727 

. 978 260 870 

. 979 166 667 

. 980 000 000 

. 980 769 231

. 981 481 481 

. 982 142 857 

. 982 758 621 

. 983 333 333 

. 983 870 968

n=2

0. 762 022 477 

. 839 318 535 

. 877 811 233 

. 901 290 959 

. 917 252 772

. 928 835 786 

. 937 618 754 

. 944 497 650 

. 950 023 795 

. 954 556 083

. 958 338 076 

. 961 540 577 

. 964 286 657 

. 966 667 087 

. 968 750 187

. 970 588 319 

. 972 222 259 

. 973 684 227 

. 975 000 007 

. 976 190 479

. 977 272 727 

. 978 260 870 

. 979 166 667 

. 980 000 000 

. 980 769 231

. 981 481 481 

. 982 142 857 

. 982 758 621 

. 983 333 333 

. 983 870 968

n=3

0. 763 410 135 

. 841 783 145 

. 880 023 802 

. 902 921 342 

. 918 346 551

. 929 531 158 

. 938 046 038 

. 944 754 255 

. 950 175 443 

. 954 644 668

. 958 389 385 

. 961 570 104 

. 964 303 567 

. 966 676 734 

. 968 755 676

. 970 591 434 

. 972 224 025 

. 973 685 226 

. 975 000 572 

. 976 190 798

. 977 272 909 

. 978 260 972 

. 979 166 724 

. 980 000 033 

. 980 769 251

. 981 481 490 

. 982 142 866 

. 982 758 623 

. 983 333 334 

. 983 870 968

n=4

0. 763 322 

. 842 800 

. 881 464 

. 904 280

. 919 475

. 930 385 

. 938 679 

. 945 206 

. 950 492 

. 954 861

. 958 536 

.961 668 

. 964 370 

. 966 721 

. 968 785

. 970 609 

. 972 237 

. 973 693 

. 975 006 

. 976 194

. 977 276 

. 978 262 

. 979 168 

. 980 001 

. 980 769

. 981 482 

. 982 143 

. 982 759 

. 983 333 

. 983 871

n=5

0. 762 71 

. 843 23 

. 882 27 

. 905 23 

. 920 37

. 931 19 

. 939 34 

. 945 74 

. 950 90 

. 955 17

. 958 76 

. 961 85 

. 964 51 

. 966 81 

. 968 86

. 970 66 

. 972 27 

. 973 72 

. 975 02 

. 976 21

. 977 28 

. 978 27 

. 979 18 

. 980 00 

. 980 77

. 981 48 

. 982 14 

. 982 76 

. 983 33 

. 983 87

w=*6

0. 7619 

.8431 

.8827 

.9058 

.9211

.9318 

.9351 

. 9426 

.9487 

.9535

.9575 

.9610 

.9638 

. 9664 

.9686

.9704 

.9721 

. 9749 

.9750 

.9762

. 9773 

.9783 

.9792 

.9800 

.9808

.9815 

.9821 

.9828 

.9833 

.9839

n=7

0.760 

.842 

.883 

.906 

.921

.932 

.940 

.946 

.952 

.956

. 959 

.963 

. 965 

.967 

.969

.971 

. 972 

.974 

.975 

.976

.977 

.978 

.979 

. 980 

. 981

.981 

.982 

.983 

.983 

.984

The values of the modified potential for Q- 0.1 

were calculated by local expansions for values of the 

relative displacement from 0 to 6 and by asymptotic 

expansions from 5 to 30; the overlapping values were 

used for adjustments.

CALCULATIONS FOR Q=+0.2

For Q= +0.2, the use of juna given by equation 21.521

is preferable to that of S% as /*«« converges to zero 

rapidly. The early values of juna are shown in table 25, 

the values of the local coefficients in table 26, and the 

values of the local ratios in table 27. In each of these 

tables, the number of retained decimal places is reduced 

as the value of n increases.
NUMERICAL EVALUATION OF THE MODIFIED POTENTIAL 

TABLE 25. Values of naa for Q= +0.2

51

u

1 ____

2-. _ . __

3 ----__

4 ___ .._.

5 ______

6-__-.._.

7_._______

8 ______

9_ _ ...._

10_____-__

11_______-

12__...__.

13 _____

14___._...

15_.____._

16__ _ ...

17____.._.

18 _____

19__... _

20.-.._._.

21___. _ .

22________

23.. ______

24 ____ _

25 _____

26 _____

27. ___ __

28_ __ -_.

29 ___ .__

30 _____

Values of /in « where 

w=l

0. 026 620 979 

. 006 422 979 

. 001 581 301 

. 000 392 689 

. 000 097 828

. 000 024 439 

. 000 006 106 

. 000 001 526 

. 000 000 381 

. 000 000 095

. 000 000 024 

. 000 000 006 

. 000 000 001 

. 000 000 000

w=2

0. 064 839 148 

. 027 676 633 

. 012 031 926 

. 005 282 767 

. 002 332 080

. 001 032 584 

. 000 457 961 

. 000 203 298 

. 000 090 295 

. 000 040 116

. 000 017 826 

. 000 007 922 

. 000 003 520 

. 000 001 565 

. 000 000 695

. 000 000 309 

. 000 000 137 

. 000 000 061 

. 000 000 027 

. 000 000 012

. 000 000 005 

. 000 000 002 

. 000 000 001 

. 000 000 000

71=3

0. 094 160 60 

. 050 847 14 

. 027 883 64 

. 015 436 50 

. 008 596 20

. 004 804 70 

. 002 691 75 

. 001 510 22 

. 000 848 11 

. 000 476 56

. 000 267 89 

. 000 150 62 

. 000 084 70 

. 000 047 64 

. 000 026 79

. 000 015 07 

. 000 008 48 

. 000 004 77 

. 000 002 68 

. 000 001 51

. 000 000 85 

. 000 000 48 

. 000 000 27 

. 000 000 15 

. 000 000 08

. 000 000 05 

. 000 000 03 

. 000 000 02 

. 000. 000 01 

. 000 000 01

w=4

0. 115 977 48 

. 071 347 20 

. 044 457 02 

. 027 939 37 

. 017 660 10

. Oil 206 39 

. 007 130 08 

. 004 544 80 

. 002 900 54 

. 001 852 76

.001 184 17 

. 000 757 17 

. 000 484 28 

. 000 309 80 

. 000 198 21

. 000 126 83 

. 000 081 16 

. 000 051 93 

. 000 033 24 

. 000 021 27

. 000 013 61 

. 000 008 71 

. 000 005 58 

. 000 003 57 

. 000 002 28

. 000 001 46 

. 000 000 94 

. 000 000 60 

. 000 000 38 

. 000 000 24

«=5

0. 132 5926 

. 088 6719 

. 059 9378 

. 040 8225 

.027 9534

. 019 2153 

. 013 2454 

.009 1486 

. 006 3282 

. 004 3819

. 003 0366 

.002 1055 

. 001 4605 

. 001 0134 

. 000 7033

. 000 4893 

. 000 3389 

. 000 2353 

.000 1634 

. 000 1135

. 000 0788 

. 000 0547 

. 000 0379 

. 000 0264 

. 000 0183

. 000 0104 

. 000 0088 

. 000 0061 

. 000 0043 

. 000 0030

w=6

0. 145 600 

. 103 221 

. 073 841 

. 053 176 

. 038 484

. 027 954 

. 020 361 

. 014 861 

. 010 865 

.007 952

. 005 826 

. 004 270 

. 003 132 

. 002 298 

.001 687

.001 239 

. 000 909 

. 000 668 

. 000 490 

. 000 360

. 000 264 

.000 194 

.000 143 

. 000 105 

. 000 077

. 000 057 

. 000 042 

. 000 031 

. 000 022 

. 000 016

n=7

0. 15607 

. 11550 

. 08616 

. 06465 

. 04873

. 03685 

. 02794 

. 02124 

. 01617 

. 01232

. 00938 

. 00718 

. 00548 

. 00419 

. 00320

. 00245 

. 00188 

. 00143 

. 00109 

.00084

.00064 

. 00049 

. 00038 

. 00029 

. 00022

. 00017 

. 00013 

.00010 

. 00007 

. 00006

n=8

0. 1646 

. 1260 

.0971 

.0751 

.0585

.0456 

.0357 

.0279 

.0219 

.0172

.0136 

.0107 

.0084 

.0066 

.0051

.0041 

.0032 

.0026 

.0020 

.0016

.0013 

.0010 

.0008 

.0006 

.0005

.0004 

.0003 

.0002 

.0002 

.0002

TABLE 26. Local coefficients Cnu /or Q=+0.#

Terminal digit of the coefficient is hi ninth decimal place. 

Signisthatof (-l)--i

u

I ...........

2 ...........

3. .......

4...........

5 ... ... ..

6. ____ ...

7 ...........

8 ...........

9 . ....

10 ..........

11 ..........

12... 

13..........

14 __ . ....

15 

16 

17 

18 

19. .

20 ..........

21. .........

22 

23.. 

24. .........

25.. 

26 

27. .

28. 

29..... ..

30 

ra=l

102662098

7548 1723

6259 8831

5470 8975

4922 3565

4511 8291

4189 4787

3927 6183

3709 4130

3523 9413

3363 7620

3223 6052

3099 6203

29889196

2889 2890

2798 9987

2716 6752

2641 2120

2571 7064

2507 4138

2447 7134

2392 0836

23400818

2291 3301

2245 5035

22023207

2161 5370

2122 9381

20863357

2051 5635

w=2

10648391

770 7575

632 5200

549 7640

4933353

451 6378

419 1372

3928410

370 9747

352 4082

3363822

322 3631

3099631

298 8925

288 9291

279 9000

271 6675

264 1212

257 1706

250 7414

244 7713

239 2084

2340082

229 1330

224 5503

2202321

216 1537

212 2938

208 6336

205 1563

w=3

1458880

105 0847

85 6569

74 0423

66 1891

60 4453

56 0098

52 4473

49 5007

470083

44 8622

429879

41 3318

398542

38 5249

37 3206

36 2226

35 2162

342895

33 4323

326362

31 8944

31 2011

30 5511

29 9400

293643

28 8205

28 3058

27 8178

273542

n=4

22 3195

16 0702

13 0557

11 2431

10 0176

9 1245

8 4386

7 8909

7 4403

70610

6 7355

6 4521

6 2022

5 9797

5 7797

55987

54340

5 2827

51436

50149

4 8955

4 7842

4 6802

4 5827

4 4910

44046

43231

42459

4 1727

4 1031

n=5

36243

26128

21199

18214

16190

14715

13584

12683

11945

11326

10797

10338

9933

9575

9253

8961

8696

8454

8230

8025

7834

7655

7488

7332

7186

7047

6917

6793

6676

6565

n=6

6109

4413

3579

3072

2726

2473

2279

2126

1999

1894

1804

1726

1658

1598

1544

1495

1450

1410

1373

1337

1305

1276

1248

1222

1198

1175

1153

1132

1113

1094

n=7

1057

765

620

532

472

428

394

367

344

326

311

297

285

274

265

257

248

241

235

229

224

219

214

209

205

201

198

194

191

188

»=8

186

135

110

95

84

75

69

65

60

57

55

53

50

48

46

45

43

42

41

40

39

38

37

37

36

35

35

34

33

33

In calculating the values of the modified potential, 

for Q=+0.2, local expansions were used for relative 

displacements from 0 to 8 and asymptotic expansions 

were used for relative displacements of 14 or more. 

For the gap between 8 and 14, with some'Werlapping, 

the asymptotic expansion of ( W) through -55/a9 was 

augmented by the quantity 355.7/a10, determined 

empirically by comparison with the direct series values 

on the applicable intervals of the basic range.

CALCULATIONS FOR Q = -0.2

For the reflection factor Q= 0.2, the early values of 

Mn«, Cnu, and \nu are shown in tables 28, 29, and 30, 

respectively. In calculating values of the modified 

potential, local expansions were used for a _=i6 and 

asymptotic expansions for a^ 7.3. On the range 

4.8_|a _S7.2, the asymptotic expansion values of W 

through _56a9 were decreased by 0.208/a10, determined 

empirically. Overlapping of ranges served as a check 

on the calculations.
52 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN 

TABLE 27. Local ratios for Q= + 0.2

u

l_-__--___._

2 ---__

3 _ _ _

4

5 __-__ 

6 --___

7__ _ ______

8 ________

9 _ _ _

10 _______

11 _______

12 ____ ___

13 -__ -

14 _____

15 _______

16 ----___

17_._____.__

18 _______

19

20__._______

21 _______

22__. _______

23 _______

24_.________

25_-_----.__

26__________

27__________

28_-_______

29 _--_ _

30 _______

Values of Xn« where 

«=1

0. 735 244 311 

. 829 324 351 

. 873 961 607 

. 899 734 730 

. 916 599 400

. 928 554 405 

. 937 495 706 

. 944 443 363 

. 949 999 728 

. 954 545 387

. 958 333 316 

. 961 538 457 

. 964 285 713 

. 966 666 667 

. 968 750 000

. 970 588 235 

. 972 222 222 

. 973 684 211 

. 975 000 000 

. 976 190 476

. 977 272 727 

. 978 260 870 

. 979 166 667 

. 980 000 000 

. 980 769 231

. 981 481 481 

. 982 142 857 

. 982 758 621 

. 983 333 333 

. 983 870 968

W=2

0. 723 825 262 

. 820 647 187 

. 869 164 696 

. 879 358 337 

. 915 478 233

. 928 038 401 

. 937 261 363 

. 944 337 740 

. 949 952 335 

. 954 524 179

. 958 323 742 

. 961 534 229 

. 964 283 829 

. 966 665 826 

. 968 749 626

. 970 588 069 

. 972 222 148 

. 973 684 178 

. 974 999. 985 

. 976 190 469

. 977 272 724 

. 978 260 869 

. 979 166 666 

. 980 000 000 

. 980 769 231

. 981 481 481 

. 982 142 857 

. 982 758 621 

. 983 333 333 

. 983 870 968

w=3

0. 7203 1048 

.8151 2303 

. 8644 0420 

. 8939 3732 

. 9132 2075

. 9266 1878 

. 9363 9529 

. 9438 2006 

. 9496 4732 

. 9543 4636

. 9582 2098 

. 9614 7509 

. 9642 4997 

. 9666 4652 

. 9687 0945

. 9705 8184 

. 9722 1861 

. 9736 8218 

. 9749 9886 

. 9761 8984

. 9772 7237 

. 9782 6066 

. 9791 6655 

. 9799 9993 

. 9807 6920

. 9814 8146 

. 9821 4284 

. 9827 5861 

. 9833 3333 

. 9838 7097

W=4

0. 7200 059 

. 8124 249 

. 8611 622 

. 8910 001 

. 9108 534

. 9248 282 

. 9350 935 

. 9428 986 

. 9490 075 

. 9539 084

. 9579 246 

. 9612 763 

.9641 175 

. 9665 588 

. 9687 431

. 9705 439 

. 9721 938 

. 9736 660 

. 9749 883 

. 9761 830

. 9772 679 

. 9782 578 

. 9791 647 

. 9799 987 

. 9807 684

. 9814 810 

. 9821 425 

. 9827 584 

. 9833 332 

. 9838 702

w=5

0. 7209 158 

. 8113 386 

. 8592 199 

. 8888 721 

. 9088 746

. 9231 324 

. 9337 095 

9418 048 

. 9481 626 

. 9532 670

. 9574 437 

. 9609 196 

. 9638 552 

. 9663 672 

. 9685 428

. 9704 422 

. 9721 215 

. 9736 142 

. 9749 513 

. 9758 518

. 9772 491 

. 9782 445 

. 9790 541 

. 9799 921 

. 9807 615

. 9814 658 

. 9821 402 

. 9827 568 

. 9833 320 

. 9838 701

W=6

0. 722 255 

. 811 141 

. 858 161

. 887 445 

. 907 372

. 921 712 

. 932 447 

. 940 725 

. 947 262 

. 952 532

. 956 850 

. 960 449 

. 963 484 

. 966 077 

. 968 317

. 970 268 

. 971 988 

. 973 511 

. 974 873 

. 976 096

. 977 205 

. 978 211 

.979 130 

. 979 973 

. 980 749

. 981 466 

. 982 132 

. 982 750 

. 983 327 

. 983 870

n=7

0. 72368 

. 81141 

. 85768 

. 88654 

. 90628

. 92059 

. 93139 

. 93976 

. 94640 

. 95178

. 95622 

. 95991 

. 96306 

. 96571 

. 96802

. 97004 

. 97178 

. 97335 

. 97476 

. 97599

. 97712 

. 97815 

. 97908 

. 97993 

. 98072

. 98144 

. 98211 

. 98273 

. 98332 

. 98387

w=8

0. 7252 

.8119 

.8574 

.8861 

.9055

. 9198 

. 9305 

. 9389 

.9456 

. 9512

. 9555 

. 9593 

. 9626 

. 9652 

. 9678

. 9697 

. 9716 

. 9731 

.9746 

. 9759

.9770 

. 9781 

. 9790 

. 9799 

. 9807

. 9814 

. 9820 

. 9826 

.9831 

. 9839

TABLE 28. Values of ( /inn) for Q = -0.2

u

! ____ _ _______________

2 __ _ _________________

3 __ __- _

4

5_ _ _

6___ ___ ____________________

7 ______________________

8_ _ _

9-_______________________.

10 ______________________

11_ _____________________

12__ _ __

13 ______________________

14___ ______________________

15 ________________ _ ___

16 ______________________

17 ___________________

18 ______________________

19 ______________________

20_ _____________________

21 ____________ _ _

22________________. ________

23 __________________ ___

24____ _________________ _

25__- ____ _ ___ _ __ _

26 _ __ ___ _ .

27___ ____ _ ___

28 _ __ ______

29 ______________________

30 ________________

Values of ( M ) where 

n=l

0. 023 632 035 

. 006 092 729 

. 001 544 678 

. 000 388 623 

. 000 0,97 489

. 000 024 389 

. 000 006 101 

. 000 001 526 

. 000 000 381 

. 000 000 095

. 000 000 024 

. 000 000 006 

. 000 000 001 

. 000 000 000

«=2

0. 054 718 586 

. 025 163 338 

. Oil 405 454 

. 005 126 353 

. 002 293 000

. 001 022 817 

. 000 455 519 

. 000 202 688 

. 000 090 142 

. 000 040 078

. 000 017 816 

. 000 007 919 

. 000 003 520 

. 000 001 565 

. 000 000 695

. 000 000 309 

. 000 000 137 

. 000 000 061 

. 000 000 027 

. 000 000 012

. 000 000 005 

. 000 000 002 

. 000 000 001 

. 000 000 000

n=3

0. 076 623 86 

. 044 579 54 

. 025 635 59 

. 014 628 72 

. 008 305 68

. 004 700 16 

. 002 654 12 

. 001 496 68 

. 000 843 23 

. 000 474 81

. 000 267 26 

. 000 150 40 

. 000 084 62 

. 000 047 61 

. 000 026 78

. 000 015 07 

. 000 008 48 

. 000 004 77 

. 000 002 68 

. 000 001 51

. 000 000 85 

. 000 000 48 

. 000 000 27 

. 000 000 15 

. 000 000 08

. 000 000 05 

. 000 000 03 

. 000 000 02 

. 000 000 01 

. 000 000 01

W=4

0. 091 865 39 

. 060 710 89 

. 039 749 59 

. 025 852 10 

. 016 733 66

. 010 794 94 

. 006 947 29 

. 004 463 58 

. 002 864 45 

. 001 836 71

. 001 177 05 

. 000 754 00 

. 000 482 87 

. 000 309 17 

. 000 197 93

.000 126 71 

. 000 081 10 

. 000 051 91 

. 000 033 22 

. 000 021 27

. 000 013 61 

. 000 008 71 

. 000 005 58 

. 000 003 57 

. 000 002 28

. 000 001 46 

. 000 000 94 

. 000 000 60 

. 000 000 38 

. 000 000 24

n=5

0. 102 8770 

. 073 6253 

. 052 2958 

. 036 9343 

. 025 9729

. 018 2058 

. 012 7307 

. 008 8860 

.006 1943 

. 004 3136

. 003 0017 

. 002 0877 

. 001 4514 

.001 0088 

. 000 7010

. 000 4859 

. 000 3383 

. 000 2350 

.000 1632 

.000 1134

. 000 0787 

. 000 0547 

. 000 0379 

. 000 0264 

. 000 0183

. 000 0104 

. 000 0088 

. 000 0061 

. 000 0043 

. 000 0030

«=6

0. Ill 142 

. 083 983 

. 063 071 

.047 137 

. 035 093

. 026 049 

. 019 290 

. 014 260 

. 010 526 

. 007 761

.005 718 

. 004 210 

. 003 098 

. 002 279 

.001 676

. 001 232 

. 000 906 

. 000 665 

. 000 489 

. 000 360

. 000 264 

.000 194 

. 000 143 

.000 105 

. 000 077

. 000 057 

. 000 042 

. 000 031 

. 000 022 

. 000 016
NUMERICAL EVALUATION OF THE MODIFIED POTENTIAL 53

TABLE 29. Local coefficients Cn ufor Q = 0.2

Terminal digit of the coefficient is in ninth decimal place.

Sign is that of (-1) »+»->.

For «>22, the value of Cnu is the same as that shown in table 24.

u

!-_-_____ _ ___ __ __ __ __

2_______ ________

3_.__ _

4 _

5_ _

6

7___ _

8_

9

10_._____ ______

ll__-_______-________ _ ________

12_._________________ _ ___ _ __

13_______________________________

14___ _

15___________ ____________________

16_____________________ _______

17______.________________________

18_______________________________

19-______________________________

20______________________ ________

21 __

22______________________________.

Values of (7n« where 

n=l

9763 6797 

7454 3045 

6240 3458 

5466 6247 

4921 3952

4511 6088 

4189 4275 

3927 6063 

3709 4102 

3523 9407

3363 7618 

3223 6052 

3099 6203 

2988 9196 

2889 2890

2798 9987 

2716 6752 

2641 2120 

2571 7064 

2507 4138

2447 7134 

2392 0836

n=2

9452 814 

7311 275 

6178 716 

5440 715 

4910 589

4507 104 

4187 545 

3926 816 

3709 078 

3523 800

3363 702 

3223 605 

3099 620 

2988 915 

2889 287

2798 999 

2716 675 

2641 212 

2571 706 

2507 414

2447 713 

2392 084

n=3

1231 168 

955 420 

811 970 

718 500 

650 799

598 736 

557 111 

522 898 

494 171 

469 636

448 382 

429 749 

413 248 

398 504 

385 229

373 194 

362 220 

352 162 

342 893 

334 321

326 362 

318 944

n=4

181 627 

140 893 

120 031 

106 547 

96 791

89 260 

83 207 

78 201 

73 975 

70 350

67 196 

64 423 

61 962 

59 760 

57 775

55 973 

54 332 

52 821 

51 432 

50 147

48 953

47 842

n=5

28 708 

22 233 

18 954 

16 854 

15 341

14 175 

13 235 

12 456 

11 796 

11 228

10 732 

10 294 

9 905 

9 555 

9 240

8 953 

8 690 

8 450 

8 228 

8 023

7 832 

7 655

n=6

4740 

3664 

3123 

2780 

2533

2343 

2191 

2065 

1957 

1864

1784 

1712 

1648 

1590 

1538

1491 

1448 

1408 

1371 

1337

1305 

1276

TABLE 30. Local ratios X nu /or Q= 0.2

u

1__ _ ...

2____. ___._.._. __ ._._.__. __ .__

3 _ .

4 ___ _____

5 _ ___ _ __--_--______.__ _ ____

6__ _ _

7 _

8________________________________

9. __ _ _ _ __ __ _

10______. _._._________.._ ________

ll___-___________________________

12_______________________________

13________________________.______

14_______________________________

15 -_-__--__-_______-_._---____

16

17_______ _________

18 ____

19 __ _

20_______________________________

21

22 __

23_ __ __ __ __

24_ _ _ __

25__-__-____-_-______-___________

26_____________ __________________

27_______________________________

28_________. _____________________

29_.____________

30_ _ _ _ _

n=l

0. 763 472 871

. 837 146 609

. 876 013 113

. 900 262 122

. 916 733 682

. 928 588 411

. 937 504 289

. 944 445 525

. 950 000 272

. 954 545 523

. 958 333 350

. 961 538 467

. 964 285 715

. 966 666 667

. 968 750 000

. 970 588 235

. 972 222 222

. 973 684 211

. 975 000 000

. 976 190 476

. 977 272 727

. 978 260 870

. 979 166 667

. 980 000 000

. 980 769 231

. 981 481 481

. 982 142 857

. 982 758 621

. 983 333 333

. 983 870 968

n=2

0. 773 449 562

. 845 094 179

. 880 557 600

. 902 563 158

. 917 833 677

. 929 098 745

. 937 737 137

. 944 550 759

. 950 047 566

. 954 566 705

. 958 342 818

. 961 542 692

. 964 287 599

. 966 667 508

. 968 750 374

. 970 588 402

. 972 222 296

. 973 684 244

. 975 000 015

. 976 190 483

. 977 272 730

. 978 260 871

. 979 166 668

. 980 000 000

. 980 769 231

. 981 481 481

. 982 142 857

. 982 758 621

. 983 333 333

. 983 870 968

Values of x». where-

^3

0. 7760 2757

. 8498 5655

. 8848 8440

. 9057 7522

. 9199 9941

. 9304 8029

. 9385 8799

. 9450 6251

. 9503 5029

. 9547 4366

. 9584 4535

. 9616 0172

. 9643 2140

. 9666 8681

. 9687 6134

. 9705 9464

. 9722 2583

. 9736 8624

. 9750 0114

. 9761 9112

. 9772 7309

. 9782 6108

.9791 6679

. 9800 0007

. 9807 6926

. 9814 8150

. 9821 4287

. 9827 5863

. 9833 3334

. 9838 7097

ra=4

0. 7757 295

. 8519 301

. 8876 637

. 9084 244

. 9222 031

. 9321 832

. 9398 448

. 9459 615

. 9509 792

.9551 763

. 9587 392

. 9617 994

. 9644 533

. 9667 742

. 9688 190

. 9706 325

. 9722 506

. 9737 024

. 9750 117

. 9761 980

. 9772 775

. 9782 639

.9791 686

. 9800 013

. 9807 700

. 9814 820

. 9821 432

. 9827 588

. 9833 335

. 9838 702

n=5

0. 7744 546

. 8525 205

. 8891 831

. 9102 436

. 9239 764

. 9337 500

. 9411 509

. 9470 093

. 9517 978

. 9558 032

.9592 119

.9621 516

. 9647 131

. 9669 645

. 9689 624

. 9707 316

. 9723 226

. 9737 541

. 9750 486

. 9762 244

. 9772 962

. 9784 252

.9791 780

. 9800 794

. 9807 769

. 9814 831

. 9821 456

. 9827 604

.9833 346

. 9338 701

n=6

0. 772 916

. 852 357

. 889 881

. 911 376

.925 259

. 935 015

. 942 308

. 948 022

. 952 654

. 956 510

. 959 787

. 962 612

. 965 079

. 967 251

.969 181

. 970 904

. 971 988

. 973 513

. 974 874

. 976 284

. 977 341

. 978 311

. 979 204

. 980 027

. 980 789

. 981 496

.982 154

. 982 768

. 983 339

.983 870
54 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

CAIjCtHLATIONS FOR Q=+0.3

For the reflection factor Q=+0.3, early values of 

Unu, Gnu, and \nu are shown in tables 31, 32, and 33, 

respectively. In calculating values of the modified 

potential on the basic range, the direct series values 

were checked by local expansions for a ^8, by 

asymptotic expansions from a=16 to a=30, and by 

asymptotic expansions of ( W) through B5/a? augmented by 6650/a10 from a=10 to a=16. The direct 

value for a=9 was checked only approximately.

For values of the relative displacement not on the 

basic range, values were obtained by local expansions 

for a<2.2, by asymptotic values for a>16, by augmented asymptotic expansion for 13<a<16, and 

by interpolation in the first order direct remainder for 

10<a<13.

TABLE 31. Values of p for Q= + 0.3

Values for j_nu where

1 __ _ ___________

2__ ____ _ ____ .

3 _---_________

4 __ ____ ____

5 -.. __ _______

6 --_-_-_.__-___

7__.______________

8 _ _ _____________

9 __________

10_ ____________

11_ ___ __ _ _ ___

12________________

13__-_-_._________

14__. _____ __ _

15___ _______ _._

16_ ______ _ ____

17________________

18 _ _ _________

19 ________________

20 _______ __

21 ________________

22__ ______ _____

23 _____________

24 _____ _______

25-. _____ _ _____

26 _________ __

27 _________ ._

28 _____________

29 __ _ _____ ___

30 _____________

n=l

0. 041 333 926

. 009 774 692

. 002 386 664

. 000 590 618

. 000 146 999

. 000 036 678

. 000 009 161

. 000 002 290

. 000 000 572

. 000 000 143

. 000 000 036

. 000 000 009

. 000 000 002

. 000 000 001

. 000 000 000

n=2

0. 1022 3805

. 0426 3397

. 0183 0979

. 0079 8699

.0035 1343

. 0015 5265

. 0006 8787

. 0003 0518

. 0001 3550

. 0000 6019

. 0000 2674

. 0000 1188

. 0000 0528

. 0000 0235

. 0000 0104

. 0000 0046

. 0000 0021

. 0000 0009

. 0000 0004

. 0000 0002

. 0000 0001

. 0000 0000

n=3

0. 1501 779

. 0791 724

. 0428 000

. 0234 895

.0130 110

. 0072 482

. 0040 522

. 0022 705

. 0012 740

.0007 155

. 0004 020

. 0002 261

.0001 270

. 0000 715

. 0000 401

. 0000 226

. 0000 127

. 0000 071

. 0000 040

. 0000 023

. 0000 013

. 0000 007

. 0000 004

. 0000 002

. 0000 001

. 0000 001

. 0000 000

n=4

0. 1865 849

. 1121 053

. 0687 922

. 0427 995

. 0268 719

. 0169 749

. 0107 672

. 0068 488

. 0043 647

. 0027 852

. 0017 790

. 0011 370

. 0007 270

. 0004 649

. 0002 974

. 0001 903

. 0001 219

. 0000 779

. 0000 498

. 0000 319

. 0000 204

. 0000 131

. 0000 084

. 0000 054

. 0000 034

. 0000 022

. 0000 014

. 0000 009

. 0000 006

. 0000 004

n=5

0. 214 783

. 140 394

. 093 423

. 062 937

. 042 766

. 029 237

. 020 075

. 013 827

. 009 545

. 006 599

. 004 568

.003 165

. 002 202

. 001 522

. 001 056

. 000 733

. 000 509

. 000 354

. 000 245

.000 170

.000 118

. 000 082

. 000 057

. 000 040

. 000 027

. 000 019

. 000 013

. 000 009

. 000 006

. 000 004

n=6

0. 23716

. 16449

. 11583

. 08247

. 05919

. 04273

. 03098

. 02253

. 01643

. 01200

. 00877

. 00643

. 00471

. 00345

. 00253

. 00186

. 00136

. 00100

. 00074

. 00054

. 00040

. 00029

. 00021

. 00016

. 00011

. 00008

. 00006

. 00005

. 00003

. 00002

n=7

0. 25530

. 18508

. 13593

. 10082

. 07532

. 05658

. 04270

. 03232

. 02452

. 01864

. 01420

. 01082

. 00825

. 00630

. 00482

. 00368

. 00282

. 00216

. 00166

. 00126

. 00096

. 00074

. 00056

. 00043

. 00033

. 00025

. 00019

. 00015

. 00011

. 00009

n=8

0. 2703

.2027

. 1539

. 1178

.0907

.0703

.0547

.0427

.0334

.0261

.0205

.0161

.0127

.0100

.0079

.0062

.0049

.0039

.0031

.0024

.0019

.0015

.0012

.0009

.0007

.0006

.0005

.0004

.0003

.0002
NUMERICAL EVALUATION OP THE MODIFIED POTENTIAL 55

TABLE 32. Local coefficients Cnu /or Q= + 0.3 

Terminal digit of the coefficient is In ninth decimal place. Sign is that of (-1)

u

1 ......

2

3. _ 

4 ......

5......

6

7... ...

8

9

10 .....

11 .....

12..-.

13 

14 

15 .....

16. ....

17 .....

18 

19 _ 

20 .....

21 .....

22 .....

23 

24.. ...

25. ....

26 

27 _ 

28 -

29 

30 

n=l

156200089

11359 9653

9397 3750

82079699

73838978

6767 8263

62842373

5891 4320

5564 1206

52859124

50456431

48354078

46494305

4483 3794

43339334

41984980

4075 0128

3961 8180

3857 5596

3761 1206

3671 5701

3588 1254

3510 1226

3436 9951

3368 2552

3303 4810

32423055

3184 4072

3129 5036

3077 3452

7i=2

frSOV 94AO (Y3KR UOUU

17494448

1431 9981

12402965

1111 3127

1016 7129

9432754

8839825

834 7307

7929344

756 8666

725 3198

697 4183

672 5085

6500907

6297750

611 2520

594 2728

5786339

564 1681

550 7355

538 2188

526 5184

515 5493

505 2383

495 5222

4863458

477 6611

469 4255

461 6018

n=3

5175 801

3642207

2932875

2518 744

2243 661

9f_U fflfrt QfiQ 9O<7

1892893

1771 439

1671 362

1586908

1514 302

1450 950

1395006

1345 110

1300 232

1259 577

1222 520

1188 553

1157 273

1128 339

1101 472

1076 439

1053 037

1031 099

1010 477

991 044

972 692

955322

938 851

923 204

n=4

1201 417

844 505

676 345

577 410

511 731

464 566

428 749

Ann QQR ftm/ OtfO

377 217

357 793

341 187

326 761

314065

302 769

292 628

283453

275097

267 444

260398

253884

247836

242 201

236 935

231 998

227358

222 985

218 856

214 947

211 241

207 721

7i=5

295 192

207 837

166 064

141 255

124 717

119 114 ft-lfl o*±U

1AQ QAO J.UO Crto

96 761

on yu QQO WWW

86 197

82 112

78582

75 487

72 742

70284

68 066

66049

64 204

62 507

60940

59486

58 133

56867

55 681

54 567

53 517

52 526

51 587

50 698

49 853

71 = 6

75159

53056

.irtOftTr 4iOD(

35963

31670

OQRQA J-OQOU

26239

O_IQQQ tuto&a

22905

21665

20614

19709

18919

18221

17596

17036

16526

16061

15635

15241

14876

14536

14219

13922

13643

13380

13132

12898

12674

12463

7i=7

IQ Ann XtfUW

IQfifiJ. looorc iinon XXUoU

Q4/14 mtrt

S9ft» OAvO

7447

6824

fiooo DOOO

5937

5608

5330

5090

4882

4698

4535

4388

4256

4135

4024

3922

3828

3740

3658

3581

3509

3441

3378

3316

3259

3205

7i=8

KQflQ a_Ua

3698

OQCfi &0OO

oertA ^soUo

2201

1QQA loOv

1819 iOifi

1679

1 KTO lo/_

1483

1407

1343

1287

1237

1194

1155

1119

1087

1057

1030

1006

982

960

940

922

904

886

870

855

841

CALCULATIONS FOR Q = -0.3

For the reflection factor Q ~ 0.3, early values of 

Una, Gnu, and XnM are shown in tables 34, 35, and 36, 

respectively. On the basic range, direct values of the 

modified potential were checked by local expansions for 

a ^6.9 and by asymptotic expansions for a s.6.4. For 

intermediate values, local expansions were used for 

a<2.2 and asymptotic expansions were used for a>10.

CALCULATIONS FOB 0.4<Q<0.9

For reflection factors Q, where 0.9^<2=5 0.4 or 

0.4 _g Q 5j= 0.9, local expansion involve so many terms that 

other methods were developed, and special devices were 

used. For the sake of conciseness, these 12 cases are 

discussed in a condensed notation. For each case, the 

values of Won the basic range were calculated by direct 

series, and the summary omits such values.

TABLE 33. Local ratios Xnu /or Q = +0.3

u

1 _ _ ._ __

2

3____ ___.-- ____ .

4__._.___ _ ._. _

5 ____ --- ____

6__ ___ . _ ______

7 -_----__-____

8 ______ _ _ -_-.

9 -_-_-_-._-__-_

lO-_. ____________

11_____ _ .___ __

12___. ____________

13________________

14. _______ _ _ _

15 _____________

16.___________.___

17________________

18 __--_._______

19-________._____

20___- ____________

21____ _____ _ _

22 ________________

23 ______ _ _____

24_ _______________

25 _____________

26___ _ ____________

27 _____________

28 _____________

29. ___ _ _________

30 _____________

Values for Xn o where 

7i=l

0. 727 270 091 

. 827 236 241 

. 873 432 201 

. 899 600 979 

. 916 565 554

. 928 545 878 

. 937 493 558 

. 944 442 821 

. 949 999 592 

. 954 545 353

. 958 333 307 

. 961 538 455 

. 964 285 713 

. 966 666 666 

. 968 750 000

. 970 588 235 

. 972 222 222 

. 973 684 211 

. 975 000 000 

. 976 190 476

. 977 272 727 

. 978 260 870 

. 979 166 667 

. 980 000 000 

. 980 769 231

. 981 481 481 

. 982 142 857 

. 982 758 621 

. 983 333 333 

. 983 870 968

7i=2

0. 7094 4337 

. 8138 9204 

. 8661 2996 

. 8960 0570 

. 9148 7558

. 9277 6966 

.9371 4147 

. 9442 8423 

. 9499 2846 

.9545 1352

.9583 1909 

. 9615 3211 

. 9642 8288 

. 9666 6540 

. 9687 4944

. 9705 8800 

. 9722 2210 

. 9736 8416 

. 9749 9998 

. 9761 9047

. 9772 7272 

. 9782 6087 

. 9791 6667 

. 9800 0000 

. 9807 6923

. 9814 8148 

.9821 4286 

. 9827 5862 

. 9833 3333 

. 9838 7097

7i=3

0. 7036 992 

. 8052 466 

. 8587 968 

. 8907 858 

. 9114 520

. 9256 250 

. 9358 364 

. 9435 054 

. 9494 701 

. 9542 464

. 9581 648 

. 9614 432 

. 9642 322 

. 9666 363 

. 9687 330

. 9705 786 

. 9722 168 

. 9736 812 

. 9749 983 

. 9761 895

. 9772 721 

. 9782 606 

.9791 665 

. 9799 999 

. 9807 692

. 9814 814 

.9821 429 

. 9827 586 

. 9833 333 

. 9838 710

n=4

0. 7029 240 

. 8008 775 

. 8537 203 

. 8862 535 

. 9078 318

. 9229 034 

. 9338 656 

. 9421 143 

. 9485 060 

. 9535 877

.9577 192 

. 9611 447 

. 9640 332 

. 9665 048 

. 9686 463

. 9705 219 

. 9721 794 

. 9736 569 

. 9749 825 

. 9761 792

. 9772 656 

. 9782 563 

.9791 637 

. 9799 980 

. 9807 681

. 9814 807 

. 9821 424 

. 9827 583 

. 9833 331 

. 9838 712

n=5

0. 704 073 

. 799 010 

. 850 604 

. 882 921 

. 904 773

. 920 305 

.931 758 

. 940 455 

. 947 228 

. 952 619

. 956 994 

. 960 615 

. 963 631 

. 966 217 

. 968 437

. 970 371 

. 972 071 

. 973 578 

. 974 927 

. 976 139

. 977 238 

. 978 237 

. 979 150 

. 979 987 

. 980 761

. 981 475 

. 982 139 

. 982 756 

. 983 331 

. 983 872

71=6

0. 70595 

. 79851 

. 84884 

. 88064 

. 90242

. 91811 

. 92981 

. 93880 

. 94586 

. 95150

. 95611 

. 95990 

. 96308 

. 96578 

. 96810

. 97010 

. 97187 

. 97343 

. 97480 

. 97605

. 97716 

. 97818 

. 97912 

. 97995 

. 98074

. 98146 

. 98213 

. 98274 

. 98332 

. 98389

w=7

0. 70805

. 79877 

. 84795 

. 87916 

. 90069

. 91637 

. 92817 

. 93730 

. 94455 

. 95039

. 95514 

. 95910 

. 96243 

. 96525 

. 96766

. 96976 

. 97158 

. 97319 

. 97460 

. 97591

. 97706 

. 97808 

. 97904 

. 97990 

. 98069

. 98142 

. 98210 

. 98272 

. 98331 

. 98389

71=8

0. 7101 

.7995 

.8476 

.8782 

.8996

.9150 

.9268 

.9360 

.9433 

.9493

.9542 

.9583 

.9617 

.9647 

.9672

.9693 

.9712 

.9729 

. 9743 

.9757

.9769 

.9780 

.9789 

.9798 

.9807

.9814 

.9820 

.9827 

.9832 

.9839
56 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN 

TABLE 34. Values of (-/*n u) for Q= 0.3

u

!__-.___. _________________

2_-_-__-_____-____________

3___ --.__ _____

4____ _ __________________

5_--_______________-______

6 __

7

8_---_--__-___--____-__-__

9_-_____-________.________

10_--___---____-_____-___-

!!___ __ ___ _ _________

12_______________.____.___

13________-____-____._____

14__.___________. _________

15-_____-.________________

16 __ _____________________

17_______.________________

18__ __ ___ ___ ____

19____--_-________________

Values for ( /* _) where 

n=l

0. 034 533 220 

. 009 028 680 

. 002 304 150 

. 000 581 464 

.000 145 981

. 000 036 564 

. 000 009 149 

. 000 002 288 

. 000 000 572 

. 000 000 143

. 000 000 036 

. 000 000 009 

. 000 000 002 

. 000 000 001 

. 000 000 000

. 000 000 000 

. 000 000 000 

. 000 000 000 

. 000 000 000

n=2

0.0791 1414 

. 0369 4084 

. 0168 9604 

. 0076 3460 

. 0034 2545

. 0015 3066 

. 0006 8238 

.0003 0381 

.0001 3516 

. 0000 6010

. 0000 2673 

.0000 1188 

. 0000 0528 

. 0000 0235 

. 0000 0104

. 0000 0046 

. 0000 0021 

. 0000 0009 

. 0000 0004

w=3

0. 1099 658 

. 0649 351 

. 0377 174 

. 0216 675 

. 0123 565

.0070 128 

. 0039 675 

. 0022 400 

. 0012 630 

.0007 116

. 0004 007 

. 0002 256 

.0001 269 

. 0000 714 

. 0000 401

.0000 226 

.0000 127 

.0000 071 

. 0000 040

71 = 4

0. 1311 404 

. 0878 799 

.0581 282 

. 0380 853 

. 0247 828

. 0160 480 

.0103 557 

.0066 660 

. 0042 835 

. 0027 492

. 0017 629 

.0011 298 

. 0007 238 

. 0004 635 

. 0002 968

.0001 900 

.0001 216 

. 0000 778 

. 0000 498

n=5

0. 146 263 

. 106 039 

. 076 077 

.054 142 

. 038 296

. 026 961 

. 018 916 

.013 235 

. 009 243 

. 006 445

. 004 490 

. 003 125 

. 002 173 

. 001 511 

. 001 051

. 000 730 

. 000 507 

. 000 352 

. 000 245

«=6

0. 15752 

. 12047 

. 09135 

. 06879 

. 05153

. 03843 

. 02856 

. 02118 

. 01567 

. 01157

. 00853 

. 00629 

. 00464 

. 00342 

. 00251

. 00184 

. 00136 

. 00100 

. 00073

W=7

0. 16620 

. 13208 

. 10424 

. 08179 

. 06387

. 04968 

. 03853 

. 02980 

. 02300 

. 01773

. 01364 

. 01049 

. 00806 

. 00618 

. 00474

. 00363 

. 00279 

. 00214 

. 00164

For M>19, the absolute value of /*»- is the same as that shown In table 31.

TABLE 35. Local coefficients Cnu /or Q= 0.3 

Terminal digit of the coefficient is in ninth decimal place. Sign is that of ( i)-+«-i

u

! .. _ ..

2. ........

3. -

4.. _ ....

6 __ -

6 

7 

8... __ .

9_ 

10 

11_ _ .

12.... _ .

13.. __ -

14.... ....

15 

16. 

17... .....

18.. _ ...

19 

20 

21 

22

23. _ -

24 ...

25 ........

26 

27 ____ .

28 ___ .. 

29. .......

30 

Values of CB « where 

n=l

144 820 017 

111 484 274 

93 533 986 

81 983 552 

73 817 347

67 673 307 

62 841 222 

58 914 050 

55 641 142 

52 859 108

50 456 427 

48 354 077 

46 494 305 

44 833 794 

43 339 334

41 984 980 

40 750 128 

39 618 180 

38 575 596 

37 611 206

36 715 701 

35 881 254 

35 101 226 

34 369 951 

33 682 552

33 034 810 

32 423 055 

31 844 072 

31 295 036 

30 773 452

n=1

2071 9932 

1625 1623 

1382 4899 

1221 0747 

1103 6285

1013 5829 

941 9838 

883 4443 

834 5048 

792 8390

756 8262 

725 3026 

697 4109 

672 5053 

650 0893

629 7744 

fill 2518 

594 2726 

578 6339 

564 1681

550 7355 

538 2188 

526 5184 

515 5493 

505 2383

495 5222 

486 3458 

477 6611 

469 4255 

461 6018

w=3

4005 154 

3155 844 

2706 420 

2407 616 

2187 476

2016 035 

1877 774 

1763 467 

1667 127 

1584 645

1513 086 

1450 295 

1394 652 

1344 918 

1300 128

1259 521 

1222 488 

1188 537 

1157 263 

1128 333

1101 470 

1076 437 

1053 037 

1031 099 

1010 477

991 044 

972 692 

955 322 

938 851 

923 204

n=4

879 720 

692 641 

596 028 

532 623 

485 990

449 481 

419 789 

395 020 

373 969 

355 818

339 981 

326 021 

313 610 

302 488 

292 454

283 345 

275 030 

267 402 

260 372 

253 868

247 826 

242 195 

236 931 

231 996 

227 356

222 985 

218 856 

214 947 

211 241 

207 721

n=5

207 458 

162 924 

140 321 

125 696 

115 022

106 679 

99 878 

94 178 

89 306 

85080

81 372 

78 089 

75 157 

72 521 

70 136

67966 

65 982 

64 153 

62 477 

60 920

59 472 

58 123 

56 861 

55 677 

54 565

53 515 

52 524 

51 687 

5069S 

49 853

w=6

51181 

40073 

34501 

30938 

28359

26*56 

24724 

23355 

22182 

21160

20261 

19460 

18743 

18096 

17508

16973 

16482 

16029 

15612 

15225

14864 

14528 

14213 

13918 

13639

13378 

13130 

12896 

12674 

12463

n=1

13025 

10169 

8745 

7844 

7198

6981 

6293 

5952 

5662 

5407

5183 

4983 

4803 

4640 

4492

4356 

4232 

4117 

4010 

3912

3820 

3734 

3654 

3577 

3507

3439 

3376 

3316 

3259 

3205

As a matter of convenience, certain expressions are 

adopted:

W[a(K)p] means that the function W(Q,a) for the specified value of Q was calculated and tabulated by 

steps of h from a=a to a=j8.

W[y(k)$\ were interpolated in W[a(h)p] means that the 

values of Won the range j^a^d were calculated 

for steps of k by interpolation in the values on the 

basic range, in which the step is h.

W[y(k)S] were interpolated in Rn[a(K)fi] means that:

1. The values of the remainder of order n, (retaining the first n terms of the series) are calculated, tabulated, and differenced on the range 

(a, j8) as far as needed to justify the interpolation.

2. The remainder table is smoothed by correction 

of errors and adjustment.

3. The range (a, /3) is extended at each end so that 

the differences will be complete over the 

range (a, 0).

4. The value of Rn is calculated by interpolation 

for each value on the range a=[y(k)8] from 

the smoothed table for the range a [a(ti)&\.
NUMERICAL EVALUATION OF THE MODIFIED POTENTIAL 

TABLE 36. Local ratios Xnu /or Q= 0.3

57

u

1- __ __-_____--_______-_

2_ ____ ___ ___. _ _ ___

3. ________________________

4_.____ _ __ ___ _ _ __

5___

6 _

7. ________________________

8---____________-_________

9_________________________

10_

11 __ ____________________

12________________________

13__ _

14________________________

15_-____________________._

16_-___.__________________

17________________________

18---__-_____-________--__

19_______________________.

20________________________

21________________________

22 .

23--_-_-_-___-_-___-_--__.

24______________________._

25___________________-_._-

26---___________________._

27________________________

28_-___________.__________

29____-___-___-______.____

30_.______________________

Values for Xn» where 

n=l

0. 769 812 598 

. 838 988 164 

. 876 510 744 

. 900 392 252 

. 916 766 981

. 928 596 887 

. 937 506 432 

. 944 446 065 

. 950 000 408 

. 954 545 865

. 958 333 359 

. 961 538 469 

. 964 285 715 

. 966 666 668 

. 968 750 000

. 970 588 235 

. 972 222 222 

. 973 684 211 

. 975 000 000 

. 976 190 476

. 977 272 727 

. 978 260 870 

. 979 166 667 

. 980 000 000 

. 980 769 231

. 981 481 481 

. 982 142 857 

. 982 758 621 

. 983 333 333 

. 983 870 968

n=2

0. 7843 4734 

. 8506 7806 

. 8832 4303 

. 9038 1738 

. 9184 0953

. 9293 6033 

. 9378 5515 

. 9446 0377 

. 9500 7132 

. 9545 7731

. 9583 4756 

. 9615 4481 

. 9642 8854 

. 9666 6793 

. 9687 5056

. 9705 8848 

. 9722 2234 

. 9736 8426 

. 9750 0002 

.9761 9049

. 9772 7274 

. 9782 6087 

. 9791 6667 

. 9800 0000 

. 9807 6923

. 9814 8148 

. 9821 4286 

. 9827 5862 

. 9833 3333 

. 9838 7097

n=3

0. 7879 457 

. 8575 898 

. 8895 941 

. 9085 655 

. 9216 264

. 9314 191 

. 9391 260 

. 9453 692 

. 9505 245 

. 9548 425

. 9585 012 

. 9616 334 

. 9643 392 

. 9666 970 

. 9687 670

.9705 978 

. 9722 276 

. 9736 872 

. 9750 017 

.9761 915

. 9772 733 

. 9782 612 

. 9791 669 

. 9800 001 

. 9807 692

. 9814 815 

. 9821 429 

. 9827 586 

. 9833 333 

. 9838 710

n=4

0. 7873 425 

.8605 152 

.8936 199 

. 9124 463 

. 9248 770

. 9339 433 

. 9409 953 

. 9467 097 

.9514 638 

. 9554 895

. 9589 411 

. 9619 293 

. 9645 369 

. 9668 279 

. 9688 535

. 9706 546 

. 9722 648 

.9737 115 

. 9750 175 

. 9762 017

. 9772 799 

. 9782 655 

. 9791 696 

. 9800 020 

. 9807 704

. 9814 823 

. 9821 433 

. 9827 589 

. 9833 335 

. 9838 708

n=5

0. 785 336 

.861 263 

. 895 773 

. 915 078 

. 927 471

. 936 248 

. 942 929 

. 948 265 

. 952 683 

. 956 424

. 959 647 

. 962 456 

. 964 925 

. 967 113 

. 969 061

. 970 804 

. 972 373 

. 973 788 

. 975 073 

. 976 241

. 977 308 

. 978 285 

. 979 184 

. 980 013 

. 980 777

. 981 487 

.982 147 

. 982 762 

. 983 335 

. 983 870

n=6

0. 78298 

. 86092 

. 89673 

. 91669 

. 92933

. 93810 

. 94462 

. 94976 

. 95396 

. 95749

. 96050 

. 96314 

. 96548 

. 96755 

. 96940

. 97106 

. 97257 

. 97394 

. 97519 

. 97633

. 97738 

. 97834 

. 97922 

. 98005 

. 98080

. 98150 

. 98215 

. 98278 

. 98334 

. 98387

n=7

0. 78069 

. 86006 

. 89693 

. 91757 

. 93057

. 93946 

. 94601 

. 95106 

. 95512 

. 95852

. 96139 

. 96391 

. 96612 

. 96807 

. 96984

. 97141 

. 97285 

. 97417 

. 97538 

. 97647

. 97748 

. 97844 

. 97930 

. 98010 

. 98085

. 98154 

. 98218 

. 98280 

.98335 

. 98385

5. The resulting values of Rn are tabulated, differenced, and smoothed, with necessary 

corrections and adjustments.

6. The value of W is calculated by evaluating the 

first n terms directly and adding the corresponding value of Rn .

The nth direct residue means the residue or remainder 

of the direct series after isolation of the first n 

terms.

The nth asymptotic residue means the residue of the 

asymptotic expansion, after isolation of the terms 

through BJa?"-1 .

Interpolation in the nth specified residue means that 

the specified residue is evaluated over the indicated 

range, that this residue is interpolated to complete 

the new range, and that the values of the isolated 

initial terms are added to the residues to obtain 

the values of the function over the new range. 

The expanded table is smoothed and corrected by 

differences.

With the foregoing condensations, the calculations of 

W(Q, a) may be summarized for each case. 

<2=+0.4:

1. The values of TF[0.00(0.05)2.50] were interpolated in the first direct residue #i[0.0(0.1)- 

2.5].

2. The values of W[0.00(0.02) 1.40] were interpolated in TF[0.00(0.05)2.50].

3. The values of TF[10.0(0.2)14.0] were interpolated in TF[10.0(0.5)14.0)].

4. The values of TF[15.0(0.5)20] were interpolated 

in the third asymptotic residue i?8 [15(l)20].

5. For a >20, asymptotic expansions are adequate. 

Q=-0.4:

1. The values of TFfO.OO(0.02) 1.30] were interpolated in the first local residue, Ri[Q.0(0.1) 1.3].

2. The values of TF[1.30(0.05)2.20] were interpolated in TF[1.3(0.1)2.2].

3. For a ^ 5.8, fifth order asymptotic expansions are 

adequate.
58 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

1. Values of IF[0.00(0.02)1.60] were interpolated 

in the first direct residue ^[0.0(0.1)1.6].

2. Values of IF[1.60(0.05)3.00] were interpolated 

in the first direct residue, ^[1.6(0.1)3.0].

3. Values of TF[15.5(1.0)29.5] were interpolated 

in the fourth asymptotic residue #4[15(1)30].

4. Values of W[W.l(0.1) 10.4(0.2) 15.8] were interpolated in -TF[10.0(0.5)16.0].

Q=-0.5:

1. Values of TF[0.00(0.02) 1.40(0.05)2.20] were interpolated in the first direct residue, 

^[0.0(0.1)2.2].

2. For a ^>6.4 asymptotic expansions are adequate. 

£=+0.6:

1. Values of TF[0.00(0.02)1.60(0.05j3.00] were 

interpolated in the first direct residue,

-i[0.0(0.1)3.4].

2. The value of W(15.5) was inserted between

-MlO.O(0.5)15] and TF[16.0(0.5)30.0] by 

smoothing.

3. Values of TF[10(0.2)16.0] were interpolated in

-TF[10.0(0.5)16.0].

4. Values of TF[16.0(0.5)30.0] were interpolated 

in the fourth asymptotic residue #4[16(1)30].

Q=-0.6:

1. Values of TF[0.00(0.02) 1.40(0.05)3.20] were interpolated in the first direct residue, ^[0.0 

(0.1)3.2].

2. For a ^6.7, asymptotic expansions are adequate.

Q=+0.7:

1. Values of -TF[0.0(0.02) 1.60(0.05)2.40] were interpolated in the first direct residue, jffJO.O 

(0.1)2.4],

2. Values of TF[15.0(0.5)30.0] were interpolated 

in -TF[15(1)30].

3. Values of TF[10.0(0.2)17.0] were interpolated 

in -TF[10.0(0.5)17.0].

Q=-0.7:

1. Values of W[0.00(0.02) 1.40(0.05) 3.20] were interpolated in the first direct residue, jff^O.O 

(0.1)3.2].

2. For a ^6.5, asymptotic expansions are adequate.

1. Values of - TF[0.00(0.02) 1.60(0.05)2.50] were 

interpolated in the first direct residue, #i[0.0 

(0.1)2.5].

2. Values of TF[15.0(0.5)30.0] were interpolated 

in the second direct residue, #2 [15(1)30].

3. Values of TF[10.0(0.02)18.0] were interpolated

in -TF[10.0(0.5)18.0]. 

Q=-0.8:

1. Values of W[0.00(0.02)1.50] were interpolated in 

the first direct residue, ^[0.0(0.1)1.5].

2. Values of W[1.50(0.05)3.40] were interpolated in 

the second direct residue, R2[1.5(Q. 1)3.4].

3. For a, ^6.6 asymptotic expansions are adequate. 

Q=+0.9:

1. Values of - TF[0.00(0.02) 1.60(0.05)2.4] were interpolated in the first direct residue, Ri[0.0 

(0.1)2.4].

2. Values of TF[15.0(0.5)33.0] were interpolated 

in the second direct residue, ^[12(1)33].

3. Values of TF[10.0(0.2)19.0] were interpolated

in -TF[10.0(0.5)19.0]. 

Q=-0.9:

1. Values of TF[0.00(0.02) 1.70(0.05)3.60] were interpolated in the first direct residue, ^[0.0 

(0.1)4.0].

2. For a, ^ 6.7, asymptotic expansions are adequate.

NUMERICAL VALUES OF THE MODIFIED 

POTENTIAL

The results of the analysis described in this report 

are summarized in tables 37 and 38. Table 37 (p. 59) 

shows the coefficients to be used for interpolating between arguments shown in table 38 (p. 59). Table 38 

shows values of the modified potential for reflection 

factors from 1 to +1 by steps of one-tenth (except 

for the trivial case of zero) and for relative displacements not exceeding 30, spaced so that fourth order 

interpolation is always adequate.

In the use of table 38, the following observations 

should be noted:

Values of db Ware complete, except for forcing errors 

of calculation. Differences are expressed in units of 

one in the eighth decimal place. Values have been 

rounded so that apparent inconsistencies occur, but 

the values of ± Ware usually within half a unit in the 

terminal digit and within one unit except in rare cases. 

The 2nd and 3rd differences are usually within 1 unit 

and the 4th difference within 2 units.
NUMERICAL VALUES OP THE MODIFIED POTENTIAL 59

TABLE 37. Difference coefficients for interpolation

V~xyt+(l-x)v<ftFtA*_l+F>&*.l+F^+...

Fv* x (1 z)/2 Ai,=yi 2{fo+IM

Fr* (l+x) x (1 z)/6 A!.I=VJ Syi+Syt y-i

Ji-O-W « tt-«) (2-*V24 A^-^+^JM+ZM

C

0.00

.01

.02

.03

.04

nf

^06

.07

.08

.09

.10

.11

.12 

.13

.14

.15 

.16

.17

.18 

.19

.20

.21 

.22 

.23

.24

.25 

.26

.27

.28 

.29

.30

.31 

.32 

.33

.34

.35 

.36

.37

.38 

.39

.40

.41 

.42 

.43

.44

.45 

.46

.47

.48 

.40

 Ft

O nnnnn . U'JvUU

.00495

.00980

.01455

.01920

.02375

.02820

.03255

.03680

.04095

.04500

.04895 

.05280 

.05655

.06020

. 06375 

.06720

.07055

.07380 

. 07695

.08000

.08295 

.08580 

.08855

.09120

.09375 

.09620

.09«55

.10080 

. 10295

.10500

. 10695 

.10880 

. 11055

.11220

. 11375 

.11520

.11655

.11780 

. 11895

.12000

.12095 

.12180 

. 12255

.12320

. 12375 

.12420

. 12455

.12480 

. 12495

-Ft

t\ U. AAAAfi UUUuU

.00167

.00333

.00500

.00666

nAQOI

. UUool

.00996

.01161

.01325

.01488

.01650

.01811 

. 01971 

.02130

.02288

.02444 

.02598

.02751

.02903 

.03052

. 03200

.03346 

.03489 

.03631

. 03770

.03906 

.04040

.04172

.04301 

.04427

.04550

.04670 

.04787 

.04901

.05012

. 05119 

.05222

.05322

.05419 

.05511

.05800

.05685 

. 05765 

.05842

. 05914

. 05981 

.06044

. 06103

. 06157 

.06206

~\~Ft

A u. AnAn uuuu 

.0008

.0016

.0025

.0033

nrui . UKrti.

.0048

.0056

.0064

.0071

.0078

.0086 

.0093 

.0100

.0106

.0113 

.0120

.0126

.0132 

.0138

.0144

.0150 

.0155 

.0161

.0166

.0171 

.0176

.0180

.0185 

.0189

.0193

.0197 

.0201 

.0205

.0208

.0211 

.0214

.0217

.0219 

.0222

.0224

.0226 

.0228 

.0229

.0231

.0232 

.0233

.0233

.0234 

.0234

X

0. 50

!52

.53

.54

.55

.56

.57

.58

.59

.60

.61 

.62 

.63

.64

.65 

.66

.67

.68 

.69

.70

.71 

.72 

.73

.74

.75 

.76

.77

.78 

.79

.80

.81 

.82 

.83

.84

.85 

.86

.87

.88 

.89

.90

.91 

.92 

.93

.94

.95 

.96

.97

.98 

.99

-Ft

'. 12495

.12480

. 12455

.12420

.12375

.12320

.12255

.12180

.12095

.12000

. 11895 

.11780 

. 11655

.11520

. 11375 

.11220

. 11055

.10880 

.10695

.10500

.10295 

.10080 

.09855

. 09620

. 09375 

.09120

.08855

.08580 

.08295

.08000

. 07695 

.07380 

. 07055

.06720

.06375 

.06020

.05655

.05280 

.04895

.04500

.04095 

.03680 

. 03255

.02820

. 02375 

.01920

. 01455

.00980 

. 00495

 Ft

06250

! 06289

. 06323

.06352

.06376

fU*QQ4

! 06406

.06413

. 06415

. 06410

.06400

.06384 

.06361 

.06333

.06298

.06256 

.06208

.06154

.06093 

.06025

.05950

.05868 

. 05779 

.05683

.05580

.05469 

.05350

.05224

.05091 

.04949

.04800

.04643 

.04477 

.04304

.04122

.03931 

. 03732

.03525

.03309 

.03084

.02850

. 02607 

.02355 

.02094

.01824

.01544 

.01254

.00955

.00647 

.00328

-\-Fi

00234

!0234

.0234

.0233

. 0233

nooo

!0231

.0229

.0228

.0226

.0224

.0222 

.0219 

.0217

.0214

.0211 

.0208

.0205

.0201 

.0197

.0193

.0189 

.0185 

.0180

.0176

.0171 

.0166

.0161

.0155 

.0150

.0144

.0138 

.0132 

.0126

.0120

.0113 

.0106

.0100

.0093 

.0086

.0078

.0071 

.0064 

.0056

.0048

.0041 

.0033

.0025

.0016 

.0008

For interpolation, use

Jf \ rf(-\\-\-(\ }-f(m-[-FA2 -I- FA3 4- FA4 J(X) xj{i)-\- ^i X)j\y)-\-£%i± i-p/'sA i -\-r 4i\ 2

with the line selected so that 0<jB<O- Except for

forcing errors, sufficient accuracy is obtainable by:

linear interpolation if|Ala £43,|ALl j£16,

J 1.2 aiiQ|ii_] <- Q

^o.

quadratic interpolation if|Al2 ^43,|Ali £16,

but|Al!|>8.

cubic interpolation if |Ala | ^43 butlAli|>16.

quartic interpolation if|Al2 |^>43.

For two additional digits in the argument, beyond

those tabulated, the values of Ft are shown in table 37.

For more than two additional digits, FK may be calculated from the formulas shown in table 37, or taken

from more extensive tables of Lagrangean interpolants.

For some purposes linear interpolation in table 37 is

adequate.

TABLE 38. Modified potential

a w A», A_j A*

-a

Q = 1.0; tabular spacing =0.02

0.00

.02

.04

.06 

.08

.10 

. 12

. 14

. 16 

.18

.20 

.22

.24

.26 

.28

.30 

.32

.34

.36 

.38

.40 

.42

.44

.46 

.48

.50 

.52

.54

.56 

.58

.60 

.62

.64

.66

.68

.70

. 72

.74

.76

.78

on 

. ou

.82

.84

.86

.88

.90

.92

.94 

.96

.98

0. 0000 0000

.0001 8025

. 0007 2030

.0016 1807 

. 0028 7008

. 0044 7157 

.0064 1643

.0086 9737

.0113 0593 

. 0142 3257

. 0174 6676 

. 0209 9710

.0248 1138

. 0288 9671 

.0332 3962

. 0378 2617 

.0426 4204

.0476 7265

. 0529 0326 

.0583 1905

. 0639 0522 

. 0696 4707

. 0755 3007

. 0815 3995 

. 0876 6275

. 0938 8485 

. 1001 9307

. 1065 7467

. 1130 1740 

. 1195 0953

. 1260 3984 

. 1325 9769

. 1391 7298

. 1457 5618

. 1523 3834

. 1589 1103

. 1654 6643

. 1719 9722

. 1784 9664

. 1849 5843

. 1913 7684

. 1977 4660

. 2040 6291

. 2103 2141

.2165 1816

. 2226 4963

. 2287 1266

. 2347 0448 

. 2406 2263

. 2464 6498

+ 3 6050

3 5980

3 5771

3 5425 

3 4946

3 4338 

3 3608

3 2762

3 1808 

3 0756

2 9614 

2 8394

2 7105

2 5758 

2 4364

2 2932 

2 1474

2 0000

1 8518 

1 7038

1 5568 

1 4115

1 2688

1 1291 

9931

8612 

7338

6113

4939 

3819

2754 

1744

+ 791 -105

-945

-1730

-2460

-3138

-3763

-4338

-4865

-5345

-5781

-6175

-6528

-6843

-7122

-7367 

-7580

-7763

-70

-209

-346

-479 -608

-730 -846

-954

-1052 

-1141

-1220 

-1289

-1347

-1395 

-1432

-1458 

-1475

-1482

-1480 

-1470

-1452 

-1428

-1397

-1360 

-1319

-1274 

-1225

-1174

-1120 

-1066

-1010 

-953

 896 -840

-785

-730

-677

-625

-575

-527

-480

-436

-394

-354

-315

-279

-245

-213 

-183

-155

-140

-139

-137

-133 -129

-122 -116

-108

-98 

-89

-79 

-69

-58

-48 

-37

-26 

-17

-7

+ 2 

10

18 

25

31

36 

41

45 

49

52

53 

54

56 

57

KT 07

56

55

55

53

52

50

48

46

45

42

40

39

36

34

32 

30

28
60 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

TABLE 38. Modified potential Continued

a w Ai, Ai, Ai2

Q = 1.0; tabular spacing=0.02 Continued

1.00

1.02

1.04

1.06

1.08

1. 10

1. 12

1. 14

1. 16

1.18

1.20

1.22

1.24 

1.26

1.28

1.30 

1. 32 

1.34 

1. 36 

1.38

1.40

1.42

1.44

1.46

1.48

1.50

1.52

1.54

1. 56

1.58

1.60

1.62 

1.64 

1.66 

1.68

1.70

1.72 

1.74 

1.76 

1. 78

1. 80

1. 82

1. 84

1.86

1.88

1. 90

1.92

1.94

1 96

1.98

2.00

2.02

2.04 

2.06 

2.08

2. 10 

2. 12

2. 14

2. 16

2. 18

0. 2522 2971

. 2579 1526

. 2635 2035

. 2690 4394

. 2744 8522

. 2798 4357

.2851 1859

.2903 1005

. 2954 1787

. 3004 4212

. 3053 8304

. 3102 4096

. 3150 1633 

. 3197 0972

. 3243 2178

. 3288 5323 

. 3333 0489 

. 3376 7763 

. 3419 7238 

. 3461 9013

. 3503 3190

. 3543 9876

. 3583 9180

. 3623 1216

. 3661 6099

. 3699 3945

. 3736 4873

. 3772 9001

. 3808 6450

. 3843 7341

. 3878 1794

. 3911 9930 

. 3945 1869 

. 3977 7730 

. 4009 7633

.4041 1696

. 4072 0035 

. 4102 2767 

. 4132 0006 

.4161 1866

. 4189 8457

. 4217 9892

. 4245 6277

. 4272 7719

. 4299 4325

. 4325 6196

. 4351 3436

. 4376 6142

.4401 4414

. 4425 8347

. 4449 8035

. 4473 3571

. 4496 5045 

. 4519 2546 

.4541 6159

. 4563 5971 

. 4585 2063

. 4606 4518

. 4627 3414

. 4647 8829

-7918

-8046

-8150

-8232

-8292

-8333

-8357

-8364

-8356

-8334

-8300

-8254

-8198 

-8133

-8060

-7979 

-7892 

-7799 

-7701

-7598

-7491

-7381

-7268

-7153

-7037

-6918

-6799

-6679

-6558

-6438

-6317

-6197 

-6077 

-5958 

-5840

-5723

-5607 

-5493 

-5380

-5268

-5158

-5049

-4942

-4837

-4734

-4632

-4533

-4435

-4339

-4245

-4152

-4062

-3974 

-3887 

-3802

-3719 

-3638

-3558

-3481

-3405

-128

-104

-82

-60

-41

-23

-7

+ 8

22

34

46

56

65 

73

81

87 

93 

98 

103 

107

110

113

115

116

118

119

120

121

120

121

120

120 

119 

118 

117

117

114 

113 

112 

110

108

107

105

103

102

100

98

96

94

92

90

88

87 

85 

83

81 

79

78

76

74

27

24

22

21

19

18

16

15

14

13

11

10

9

8

8

7 

6 

5 

5

4

3

3

2

1

2

1

0

1

0

0

0 -1 -1 -1 -1

-1

-2 

-1 

-1 

-2

-2

-2

-2

-2

-2

-2

-2

-2

-2

_ 2

_ 2

-2

_ 2 

-2 

-2

_ 2 

_ 2

-2

_ i

-2

TABLE 38. Modified potential Continued

a W Aii Al, Ai2

Q = 1.0; tabular spacing =0.02 Continued

2.20

2.22

2.24

2.26

2.28

2. 30

2. 32

2.34

2. 36

2. 38

2. 40

2. 42

2. 44 

2. 46

2. 48

2.50 

2.52 

2. 54 

2. 56 

2.58

2. 60

2. 62

2. 64

2. 66

2. 68

2. 70

2. 72

2.74

2. 76

2. 78

2.80 

2.82 

2. 84 

2. 86

2.88

2.90 

2. 92 

2.94 

2. 96

2.98

3.00

3.02

3.04

3.06

3.08

3. 10

3. 12

3. 14

3. 16

3. 18

3.20 

3.22 

3.24

3.26 

3.28

3. 30

0. 4668 0839

. 4687 9519

. 4707 4939

. 4726 7172

. 4745 6287

. 4764 2350

. 4782 5428

. 4800 5586

. 4&18 2884

. 4835 7387

. 4852 9152

. 4869 8239

. 4886 4705 

. 4902 8605

. 4918 9993

. 4934 8923 

. 4950 5446 

. 4965 9612 

. 4981 1472 

. 4996 1072

. 5010 8460

. 5025 3681

. 5039 6781

. 5053 7802

. 5067 6786

. 5081 3777

. 5094 8813

. 5108 1934

. 5121 3178

. 5134 2584

. 5147 0188 

. 5159 6026 

. 5172 0132 

. 5184 2540

. 5196 3284

. 5208 2395 

. 5219 9907 

. 5231 5849 

. 5243 0252

. 5254 3146

. 5265 4558

. 5276 4517

. 5287 3051

. 5298 0186

. 5308 5948

. 5319 0363

. 5329 3456

. 5339 5250

. 5349 5771

. 5359 5040

. 5369 3082 

. 5378 9917 

. 5388 5568

. 5398 0055 

. 5407 3400

. 5416 5623

-3331

-3259

-3188

-3119

-3051

-2985

-2921

-2858

-2797

-2737

-2678

-2621

-2566 

-2512

-2459

-2407 

-2356 

-2307 

-2259 

-2212

-2167

-2122

-2079

-2036

-1994

-1954

-1915

-1876

-1839

-1802

-1767 

-1732 

-1698 

-1664

-1632

-1600 

-1569 

-1539 

-1510

-1481

-1453

-1426

-1399

-1373

-1347

-1322

-1298

-1274

-1251

-1228

-1206 

-1184 

-1163

-1142 

-1122

72

71

69

68

66

64

63

62

60

58

57

56

54 

53

52

51

49 

48 

47 

46

45

43

43

42

40

39

39

38

37

36

35 

34 

33 

32

32

31 

30 

30 

29

28

27

27

26

26

25

24

24

23

23

22

22 

21

21

20 

19

 2

_ 1

-2

-2

-2

-2

-2

-1

_ 2

-2

-2

-1

_ i

 1

-1

-1

-2 

_ i

-1 

-1

-1

 1

-1

-1

_ 2

_ 1

_ -^

-1

-1

-1

0

-1 

_ j

-1

0

-1 -1

0 -1

-1

0

0 1

0 -1

_ J

0 -1

_ 1

0

-1

0 

0 -1 -1
NUMERICAL VALUES OF THE MODIFIED

TABLE 38. Modified potential Continued

a w *i Al, A!,

Q= 1.0; tabular spacing =0.1

3.3

3.4

3.5

3.6

3.7

3.8

3.9

40

4 1

4. 2

4.3

4 4

45

4 6

47

4 8

4.9

5.0

5. 1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9

6.0

6. 1

6.2

6.3

6.4

6.5

6.6

6.7

6. 8

6.9

7.0

7. 1

7.2

7.3

7. 4

7.5

7.6

7.7

7.8

7.9

8.0

a i

8.2

8.3

&4

8. 5

8. 6

8. 7

8.8

8.9

9.0

9. 1

9.2

9.3

9.4

0. 5416 5623

. 5461 0578

. 5503 0258

. 5542 6733

. 5580 1856

. 5615 7293

. 5649 4544

.5681 4962

. 5711 9772

. 5741 0083

. 5768 6903

. 5795 1148

. 5820 3655

. 5844 5187

. 5867 6445

. 5889 8069

. 5911 0649

. 5931 4728

. 5951 0804

. 5969 9339

. 5988 0760

. 6005 5462

. 6022 3811

. 6038 6148

. 6054 2789

. 6069 4029

. 6084 0142

. 6098 1385

.6111 7997

. 6125 0202

. 6137 8210

. 6150 2218

. 6162 2410

.6173 8960

. 6185 2032

.6196 1777

. 6206 8341

. 6217 1861

. 6227 2465

. 6237 0274

. 6246 5403

. 6255 7961

. 6264 8052

. 6273 5771

.6282 1212

. 6290 4462

. 6298 5604

. 6306 4718

.6314 1878

. 6321 7157

. 6329 0622

. 6336 2337

. 6343 2365

. 6350 0765

. 6356 7592

. 6363 2900

. 6369 6740

. 6375 9162

. 6382 0213

. 6387 9936

. 6393 8374

. 6399 5569

-2 7587

-2 5274

-2 3206

-2 1352

-1 9686

-1 8186

-1 6833

-1 5609

-1 4499

-1 3491

-1 2574

-1 1738

-1 0974

-1 0275

-9633

-9044

-8502

-8002

-7541

-7114

-6719

-6353

-6012

-5696

-5401

-5127

-4870

-4631

-4407

-4197

-4000

-3816

-3642

-3479

-3326

-3181

-3045

-^916

-2795

-2680

-2571

-2468

-2371

-2278

-2191

-2108

-2029

-1953

-1882

-1814

-1749

-1687

-1629

-1572

-1519

-1468

-1419

-1372

-1327

-1284

-1243

-1204

2313

2068

1854

1666

1500

1353

1224

1110

1008

917

836

764

700

641

589

542

500

461

427

395

366

340

316

295

274

256

240

224

210

197

185

173

163

154

145

136

129

122

115

109

103

98

92

88

83

79

75

71

68

65

62

59

56

54

51

49

47

45

43

41

39

38

-276

-245

-214

-188

-166

-147

-129

-114

-102

-91

-81

-72

-64

-58

-52

-47

-42

-38

-34

-32

-29

-26

-24

-22

-20

-18

-17

-16

-14

-13

-12

-11

-10

-10

-9

-8

_ o

-7

-6

-6

-6

-5

 £

-5

-4

-4

-4

-4

-3

-3

-3

-3

-3

-2

-2

-2

-2

-2

-2

, _ o

-2

-2

TABLE 38. Modified potential Continued

a w Ai, Ai, Ai,

Q= 1.0; tabular spacing =0.1 Continued

9.5

9.6

9.7

9.8

9. 9

10.0

0. 6405 1560

. 6410 6385

. 6416 0079

. 6421 2677

. 6426 4213

. 6431 4718

-1166

-1130

-1096

-1063

-1031

36

35

33

32

31

-2

_ i

. _ j

-1

-1

Q= 1.0; tabular spacing =0.2

10.0

10. 2

10. 4

10. 6

10.8

11.0

11.2

11. 4

11.6

11.8

12. 0

12. 2

12. 4

12. 6

12. 8

13.0

13.2

13.4

13.6

13.8

140

14. 2

14 4

14 6

14 8

15.0

15.2

15. 4

15. 6

15.8

16.0

16.2

16. 4

16. 6

16.8

17.0

0. 6431 4718

. 6441 2757

. 6450 7026

. 6459 7737

. 6468 5088

. 6476 9264

. 6485 0432

. 6492 8753

. 6500 4373

.6507 7430

. 6514 8051

. 6521 6357

. 6528 2460

. 6534 6464

. 6540 8468

. 6546 8564

. 6552 6839

. 6558 3375

. 6563 8248

. 6569 1530

. 6574 3290

. 6579 3591

. 6584 2496

. 6589 0060

. 6593 6340

. 6598 1385

. 6602 5244

. 6606 7965

. 6610 9590

. 6615 0161

. 6618 9718

. 6622 8298

. 6626 5938

. 6630 2670

. 6633 8528

. 6637 3542

-4002

-3771

-3557

-3360

-3176

-3006

-2848

-2701

-2563

-2435

-2316

-2203

-2098

-2000

-1908

-1821

-1740

-1663

-1590

-1522

-1458

-1397

-1340

-1286

-1234

-1185

-1139

-1095

-1054

-1014

-977

-941

-907

-875

-844

231

214

198

183

170

158

147

137

128

120

112

105

98

92

87

81

77

72

68

64

61

58

54

51

49

46

44

41

40

37

36

34

32

31

29

-19

-17

-16

-14

-13

-12

-11

-10

_ g

-8

-8

_ 7

_ _ fi

-6

-5

~ o

 5

-4

-4

-4

-4

_ Q

-3

~ o

. _ o

-3

-2

_ g

-2

-2

-2

-2

_ 2

-2

-2

Q= 1.0; tabular spacing =0.5

17.0

17.5

18.0

18.5

19.0

19. 5

20.0

20.5

21.0

21.5

22.0

22.5

23.0

0. 6637 3542

. 6645 7575

. 6653 6940

. 6661 2015

. 6668 3139

. 6675 0616

. 6681 4718

. 6687 5694

. 6693 3766

. 6698 9137

. 6704 1991

. 6709 2496

. 6714 0805

-5093

-4668

-4290

-3951

-3647

-3374

-3127

-2904

-2701

-2517

-2349

-2196

-2056

424

378

339

304

274

247

223

203

184

168

153

140

128

-53

-46

-40

-34

-31

-27

-23

-21

-18

-16

-14

-13

-12

547123 60 5
62 APPARENT RESISTIVITY OF A SINGLE "UNIFORM OVERBURDEN

TABLE 38. Modified potential Continued

a W Mi Alt Al2

Q= 1.0; tabular spacing =0.5 Continued

23.5 

24.0

245

25.0 

25.5

26.0 

26.5 

27.0

27.5 

28.0

28.5 

29.0 

29.5

30.0

0. 6718 7058 

.6723 1385 

. 6727 3902

. 6731 4718 

. 6735 3934

. 6739 1641 

.6742 7926 

. 6746 2866

. 6749 6536 

. 6752 9004

. 6756 0332 

. 6759 0580 

.6761 9803

. 6764 8051

-1927 

-1809 

-1701

-1601 

-1508

-1423 

-1344 

-1270

-1202 

-1139

-1080 

-1025 

-974

118 

109 

100

92 

85

79 

74 

68

63 

59

55 

51

48

-10 

-10 

-9

-8

-7

-6 

-5 

-5

-5

-4

-4 

-4 

-3

Q= 0.9; tabular spacing =0.02

0.00

.02

.04

.06

.08

. 10

. 12

- . 14

. 16

.18

.20

.22

.24

.26

.28

.30

.32

.34

.36

.38

.40

.42

.44

.46

.48

.50 

.52

.54

,56

.58

.60

.62

.64

.66

.68

.70

.72

.74

.76

.78

.80

.82

.84

.86

.88

0. 0000 0000

. 0001 6368

. 0006 5407

. 0014 6930

. 0026 0624

. 0040 6058

. 0058 2681

. 0078 9836

. 0102 6758

.0129 2587

. 0158 6373

. 0190 7086

. 0225 3624

. 0262 4824

.0301 9469

. 0343 6301

. 0387 4027

.0433 1330

. 0480 6879

. 0529 9334

. 0580 7359

. 0632 9625

. 0686 4819

. 0741 1651

. 0796 8859

. 0853 5211 

. 0910 9515

. 0969 0617

. 1027 7410

. 1086 8830

. 1146 3862

. 1206 1542

. 1266 0954

. 1326 1234

. 1386 1568

. 1446 1195

. 1505 9400

. 1565 5521

. 1624 8941

. 1683 9090

. 1742 5446

. 1800 7526

. 1858 4894

. 1915 7148

. 1972 3931

3 2735

3 2672

3 2483

3 2171

3 1739

3 1190

3 0531

2 9768

2 8907

2 7957

2 6927

2 5825

2 4662

2 3446

2 2186

2 0894

1 9577

1 8245

1 6907

1 5570

1 4241

1 2928

1 1638

1 0375

9145

7952 

6799

5690

4627

3613

2647

1732

868

+ 55 -708

-1421

-2085

-2701

-3270

-3794

-4275

-4714

-5112

-5473

-5797

-63

-189

-312

-432

-549

-659

-764

-86L

-950

-1030

-1102

-1163

-1216

-1259

- 1293

-1317

-1332

-1338

-1337

-1328

-1313

-1291

-1263

-1230

-1193

-1153 

-1109

-1062

-1015

-966

-915

-864

-813

-763

-713

-664

-616

-569

-524

-481

-438

-399

-361

-324

-290

-126

-126

-123

-120

-116

-111

-104

-97

-89

-80

-72

-62

-53

-43

-34

-24

-15

-6

+ 1

9

16

22

28

32

37

40 

44

46

48

49

50

51

51

50

50

49

48

47

45

43

42

40

38

37

34

TABLE 38. Modified potential Continued

a w Al, ' Ai, A!,

Q-- 0.9; tabular spacing =0.02 Continued

0.90

.92

. 94

.96

. 98

1. 00 

1.02 

1.04

1.06 

1. 08

1. 10 

1. 12 

1. 14

1. 16

1. 18

1. 20

1.22

1. 24

1. 26

1.28

1.30

1.32

1.34

1. 36

1. 38

1. 40

1. 42

1. 44

1. 46

1. 48

1. 50

1.52

1.54

1.56

1.58

1.60

1.62

1. 64

1. 66

1.68

1.70

0. 2028 4916 

. 2083 9815 

. 2138 8372

. 2193 0358

. 2246 5579

. 2299 3862 

. 2351 5064 

. 2402 9063

. 2453 5761 

. 2503 5078

. 2552 6957 

. 2601 1353 

. 2648 8243

. 2695 7614

. 2741 9471

. 2787 3827

. 2832 0710

. 2876 0157

. 2919 2214

. 2961 6937

. 3003 4386

. 3044 4633

. 3084 7751

. 3124 3823

. 3163 2933

.3201 5170

. 3239 0630

. 3275 9406

. 3312 1599

. 3347 7310

. 3382 6641

. 3416 9696

. 3450 6582

.3483 7404

. 3516 2269

.3548 1284

. 3579 4556

.3610 2191

. 3640 4297

. 3670 0978

. 3699 2340

-6086 

-6343 

-6569

-6767

-6937

-7082 

-7203 

-7302

-7380 

-7440

-7482 

-7507 

-7518

-7515

-7500

-7473

-7436

-7390

-7335

-7272

-7203

-7128

-7047

-6962

-6872

-6779

-6682

-6584

-6482

-6380

-6275

-6170

-6064

-5957

-5850

-5743

-5636

' -5530

-5424

-5320

-257 

-226 

-197

-170

-145

-121' 

-99

-78

-60 

-42

-26 

-11

+ 3

15

27

37

46

55

63

69

7581'

85

93' 90

96

99

101

103

104;

106'

106

107 107'

107;

107

106

106

105

105

33 

31 

29

27

25

24 

22 

20

19 

18

16 

15 

14

13

11

10

9

9

8

7

6

6

5

4

4

3

3

2

2

1

1

1

+ 1

0

0

0

0

0 1

0

Q=-0.9; tabular spacing =0.05

1.70

1.75

1.80

1. 85

1.90

1.95

2.00

2.05

2. 10

2. 15

2.20

2. 25

2.30

2.35

2.40

2.45

0. 3699 2340

. 3769 8155

. 3837 2976

.3901 8354

. 3963 5791

. 4022 6725

. 4079 2535

. 4133 4538

. 4185 3983

. 4235 2060

. 4282 9894

. 4328 8550

. 4372 9035

. 4415 2299

. 4455 9238

. 4495 0698

-3 2596

-3 0995

-2 9442

-2 7942

-2 6502

-2 5123

-2 3808

-2 2557

-2 1369

-2 0243

-1 9178

-1 8171

-1 7221

-1 6325

-1 5480

-1 4684

1600

1553

1499

1440

1379 1315;

1252

1188

1126

1065

1007

950

896

844

796

750

-37

-47

-54

-59

-62

-64

-64

-63

-62

-61

-59

-56

-54

-52

-49

-46
NUMERICAL VALUES OF THE MODIFIED POTENTIAL 63

TABLE 38. Modified potential Continued

a w A!, Al: Ai3

Q= 0.9; tabular spacing = 0.05 Continued

2.50

2.55

2.60

2.65

2.70

2.75

2.80

2.85

2.90

2.95

3.00

3.05

3. 10

3. 15

3.20

3.25

3.30

3.35

3.40

3.45

3.50

3.55

3.60

0. 4532 7472

. 4569 0312

. 4603 9923

. 4637 6968

. 4670 2073

.4701 5828

.4731 8785

. 4761 1466

. 4789 4364

. 4816 7940

. 4843 2630

. 4868 8847

. 4893 6977

. 4917 7386

.4941 0420

. 4963 6404

. 4985 5646

. 5006 8437

. 5027 5053

. 5047 5752

. 5067 0782

. 5086 0376

. 5104J4754

-1 3935

-1 3229

-1 2565

-1 1940

-1 1351

-1 0797

-1 0276

-9784

-9321

-8885

-8474

-8086

-7721

-7376

-7050

-6742

-6451

-6176

-5916

-5670

-5436

-5215

706

664

625

589

554

522

491

463

436

411

388

366

345

326

308

291

275

260

246

233

221

210

-44

-42

-39

-37

-35

-32

-30

-28

-27

-25

-23

-22

-21

-19

-18

-17

-16

-14

-14

-13

-12

-12

Q= 0.9; tabular spacing =0.1

3.6 

3.7

3.8 

3.9

4.0

4.1 A 0

rz* &

4.3 

4.4

4.5 

4.6

4.7

4.8 

4.9

5.0 

5. 1 

5.2

5.3 

5.4

5.5 

5.6 

5.7

5.8

5.9

6.0

6.1

6.2

6.3

6.4

6.5

6.6

6.7

6.8

6.9

0. 5104 4754 

. 5139 8693 

.5173 4149 

. 5205 2519

. 5235 5066

. 5264 2933

COQ1 71 CQ . ozyi / 100 

. 5317 8683 

. 5342 8366

. 5366 6991 

. 5389 5275 

.5411 3873

. 5432 3388 

. 5452 4373

.5471 7338 

. 5490 2754 

. 5508 1052

.5525 2636 

.5541 7876

. 5557 7118 

. 5573 0683 

. 5587 8868

. 5602 1952

. 5616 0193

. 5629 3831

. 5642 3094

. 5654 8192

. 5666 9325

. 5678^6676

. 5690 0421

. 5701 0723

.5711 7736

. 5722 1605

. 5732 2465

-2 0033 

-1 8483 

-1 7086 

-1 5823

 1 4680

-1 3642

-1 2700 

 1 1842 

-1 1058

-1 0342 

-9685 

-9083

-8530 

-8020

-7550 

-7117 

-6715

-6343 

-5998

-5677 

-5379 

-5102

-4843

-4602

-4376

-4164

-3966

-3780

-3607

-3443

-3289

-3144

-3008

-2879

1551 

1397 

1262 

1144

1038

942

O CO 85» 

783

717

656 

602 

554

510 

470

434 

401 

372

345 

321

298 

277 

259

241

226

212

198

186

174

163

154

145

137

129

121

-173 

-154 

-134 

-119

-106

-95

-84 

-75 

-67

-60 

-54 

-49

-44 

-40

-36 

-32

-29

-27 

-24

-23

-21 

-18

-17

-15

-14

-13

-13

-12

-10

-9

-9

-8

 o0

 -7

TABLE 38. Modified potential Continued

a W Ai, A!, '- Ai,

#= 0.9; tabular spacing =0.1 Continued

7.0

7.1

7.2

7.3

7.4

7.5

7.6

7.7

7.8

7.9

8.0

8.1

8.2

8.3

8.4

8.5

8.6

8.7

8.8

8.9

9.0

9. 1

9. 2

9.3

9.4

9.5 

9.6 

9. 7 

9.8 

9.9

10.0

0. 5742 0447

.5751 5672

. 5760 8253

. 5769 8301

.5778 5917

. 5787 1198

. 5795 4238

. 5803 5121

.5811 3933

. 5819 0750

. 5826 5649

. 5833 8700

. 5840 9970

. 5847 9524

.5854 7422

.5861 3725

. 5867 8486

.5874 1760

. 5880 3596

. 5886 4043

. 5892 3148

. 5898 0955

.5903 7505

. 5909 2840

. 5914 6999

. 5920 0018 

.5925 1932 

. 5930 2777 

. 5935 2584 

.5940 1386

. 5944 9213

-2758

-2643

-2534

-2432

-2334

-2242

-2155

-2072

-1994

-1919

-1848

-1780

-1716

-1655

-1597

-1541

-1488

-1437

-1389

-1343

-1298

-1256

-1215

-1177

-1140

-1104 

-1070 

-1037 

-1006 

-976

115

108

103

97

92

87

83

79

74

71

68

64

61

58

56

53

51

48

46

44

42

41

39

37

35

34 

33 

32 

30 

29

-7

-6

-6

-6

-5

-5

-4

_ 4

-4

-3

 3

_ 4

-3

, _ o

__ 0

_ o

-2

-2

-2

 2

-2

-2

-2

-2

-2

-1 

_ o

-1 

-2 

-1

Q= 0.9; tabular spacing =0,2

10.0 

10.2

10.4 

10.6

10.8

11.0 

11.2

11.4 

11.6 

11.8

12.0 

12. 2

12.4 

12.6 

12.8

i q f\ id. U

0. 5944 9213 

. 5954 2053

. 5963 1324

.5971 7228 

. 5979 9951

. 5987 9668 

. 5995 6538

. 6003 0712 

. 6010 2329 

.6017 1519

. 6023 8404 

. 6030 3096

. 6036 5702 

. 6042 6321 

. 6048 5046

. 6054 1965

-3788 

-3569

-3367 

-3180 

-3007

 2846 

-2696

-2557 

-2427 

-2306

-2192 

-2086

, - 1987 

-1894 

-1806

218 

202

187 

173 

161

150 

139

130 

121 

113

106 

99

93 

88 

82

-18 

-16

-15 

-14 

-12

-11 

-10

-10

-8 

-8

. _ 7 

_ 7

-7 

-5 

-5

Q = 0.9 ; tabular spacing = 0.5

13. 0

13.5

14.0

14.5

15.0

15.5

16.0

16.5

17.0

0. 6054 1965

. 6067 6886

. 6080 2171

. 6091 8818

. 6102 7690

.6112 9539

. 6122 5023

.6131 4722

. 6139 9145

-1 0791

-9635

-8639

-7775

-7023

-6364

-5786

-5276

-4824

115&

996

864

752

658

578

511

452

402

-193

-159

-133

-112

-94

-80

-68

-59

-50
64 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

TABLE 38. Modified potential Continued

a W Al, Al, Al2

Q= 0.9; tabular spacing =0.5 Continued

17.5

18.0

18.5

19.0

19.5

20.0

20.5

21.0

21.5

22.0

22.5

23.0

23.5

24.0

24.5

25.0

25.5

26.0

26.5

27.0

27.5

28.0

28.5

29.0

29.5

30.0

0. 6147 8745

. 6155 3923

. 6162 5038

. 6169 2410

. 6175 6328

.6181 7050

. 6187 4810

. 6192 9820

. 6198 2272

. 6203 2340

. 6208 0183

. 6212 5946

. 6216 9762

. 6221 1752

. 6225 2028

. 6229 0694

.6232 7843

. 6236 3564

. 6239 7936

.6243 1036

. 6246 2932

. 6249 3689

. 6252 3367

. 6255 2022

. 6257 9705

. 6260 6466

-4422

-4063

-3742

-3455

-3196

-2962

-2750

-2558

-2384

-2225

-2080

-1947

-1826

-1714

-1611

-1516

-1429

-1348

-1273

-1204

-1139

-1079

-1023

-971

-923

358

321

288

259

234

211

192

174

159

145

133

122

112

103

95

87

81

75

69

64

60

56

52

48

46

-44

-38

-33

-28

-25

-22

-19

-18

-15

-14

-12

-11

-10

-9

-8

-7

-6

-6

-6

 ~ o

-4

-4

-4

-4

-3

Q= -0.8; tabular spacing =0.92

0.00

.02

.04

.06

.08

. 10

. 12

. 14

. 16

. 18

.20

.22

.24

.26

.28

.30

.32

.34

.36

» 38V(J

.40

.42

.44

.46

.48

.50

.52

.54

.56

.58

0. 0000 0000

.0001 4683

.0005 8675

.0013 1808

. 0023 3805

. 0036 4279

. 0052 2742

. 0070 8605

. 0092 1190

. 0115 9728

. 0142 3372

.0171 1204

. 0202 2243

. 0235 5450

. 0270 9741

. 0308 3993

.0347 7054

. 0388 7748

. 0431 4888

. U^E 04.7^ i O 797Q i & 1 U

.0521 3728

.0568 3049

. 0616 4070

. 0665 5639

. 0715 6628

. 0766 5939

. 0818 2504

. 0870 5294

.0923 3318

. 0976 5624

2 9365

2 9309

2 9141

2 8863

2 8478

2 7989

2 7401

2 6720

2 5953

2 5107

2 4188

2 3206

2 2168

2 1084

1 9961

1 8808

1 7634

1 6446

1 5251

1 4058

1 2872

1 1700

1 0548

9420

8321

7255

6225

5233

4283

3375

-56

-168

-278

-386

-489

-587

-681

-767

-847

-918

-982

-1038

-1084

-1123

-1153

-1175

-1188

-1194

-1194

1 lloO ISfi 

-1172

-1152

-1128

-1099

-1066

-1030

-992

-950

-908

-864

-112

-112

-110

-108

-103

-98

-93

-86

-80

-72

-64

-56

-47

-38

-30

-22

-14

 6

+ 1

14

19

25

29

33

36

39

41

43

44

TABLE 38. Modified potential Continued

a w Al, Al, Ai,

Q=-0.8; tabular spacing =9.02 Continued

0.60

.62

.64

.66

.68

.70

.72

.74

.76

.78

.80

.82

.84

.86

.88

.90

.92.

.94

.96

.98

1.00

1.02

1.04

1.06

1.08

1.10

1. 12

1. 14

1. 16 

1. 18

1. 20

1.22

1. 24

1. 26

1. 28

1.30

1.32

1.34

1.36

1.38

1. 40

1.42

1. 44

1.46

1.48

1.50

0. 1030 1304

. 1083 9496

.1137 9380

. 1192 0181

. 1246 1171

. 1300 1665

. 1354 1024

. 1407 8651

. 1461 3994

. 1514 6540

. 1567 5817

. 1620 1393

. 1672 2871

. 1723 9892

.1775 2129

. 1825 9289

. 1876 1108

. 1925 7354

. 1974 7819

..2023 2324

.2071 0711

. 2118 2847

. 2164 8619

. 2210 7936

. 2256 0721

. 2300 6919

. 2344 6486

. 2387 9397

. 2430 5637 

. 2472 5204

. 2513 8109

.2554 4370

. 2594 4018

. 2633 7091

.2672 3632

. 2710 3695

. 2747 7338

. 2784 4625

. 2820 5625

. 2856 0410

. 2890 9059

. 2925 1651

. 2958 8269

. 2991 8998

. 3024 3928

. 3056 3146

2511

1692

918

+ 189 -496

-1135

-1732

-2285

-2797

-3269

-3702

-4097

-4458

-4784

-5077

-5340

-5574

-5780

-5961

-6117

-6251

-6364

-6456

-6531

-6588

-6630

-6657

-6671

-6672 

--6663

-6643

-6614

-6576

-6531

-6478

-6420

-6356

-6287

-6214

-6137

-6057

-5974

-5888

-5800

-5711

-819

-774

-729

-684

-640

-596

-553

-512

-472

-433

-396

-360

-326

-294

-263

-234

-207

-180

-156

-134

-113

-93

-74

 UOCO 

-42

-27

-14

 * 1

+ 10 

20

29

38

45

52

58

64

69

73

77

80

83

86

88

89

91

45

45

45

45

44

44

43

41

40

39

37

36

34

32

31

29

27

26

24

22

21

20

18

17

16

14

14

12

11 

10

10

9

7

7

6

6

5

4

4

3

3

3

2

1

2

Q = -0.8; tabular spacing =0.05

, 1.50

1.55

1.60

1. 65

1.70

1.75

1.80

1.85

1.90

1.95

0. 3056 3146

.3133 6804

.3207 6778

. 3278 4528

.3346 1510

.3410 9161

.3472 8890

. 3532 2065

. 3589 0012

. 3643 4002

-3 5125

-3 3683

-3 2225

-3 0768

-2 9330

-2 7922

-2 6553

-2 5230

-2 3956

-2 2735

1441

1459

1456

1438

1408

1369

1324

1274

1221

1167

43

+ 17 -2

-18

-30

-39

-46

-50

-53

-54
NUMEBtCAL VALUES OF THE MODIFIED POTENTIAL 65

TABLE 38. Modified potential Continued

a W Ail Al, Ai,

Q=-0.8; tabular sp*dng=8.05 Continued

2.00

2.05

2. 10

2. 15

2.20

2.25

2.30

2.35

2.40

2.45

2.50

2.55

2.60

2.65

2. 70

2. 75

2.80

2.85

2.90

2.95

3.00

3.05

3. 10

3. 15

3.20

3.25

3.30

3.35

3. 40

0. 3695 5258

. 3745 4945

. 3793 4176

. 3839 4006

.3883 5439

. 3925 9425

. 3966 6868

. 4005 8601

. 4043 5444

. 4079 8147

. 4114 7426

. 4148 3954

. 4180 8366

.4212 1260

. 4242 3201

.4271 4720

. 4299 6319

. 4326 8468

.4353 1614

. 4378 6176

.4403 2550

.4427 1111

. 4450 2210

. 4472 6182

. 4494 3340

. 4515 3982

. 4535 8390

. 4555 6829

. 4574 9550

-2 1568

-2 0457

-1 9400

-1 8397

-1 7447

-1 6548

-1 5699

-1 4896

-1 4139

-1 3425

-1 2751

-1 2116

-1 1517

-1 0954

-1 0422

-9921

-9449

-9004

-8584

-8188

-7814

-7461

-7128

-6813

-6516

-6235

-5969

-5717

1112

1057

1003

950

899

850

802

757

714

674

635

598

564

532

501

472

445

420

396

374

353

333

314

297

281

266

252

238

-55

-55

-54

-53

-51

-49

-47

-45

-43

-41

-39

-36

-35

-32

-30

-29

-27

-25

-24

-22

-21

-19

-19

-17

-16

-16

-14

-13

Q= 0.8; tabular spacing=0.1

3.4

3.5

3.6

3.7

3.8

3.9

4.0

4. 1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

5.0

5.1

5.2

5.3

5. 4

5.5

5.6

5.7

5.8

5.9

0. 4574 9550

. 4611 8783

. 4646 7846

. 4679 8324

.4711 1642

. 4740 9086

. 4769 1820

. 4796 0900

. 4821 7282

. 4846 1839

. 4869 5365

.4891 8586

. 4913 2166

. 4933 6714

. 4953 2788

. 4972 0901

.4990 1529

. 5007 5108

. 5024 2041

.5040 2702

.5055 7436

. 5070 6565

. 5085 0389

. 6098 9185

.5112 3210

. 5125 2707

-2 1928

-2 0169

-1 8586

-1 7160

-1 5874

-1 4710

-1 3655

-1 2697

-1 1825

-1 1031

-1 0305

-9641

-9032

-8474

-7960

-7486

-7049

-6645

-6272

-5927

-5605

-5306

-5029

-4770

-4528

-4303

1759

1583

1426

1287

1164

1055

958

872

795

725

664

609

558

514

474

437

404

373

346

322

299

277

258

242

226

210

-203

-176

-157

-139

-123

-109

-96

-87

-77

-69

-61

-55

-50

-45

-40

-37

-33

-31

-27

-24

-23

-22

-18

-17

-16

-15

TABLE 38. Modified potential Continued

a W Ai. A* Ai,

Q=-0.8: tabular sp*dng«=0.1 Continued

6.0

6.1

6.2

6.3

6.4

6.5

6.6

6.7

6.8

6.9

7.0

7. 1

7.2

7.3

7.4

7.5

7.6

7.7

7.8

7.9

8.0

8. 1

8.2

8.3

8.4

8.5

8.6

8.7

8.8

8.9

9.0

9. 1

9.2

9.3

9.4

9.5

9.6

9.7

9.8

9.9

10.0

0.5137 7901

. 5149 9002

.5161 6208

. 5172 9704

. 5183 9663

. 5194 6246

.5204 9608

. 5214 9892

. 5224 7232

.5234 1758

. 5243 3589

. 5252 2839

.5260 9614

.5269 4017

. 5277 6143

. 5285 6083

. 5293 3928

. 5300 9745

. 5308 3626

. 5315 5639

. 5322 5855

. 5329 4340

.5336 1158

. 5342 6368

. 5349 0027

. 5355 2190

.5361 2910

. 5367 2236

.5373 0214

. 5378 6892

. 5384 2312

. 5389 6516

. 5394 9542

.5400 1430

.5405 2214

. 5410 1931

. 5415 0613

. 5419 8293

. 5424 5000

. 5429 0764

. 5433 5615

-4092

-3896

-3710

-3537

-3375

-3222

-3078

-2943

-2815

-2695

-2582

-2474

-2373

-2277

-2186

-2100

-2018

-1941

-1867

-1797

-1731

-1668

-1608

-1550

-1496

-1444

-1394

-1347

-1301

-1258

-1217

-1177

-1130

-1103

-1068

-1035

-1003

-972

-943

-914

197

185

173

162

153

144

135

128

120

113

107

102

96

91

86

82

77

74

70

66

63

80

57

55

52

50

47

46

43

41

40

38

36

35

33

32

31

30

28

27

-14

-12

-12

-11

-9

-10

_ g

-7

-8

-7

-6

-6

-6

-5

* " 0

-4

-4

-4

-4

-3

__ 0

-3

tji o

( __ O

-2

.^ O

-2

-2

-2

_ 2

_ o

. _ o

-2

-1

 2

_ *

~-2

-1

, __ 1

* 1

Q= 0.8; tabular spacing=0.2

10.0

10.2

10. 4

10.6

10.8

11.0

11.2

11.4

11.6

11.8

12.0

12.2

12. 4

12.6

12. 8

13.0

0. 5433 5615

. 5442 2680

. 5450 6399

. 5458 6962

. 5466 4543

.5473 9306

.5481 1401

. 5488 0968

. 6494 8138

. 5501 3032

. 5507 5765

. 5513 6442

. 5519 5164

. 5525 2022

.5530 7104

. 5536 0493

-3550

-3346

-3156

-2981

-2819

-2668

-2528

-2397

-2275

-2162

-2055

-1956

-1863

-1776

-1694

206

189

175

163

151

140

131

122

114

106

99

93

87

82

77

-17

-16

-14

-12

-12

-11

-10

-9

-8

__ 0

-7

-6

-6

-6

-5
66 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

IABLE 68. Modified potential Continued

a w Ai, Ai, Ai,

Q= 0.8; tabular spacing =0.5

13.0

13.5

14.0

14.5

15.0

15. 5

16.0

16.5

17.0

17.5

18.0

18.5

19.0

19.5

20.0

20. 5

21.0

21.5

22.0

22.5

23.0

23.5

24.0

24.0

25.0

25.5

26.0

26.5

27. 0

27. 5

28.0

28.5

29. 0

29.5

30. 0

0. 5536 0493

. 5548 7048

. 5560 4567

.5571 3986

. 5581 6113

.5591 1654

.5600 1226

. 5608 5371

. 5616 4568

. 5623 9241

, 5630 9767

. 5637 6482

.5643 9686

. 5649 9650

. 5655 6616

. 5661 0805

. 5666 2413

.5671 1621

.5675 8594

. 5680 3479

.5684 6412

. 5688 7520

. 5692 6914

. 5696 4701

. 5700 0977

. 5703 5830

. 5706 9343

. 5710 1592

. 5713 2646

. 5716 2571

. 5719. 1428

.5721 9272

. 5724 6156

. 5727 2129

. 5729 7236

-1 0118

-9035

-8101

-7292

-6586

-5969

-5427

-4948

-4524

-4147

-3811

-3510

-3240

-2998

-2778

-2580

-2400

-2236

-2087

-1951

-1827

-1712

-1608

-1511

-1422

-1340

-1264

-1194

-1129

-1069

 1012

-960

-911

-866

1083

934

810

705

617

542

479

424

377

336

301

270

243

219

198

180

164

149

136

125

114

105

96

89

82

76

70

65

61

56

52

49

46

43

-181

-149

-124

 104

-88

-75

-64

-55

-47

-40

-35

-31

-27

-24

-21

-18

-16

-15

-13

-11

-11

-9

-8

 oo

_ 7

-6

-6

-5

-4

-4

-4

-3

-3

-3

Q= 0.7; tabular spacing=0.02

0.00

. 02

.04

.06

.08

. 10

. 12

. 14

. 16

. 18

.20

.22

.24

.26

.28

.30

.32

.34

.36

.38

.40

,42

0. 0000 0000

.0001 2969

.0005 1828

.0011 6428

. 0020 6525

. 0032 1783

. 0046 1770

. 0062 5973

.0081 3793

. 0102 4558

. 0125 7524

.0151 ]885

.0178 6779

. 0208 1296

.0239 4484

. 0272 5357

. 0307 2902

. 0343 6088

. 0381 3872

. 0420 5204

. 0460 9036

. 0502 4327

2 5938

2 5889

2 5742

2 5498

2 5160

2 4730

2 4215

2 3618

2 2944

2 2202

2 1395

2 0533

1 9623

1 8671

1 7685

1 6672

1 5641

1 4598

1 3548

1 2500

1 1458

1 0428

-49

-148

-244

-338

-429

-516

-597

-673

-743

-806

-862

-911

-952

-986

-1012

-1031

-1044

-1049

-1048

-1042

-1030

-1013

-98

-98

-97

-94

-91

-86

-82

-76

-69

-64

-56

-49

-41

-34

-26

-19

-12

-6

+ 1

7

12

17

TABLE 38. Modified potential uontmuea

a W Ai, Ai, Ai,

Q= 0.7; tabular spacing=0.02 Continued

0.44

.46

.48

.50

.52

.54

.56

.58

.60

.62

.64

.66

.68

.70

. 72

. 74

. 76

.78

.80

.82

.84

.86

.88

.90

.92

.94

.96

.98

1.00

1.02

1. 04

1.06

1.08

1. 10

1. 12

1. 14

1. 16 

1. 18

1.20

1. 22

1.24

1. 26

1. 28

1.30

1.32

1.34

1.36

1.38

1. 40

0. 0545 0045

. 0588 5180

. 0632 8737

.0677 9753

.0723 7287

. 0770 0435

. 0816 8323

.0864 0114

.0911 5010

. 0959 2248

. 1007 1106

. 1055 0901

. 1103 0991

. 1151 0772

.1198 9677

. 1246 7180

. 1294 2793

. 1341 6060

. 1388 6564

. 1435 3921

. 1481 7780

. 1527 7819

. 1573 3750

. 1618 5309

. 1663 2261

. 1707 4398

. 1751 1533

. 1794 3505

. 1837 0171

. 1879 1410

. 1920 7119

. 1961 7213

.2002 1622

. 2042 0292

.2081 3183

.2120 0266

. 2158 1525 

.2195 6957

. 2232 6564

. 2269 0363

. 2304 8374

. 2340 0627

. 2374 7159

. 2408 8013

. 2442 3237

. 2475 2883

. 2507 7010

. 2539 5678

. 2570 8954

9415

8424

7457

6519

5613

4740

3903

3104

2342

1620

938

+ 294

-310

-875

-1402

-1891

-2345

-2763

-3147

-3499

-3819

-4109

-4371

-4606

-4816

-5001

-5164

-5305

-5427

-5530

-5615

-5685

-5739

-5780

-5807

-5823

-5828 

-5824

-5810

-5787

-5758

-5721

-5678

-5630

-5577

-5520

-5458

-5393

-992

-966

-938

-906

-873

-837

-800

-761

-722

-683

-643

-604

-565

-527

-490

-454

-418

-384

-352

-320

-290

-262

-235

-210

-185

-163

-142

-122

-103

-85

-69

-54

-41

-28

-16

-5

+5 

14

22

30

36

43

48

53

57

62

65

68

21

25

28

32

34

36

37

38

39

40

39

39

39

38

37

36

36

34

33

31

30

29

27

25

24

23

21

20

18

18

16

15

14

13

12

11

10 

9

8

7

7

6

6

5

4

4

3

3

Q=-0.7; tabular spacing =0.05

1. 40

1. 45

1. 50

1.55

1. 60

1.65

1. 70

0. 2570 8954

. 2646 8998

. 2719 6890

. 2789 3823

. 2856 1031

. 2919 9768

. 2981 1287

-3 3277

 3 2152

-3 0959

-2 9725

-2 8472

 2 7217

-2 5976

1124

1193.

1234

1253

1254

1241

1218

103

69

41

19

+ 1

-13

-23
NUMERICAL VALUES OF THE MODIFIED POTENTIAL 67

TABLE 38. Modified potential Continued

a W Al, A*. Ai,

Q=-0.7; tabular spacing =0.05 Continued

1.75

1.80

1.85

1.90

1.95

2.00

2.05

2. 10

2. 15

2.20

2.25

2.30

2.35

2. 40

2.45

2.50

2.55

2.60

2.65

2.70

2.75

2.80

2.85

2.90

2.95

3.00

3.05

3. 10

3. 15

3.20

0. 3039 6830

. 3095 7614

. 3149 4827

. 3200 9616

. 3250 3091

. 3297 6314

. 3343 0302

. 3386 6026

. 3428 4412

. 3468 6336

. 3507 2634

. 3544 4093

.3580 1462

. 3614 5448

. 3647 6717

.3679 5902

. 3710 3598

. 3740 0367

. 3768 6739

. 3796 3215

. 3823 0268

. 3848 8342

. 3873 7858

. 3897 9215

.3921 2785

. 3943 8924

. 3965 7966

. 3987 0225

. 4007 6002

. 4027 5578

-2 4758

-2 3572

-2 2423

-2 1315

-2 0252

-1 9234

-1 8264

-1 7339

-1 6461

-1 5627

-1 4838

-1 4090

-1 3384

-1 2716

-1 2085

-1 1489

-1 0927

-1 0397

-9896

-9424

-8978

-8558

-8160

-7786

-7432

-7097

-6782

-6483

-6201

1186

1149'

1108'

1063

1017

971

924 878'

833

790

747

707

668

631

596

562

530

500

472

446

421

397

375

354

334

316

299

282

267

-31

-38

-41

 44

-46

-46

-46

-46

-45

-44

-42

-40

-39

-37

-35

-33

-32

-30

-28

-27

-25

-24

-22

-21

-20

-19

-17

-16

-15

Q=-0.7; tabular spacing=0.1

3.2 

3.3 

3. 4

3. 5 

3.6

3.7 

3.8 

3.9

4. 0 

4. 1

4. 2 

4.3

44 

4.5

4 6

47

4.8

4. 9

5.0

5. 1

5. 2

5.3

5.4

5.5

5. 6

0. 4027 5578 

. 4065 7182 

.4101 7007

. 4135 6821 

. 4167 8211

. 4198 2608 

.4227 1303 

. 4254 5461

. 4280 6138 

. 4305 4295

. 4329 0802 

.4351 6452

. 4.Q7Q rrO i O 1Q71 Xy / X

.4393 8019

. 4413 5204

. 4432 4083

. 4450 5165

. 4467 8922

. 4484 5786

. 4500 6158

. 4516 0408

. 4530 8880

.4545 1890

. 4558 9733

. 4572 2683

-2 3749 

-2 1779 

-2 0011

-1 8423 

-1 6993

-1 5703 

-1 4537 

-1 3480

-1 2521 

-1 1650

-1 0856 

-1 0132 

Q470 «7^ \J

-8863

-8307

-7796

-7326

-6892

-6492

-6122

-5779

-5462

-5167

-4892

-4637

1970 

1767 

1588

1430 

1290

1166 

1056 

959

871 

793

724 

662

Cf\>7 6U7 

556

511

470

433

400

370

343

318

295

274

255

238

-229 

-203 

-179

-158 

-140

-124 

-110 

-97

-88 

-78

-69 

-62

_ ccoo 

-50

-45

-41

-37

-33

-30

-27

-25

-23

-20

-19

-17

TABLE 38. Modified potential Continued

a W Aii A!, Al2

Q=-0.7; tabular spacing=0.1 Continued

5.7

5.8

5.9

6. 0

6. 1

6.2

6. 3

6.4

6.5

6. 6

6.7

6.8

6.9

7. 0

7. 1

7.2

7.3

7.4

7.5

7.6

7.7

7.8

7.9

8.0

8. 1

8. 2

8. 3

8.4

8. 5

8. 6

8.7

8.8

8 Q O* »7

9.0 

9. 1 

9.2

9.3 

9.4

9.5 

9.6 

9.7

9.8 

9.9

10.0

0. 4585 0996

. 4597 4910

. 4609 4646

. 4621 0412

. 4632 2401

. 4643 0795

. 4653 5763

. 4663 7466

. 4673 6054

. 4683 1666

. 4692 4435

. 4701 4486

. 4710. 1937

. 4718 6899

. 4726 9476

. 4734 9766

. 4742 7864

. 4750 3858

. 4757 7832

. 4764 9865

. 4772 0032

. 4778 8406

. 4785 5053

. 4792 0039

. 4798 3424

. 4804 5268

. 4810 5624

. 4816 4547

. 4822 2087

. 4827 8292

. 4833 3207

. 4838 6878

.4843 9344

. 4849 0648 

. 4854 0826 

. 4858 9915

. 4863 7950 

. 4868 4966

. 4873 0994 

. 4877 6064 

. 4882 0206

. 4886 3450 

. 4890 5822

.4894 7347

-4400

-4178

-3970

-3777

-3595

-3425

-3266

-3116

-2975

-2843

-2718

-2600

-2489

-2385

-2286

-2193

-2104

-2020

-1941

-1866

-1794

-1726

-1662

-1600

-1542

-1486

-1434

-1383

-1335

-1289

-1245

-1203

1 i a A

-1125 

-1089 

-1053

-1020 

-988

-957 

-928 -899

-872 

-846

222

207

194

181

170

160

150

141

132

125

118

111

105

99

93

89

84

79

75

72

68

64

61

58

56

52

51

48

46-

44

42

40

OQOO

37 

35 

33

32 

31

30 

29

27

26 

26

-16

-15

-13

-13

-11

-11

-10

-9

-8

__ Q

-7

-7

-6

-6

-5

-4

_ ^

-5

 4

-3

-4

-4

-3

_ 3

-3

-3

-2

_ 2

-3

_ 2

2

-2

-2 

-1 

-2

-2 

_ 1

_ 1

-1 

-2

-1 

0

Q= 0.7; tabular spacing =0.2

10.0

10.2

10.4

10.6

10.8

11.0

11.2

11.4

11.6

11.8

0. 4894 7347

.4902 7959

. 4910 5475

. 4918 0070

.4925 1907

.4932 1135

. 4938 7894

. 4945 2313

.4951 4514

.4957 4608

-3285

-3096

-2921

-2759

-2609

 2469

-2339

^ 2219

-2106

-2001

190

175

162

150

140

130.

121

113

105

98

-15

:-15

-13

-12

-11

-10

-9

-8

-7

-7
68 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

TABLE 38. Modified potential Continued

W

Q= 0.7; tabular spacing=0.2 Continued

12.0

12.2

12.4

12.6

12.8

13.0

0. 4963 2702

. 4968 8893

. 4974 3274

. 4979 5929

. 4984 6942

. 4989 6386

-1902

-1811

-1724

-1644

-1568

92

86

81

76

71

__ "7

-5

-6

-5

-5

Q= 0.7; tabular (spacing=0.5

13.0

13.5

14.0

14.5

15.0

15.5

16.0

16.5

17.0

17.5

18.0

18.5

19.0

19.5

20.0

20.5

21.0

21.5

22.0

22.5

23.0

23.5

24.0

24.5

25.0

25.5

26.0

26.5

27.0

27.5

28.0

28.5

29.0

29.5

30.0

0. 4989 6386

. 5001 3592

. 5012 2433

. 5022 3774

.5031 8364

. 5040 6856

. 5048 9820

. 5056 7760

.5064 1117

. 5071 0285

. 5077 5612

. 5083 7409

. 5089 5955

. 5095 1500

. 5100 4270

.5105 4466

.5110 2272

.5114 7856

. 5119 1369

.5123 2948

. 5127 2720

.5131 0800

. 5i34 7294

. 5138 2298

.5141 5903

. 5144 8191

. 5147 9236

.5150 9111

. 5153 7879

. 5156 5602

. 5159 2335

.5161 8129

.5164 3034

. 5166 7096

. 5169 0355

-9367

-8365

-7500

-6751

-6098

-5527

-5025

-4582

-4190

-3841

-3530

-3251

-3001

-2776

-2573

-2389

-2223

-2071

-1933

-1807

-1692

-1586

-1489

-1400

-1318

-1242

-1171

-1106

-1046

-990

-938

-889

-844

-802

1002

864

749

653

571

502

443

392

349

311

279

250

225

203

184

167

152

138

126

115

105

97

90

82

76

70

65

60

56

52

49

45

42

39

-167

-138

-115

-97

-82

-69

-59

-51

-44

-37

-33

-29

-25

-22

-19

-17

-15

-14

-12

-11

-10

-8

-8

-7

-6

-6

-5

-5

 £J£

-4

-3

-3

-3

-3

Q- O.G; tabular spacing=0.02

0.00

.02

.04

.06

.08

.10

. 12

. 14

. 16

.18

.20

.22

0. 0000 0000

.0001 1225

. 0004 4859

. 0010 0774

. 0017 8760

. 0027 8528

. 0039 9708

. 0054 1856

. 0070 4461

. 0088 6942

. 0108 8661

. 0130 8926

2 2451

2 2408

2 2281

2 2072

2 1781

2 1412

2 0969

2 0456

1 9877

1 9238

1 8545

1 7804

-42

-127

-210

-291

-369

-443

-513

-579

-639

-693

-741

-783

-85

-84

-83

-81

-78

-74

-70

-65

-60

-54

-48

-42

TABLE 38. Modified potential Continued

a W Aii Ai, Ai,

Q=-0.6; tabular spacing=0.02 Continued

0.24

.26

.28

.30

.32

.34

.36

.38

.40

.42

.44

.46

.48

.50

.52

.54

.56

.58

.60

.62

.64

. 66

.68

.70

.72

.74

.76

.78

.80

.82

.84

.86

.88

.90

.92

.94

. 96

.98

1.00

1.02

1.04

1.06

1.08

1. 10

1. 12

1. 14

1. 16

1.18

1. 20

1. 22

1.24

1.26

1.28

1.30

1.32

1.34

1.36

1.38

1. 40

0. 0154 6994

. 0180 2083

. 0207 3374

, 0236 0020

. 0266 1148

. 0297 5873

. 0330 3296

. 0364 2515

. 0399 2627

. 0435 2735

.0472 1954

. 0509 9409

. 0548 4248

. 0587 5637

. 0627 2769

. 0667 4862

. 0708 1163

. 0749 0951

. 0790 3535

. 0831 8259

. 0873 4499

.0915 1664

. 0956 9198

. 0998 6580

. 1040 3320

. 1081 8961

. 1123 3079

. 1164 5281

. 1205 5204

. 1246 2514

. 1286 6904

. 1326 8096

. 1366 5836

. 1405 9896

. 1445 0069

. 1483 6172

. 1521 8040

. 1559 5532

. 1596 8521

. 1633 6898

. 1670 0573

. 1705 9468

. 1741 3520

. 1776 2677

. 1810 6903

. 1844 6169

. 1878 0459

. 1910 9766

. 1943 4088

. 1975 3437

. 2006 7828

. 2037 7283

.2068 1831

. 2098 1505

. 2127 6345

. 2156 6393

. 2185 1697

. 2213 2306

. 2240 8274

1 7021

1 6202

1 5354

1 4484

1 3596

1 2698

1 1796

1 0893

9996

9110

8237

7383

6551

5742

4961

4208

3487

2797

2139

1516

926

+370

-153

-642

-1098

-1523

-1916

-2279

-2613

-2919

-3198

-3452

-3681

-3887

-4070

-4234

-4377

-4503

-4611

-4703

-4780

-4843

-4894

-4932

 4959

-4976

-4984

-4983

-4974

-4958

-4936

-4907

-4873

-4835

-4792

-4745

-4694

-4641

-819

-848

-871

-887

-898

-903

-902

-897

-887

-872

-854

-833

-808

-782

-752

-722

-690

-657

-624

-590

-556

-522

-489

-456

-424

-393

-363

-334

-306

-279

-254

-229

-206

-184

-163

-144

-125

-108

-92

-77

-63

-50

-38

-27

-17

__ Q

+ 1

9

16

22

28

34

39

43

47

50

54

56

-35

-29

-23

-17

-10

-5

0

+ 6

10

14

18

22

24

27

29

31

32

33

33

34

34

34

33

33

32

31

30

29

28

27

26

25

23

22

21

20

18

17

16

15

14

13

12

11

10

9

9

8

; 7

6

6

6

5

4

4

3

3

2
NUMERICAL VALUES OF THE MODIFIED POTENTIAL 69

TABLE 38. Modified potential Continued

a w Al, Ai, Ai2

Q= 0.6; tabular spacing =0.05

1.40 

1.45

1.50 

1.55 

1.60 

1. 65 

1.70

1.75 

1.80 

1.85 

1. 90 

1. 95

2.00 

2.05 

2. 10 

2. 15 

2.20

2.25 

2.30 

2.35 

2.40 

2.45

2.50 

2. 55 

2.60 

2.65 

2. 70

2.75 

2.80

2.85 

2.90 

2.95

3.00

3.05

3. 10 

3. 15

3.20

0. 2240 8274 

. 2307 8262

. 2372 0533 

. 2433 6082 

. 2492 5941 

.2549 1164 

. 2603 2807

.2655 1918 

. 2704 9526 

. 2752 6639 

. 2798 4233 

. 2842 3248

. 2884 4592 

. 2924 9133 

. 2963 7700 

.3001 1085 

. 3037 0041

.3071 5284 

. 3104 7492 

. 3136 7309 

.3167 5345 

. 3197 2175

. 3225 8345 

. 3253 4371 

. 3280 0739 

. 3305 7910 

. 3330 6319

. 3354 6377 

. 3377 8472

. 3400 2970 

. 3422 0219 

. 3443 0545

. 3463 4259

.3483 1654

. 3502 3005 

. 3520 8575

. 3538 8614

-2 8650 

-2 7717

-2 6722 

-2 5689 

-2 4637 

-2 3580 

 2 2532

-2 1502 

-2 0496 

-1 9520 

-1 8578 

-1 7672

-1 6803 

-1 5974 

-1 5182 

-1 4429 

-1 3713

-1 3034 

-1 2391 

-1 1782 

-1 1205 

-1 0660

-1 0144 

-9657 

-9197 

-8762 

-8351

-7963 

-7596

-7250 

-6922 

-6612

-6320

-6043

-5781 

-5532

933 

995

1033 

1053 

1056 

1048 

1030

1006 

976 

942 

906 

868

830 

791 

753 

716 

679

643 

609 

576 

545 

516

487 

460 

435 

411 

388

366 

347

328 

310 

293

277

262

248 

235

92 

62

38 

19

+ 4 

_ g

-18

-24 

-30 

-34 

-36 

-38

-39 

-38 

-38 

-37 

-37

-35 

-34 

-33 

-31 

-30

-28 

-27 

-26 

-24 

-23

-22 

-20

-19 

-18 

-17

-16

-14

-14 

-14

Q= 0.6; tabular gpacing--=0.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4.0 

4. 1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

5.0

5. 1

0. 3538 8614

. 3573 3019

. 3605 7956

. 3636 4985

. 3665 5508

. 3693 0794

.3719 1986

. 3744 0119

0*7 r**7 f*-\ o A . o7o/ olo4 

. 3790 0885

.3811 5147

.3831 9630

.3851 4980

.3870 1791

. 3888 0605

.3905 1920

.3921 6195

. 3937 3851

. 3952 5279

. 3967 0838

-2 1203

-1 9468

-1 7909

-1 6505

-1 5238

-1 4094

-1 3058

-1 2119

-1 1265 

-1 0488

-9780

-9132

-8540

-7997

-7499

-7040

-6618

-6229

-5869

-5536

1735

1560

1404

1266

1145

1036

940

854

777 

708

647

593

543

498

458

422

389

360

333

308

-198

-176

-156

-137

-122

-109

-96

-86

-77 

-69

-61

-55

-50

-45

-40

-36

-33

-30

-27

-25

TABLE 38. Modified potential Continued

a W Ai, Al,

Q= 0.6; tabular spacing 0.1 Continued

5.2 

5.3

5.4

5.5 

5.6 

5.7 

5.8 

5. 9

6.0 

6. 1 

6.2 

6.3 

6.4

6.5 

6.6 

6.7 

6.8 

6.9

7.0

7. 1 

7.2 

7.3

7.4

7.5 

7.6

7.7 

7.8 

7.9

8.0

8. 1 

8.2 

8.3 

8.4

8.5

8.6 

8.7 

8.8

8. 9

9.0 

9. 1

9.2

9.3

9.4

9.5

9.6

9.7

9.8

9. 9

10.0

0.3981 0860 

. 3994 5654 

. 4007 5506

. 4020 0681 

. 4032 1427 

. 4043 7974 

. 4055 0535 

. 4065 9311

. 4076 4490 

. 4086 6247 

. 4096 4744 

. 4106 0136 

. 4115 2567

. 4124 2172 

. 4132 9079 

. 4141 3407 

. 4149 5269

. 4157 4772

. 4165 2015 

. 41 72 7094 

. 4180 0098 

. 4187 1112 

. 4194 0216

. 4200 7486 

. 4207 2993 

. 4213 6806 

. 4219 8990 

. 4225 9606

. 4231 8713

. 4237 6366 

. 4243 2618 

. 4248 7520 

. 4254 1120

. 4259 3463

. 4264 4592 

. 4269 4551 

. 4274 3377

.4279 1110

XOQO 77QC

. tAOO 1 I OO

. 4288 3438

. 4292 8101

. 4297 1807

. 4301 4585

. 4305 6465

. 4309 7475

. 4313 7642

. 4317 6991

. 4321 5547

. 4325 3334

-5228 

-4942 

-4676

-4430 

-4200 

-3985 

-3785 

-3598

-3423 

-3259 

-3105 

-2961 

-2826

-2699 

-2579 

-2466 

-2359 

-2259

-2165 

-2075 

-1990 

-1910 

-1834

-1762 

-1694 

-1629 

-1567 

-1509

-1454

-1401 

-1350 

-1302 

-1257

-1213

-1171 

-1132 

-1094

-1057

 1O99 i\j44

-989

-958

-927

-898

-870

-844

-818

-793

-769

286 

266 

247

230 

214 

200 

187 

175

164 

154 

144 

135 

127

120 

113 

106 

100 

95

90 

85 

79 

76

72

68 

65 

62 

58 

55

53

51 

48 

45 

44

42

40 

38 

36

35

00

OO

31

31

30

28

26

26

25

24

23

Ai,

-22 

-20 

-19

-17 

-15 

-14 

-13 

-12

-11 

-10 

-10

-9

-8

-7 

-7 

-7 

-6 

-5

-4 

_ 5

-5 

-3

-4

-4

-3

_ 4.

-4 

-3

-2

-2 

-3

-2 

-2

-2

-2 

-2 

-1

-2

-2

-1

-1

-2

-1

_ i

-1

_ i

-1

Q = -0.6; tabular spacing=0.2

10.0

10. 2

10.4

10.6

10.8

11.0

11.2

11.4

11.6

11.8

0. 4325 3334

. 4332 6690

. 4339 7232

. 4346 5118

. 4353 0494

. 4359 3498

. 4365 4257

.4371 2888

. 4376 9501

. 4382 4198

-2987

-2815

-2656

-2509

-2372

-2246

-2128

-2018

-1916

-1820

172

159

147

136

127

118

110

102

95

90

-14

-13

-12

-10

-10

-9

-8

-7

_ >Y

 S

547123
70 APPAEENT EESISTIVITY OF A SINGLE UNIFOKM OVERBUKDEN

TABLE 38. Modified potential Continued

a W Ail Ai, *-*

Q= 0.6; tabular spacing = 0.2 Continued

12.0

12.2

12. 4

12.6

12.8

13.0

0. 4387 7075

. 4392 8221

. 4397 7720

. 4402 5650

. 4407 2084

. 4411 7092

-1731

-1647

-1569

-1496

-1427

84

78

73

69

65

-6

-6

"~-~ ^

-4

-4

Q= 0.6; tabular spacing=0.5

13.0

13.5

14.0

14.5

15.0

15.5

16.0

16.5

17.0

17.5

18.0

18.5

19.0

19. 5

20.0

20.5

21.0

21. 5

22.0

22.5

23.0

23. 5

24.0

24.5

25.0

25. 5

26.0

26. 5

27.0

27.5

28.0

28. 5

29.0

29.5

30.0

0. 4411 7092

. 4422 3785

. 4432 2867

. 4441 5123

. 4450 1236

. 4458 1799

. 4465 7332

. 4472 8291

. 4479 5080

. 4485 8055

. 4491 7534

. 4497 3800

. 4502 7108

. 4507 7682

. 4512 5730

. 4517 1436

. 4521 4966

. 4525 6473

. 4529 6093

. 4533 3954

. 4537 0170

. 4540 4844

. 4543 8075

. 4546 9950

. 4550 0551

. 4552 9952

. 4555 8222

. 4558 5426

. 4561 1623

. 4563 6868

. 4566 1211

. 4568 4700

. 4570 7380

. 4572 9291

. 4575 0471

-8523

-7611

-6825

-6144

-5550

-5030

-4574

-4170

-3814

-3496

-3213

-2959

-2732

-2527

-2342

-2175

-2024

-1886

-1760

-1645

-1540

-1444

-1356

-1274

 1200

-1130

-1066

-1007 -952

-901

-854

-810

-769

-730

912

786

682

594

519

457

403

357

317

284

254

227

205

185

167

152

138

126

115

105

96

88

81

75

69

64

59

55

51

48

44

41

39

36

-152

-126

-104

-88

-74

-62

-54

-46

-39

-34

-30

-26

-22

-20

-18

-16

-14

-12

-11

-10

-9

-8

-7

-6

-6

-5

""" O

_ 4

-4

-3

_ 4

-3

-3

-3

Q= 0.5; tabular spacing =0.02

0.00 

.02 

.04 

.06 

.08

.10 

.12 

. 14 

.16 

.18

.20 

.22

. 24

0. 0000 0000 

.0000 9449 

. 0003 7760 

. 0008 4829 

. 0015 0479

. 0023 4467 

. 0033 6486 

. 0045 6164 

.0059 3073 

. 0074 6730

.0091 6599 

. 0110 2102

. 01 30 2620

1 8898 

1 8862 

1 8757 

1 8582 

1 8338

1 8030 

1 7660 

1 7231 

1 6747 

1 6213

1 5634 

1 5014

1 4359

-36 

-106 

-175 

-243 -308

-370 

-429 

-484 

-534 -579

-620 -655

684

-71 

-70 

-70 

-68 

-65

-62 

-58 

-55 

-50 

-46

-40 

-35

an

TABLE 38. Modified potential Continued

a w Ait A!, Ai.

Q = 0.5; tabular spacing=0.02 Continued

0. 26

.28

.30

.32

.34

.36

.38

.40

.42

.44

.46

.48

.50

.52

.54

.56

.58

.60

.62

.64

.66

.68

.70

.72

.74

.76

.78

.80

. 82

. 84

.86

.88

.90

.92

. 94

.96

.98

1.00

1.02

1.04

1.06

1.08

1.10

1.12

1. 14

1.16 1 I Q

1. JLo

1. 20 

1.22 

1.24 

1.26 

1.28

1.30 

1.32 

1.34 

1.36 

1.38

1.40

0. 0151 7497

. 0174 6049

. 0198 7566

. 0224 1322

. 0250 6572

. 0278 2568

. 0306 8552

. 0336 3770

. 0366 7472

.0397 8914

. 0429 7368

.0462 2116

. 0495 2463

. 0528 7730

. 0562 7263

. 0597 0431

.0631 6628

. 0666 5275

.0701 5821

. 0736 7741

. 0772 0541

. 0807 3753

. 0842 6936

. 0877 9681

.0913 1601

.0948 2340

. 0983 1565

.1017 8970

. 1052 4274

. 1086 7215

.1120 7559

.1154 5091

. 1187 9614

. 1221 0955

. 1253 8956

. 1286 3477

.1318 4396

. 1350 1604

.1381 5008

.1412 4526

. 1443 0092

. 1473 1651

. 1502 9156

. 1532 2572

. 1561 1874

. 1589 7045

1 d'i *T OA^45 .1617 oU7o

. 1645 4964 

. 1672 7714 

. 1699 6337 

. 1726 0849 

. 1752 1271

. 1777 7629 

. 1802 9954 

. 1827 8278 

. 1852 2639 

. 1876 3076

. 1899 9632

1 3675

1 2966

1 2238

1 1496

1 0744

9989

9234

QJQA O^EOT:

7741

7011

6295

5598

4921

4266

3635

3029

2450

1899

1375

879

+ 412

-28

-440

-824

-1182

-1513

-1820

-2102

-2361

-2598

-2813

-3008

-3183

-3340

-3480

-3603

-3710

-3804

-3885

-3952

-4008

-4053

-4088

-4114

-4131

-4140

_ 4.1 4.9 TrJ. ^r &

-4138 

-4127 

-4111 

-4090 

-4064

-4034 

-4000 

-3964 

-3924 

-3881

-709

-728

-742

-751

-755

-755

-750

-742

-730

-715

-697

-677

-655

-631

-605

-579

-552

-524

-496

-468

-439

-412

-384

-358

-332

-307

-282

-259

-236

-215

-195

-175

-157

-140

-123

-108

-94

-81

-67

-56

-45

-35

-26

-17

-9

-2

T^O 1 e

11

16 

21 

26 

30

34 

37 

40 

42 

45

-25

-19

-14

-9

-4

0

+ 5

8

12

15

18

20

22

24

26

27

27

28

28

28

28

28

27

27

26

25

24

23

22

21

20

20

18

17

17

15

14

13

13

11

11

10

9

9

8

7

7

6 

6 

5 

5 

4

4 

3 

3 

2 

2
NUMERICAL VALUES OF THE MODIFIED POTENTIAL 71

TABLE 38. Modified potential Continued

a w Ali Ai, At.

Q = 0.5; tabular spacing =0.05

1.40

1.45

1.50

1. 55

1. 60

1. 65

1.70

1.75

1.80

1.85

1. 90

1.95

2.00

2.05

2. 10

2. 15

2.20

0. 1899 9632

. 1957 4343

. 2012 5832

. 2065 4905

.2116 2400

. 2164 9175

.2211 6093

. 2256 4014

. 2299 3784

. 2340 6231

. 2380 2158

. 2418 2345

. 2454 7541

. 2489 8466

. 2523 5809

. 2556 0227

. 2587 2348

-2 3971

-2 3221

-2 2417

-2 1578

-2 0720

-1 9856

-1 8998

-1 8151

-1 7323

-1 6519

-1 5741

-1 4991

-1 4271

-1 3582

-1 2924

-1 2297

749

804

839

858

863

859

846

828

804

778

750

720

689

658

627

596

79

55

35

18

+ 6

_ 4

-13

-18

-23

-26

-29

-30

-31

-31

-31

-30

Q = 0.5; tabular spacing =0.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3.0

3. 1

3.2

3. 3

3.4

3. 5 

3.6 

3.7 

3.8 

3.9

4.0

4 1 

4! 2

4. 3

44

4.5 

4 6 

4.7

48 

49

5.0

5. 1

5. 2

5. 3

5. 4

5.5

5.6

5.7

5.8

5.9

0. 2587 2348

. 2646 2054

. 2700 9347

. 2751 8206

. 2799 2198

. 2843 4516

. 2884 8021

. 2923 5271

. 2959 8556

. 2993 9928

. 3026 1228

. 3056 4105

. 3085 0047

.3112 0390 

. 3137 6341 

. 3161 8990 

. 3184 9325 

. 3206 8239

. 3227 6544

. 3247 4981 

.3266 4223

. 3284 4886

. 3301 7530

. 3318 2674 

. 3334 0788 

. 3349 2308

. 3363 7634 

. 3377 7134

. 3391 1149

. 3403 9993

. 3416 3958

. 3428 3314

. 3439 8311

. 3450 9182

. 3461 6145

.3471 9401

. 3481 9138

. 3491 5532

-4 6834

-4 2412

-3 8434

-3 4868

-3 1673

-2 8813

-2 6255

-2 3965

-2 1913

-2 0073

-1 8422

-1 6936

-1 5598

-1 4392 

-1 3302 

-1 2315 

-1 1420 

-1 0608

-9869

_ Q1 yiyoQC

QCQfl

-8017

-7502

-7029 

-6594 

-6194

-5826 

-5485

-5170

-4879

-4610

-4358

-4126

-3909

-3707

-3519

-3343

-3178

4421

3978

3567

3195

2860

2558

2290

2052

1840

1652

1485

1338

1206

1090

987 

895 

812 

739

674

a i K olo 

562

516

473

434 

400 

369

340 

315

291

270

251

233

217

202

188

176

164

154

-473

-444

-411

-372

-335

-301

-268

-238

-212

-188

-166

-147

-132

-116 

-103 

-92 

-83 

-73

-65

KQ

-52

-47

-43

-38 

-34 

-32

-28 

-25

-24

-21

-19

-18

-16

-15

-14

-12

-12

-10

TABL& 38. Modified potential Continued

a w Ai, Aii Ai,

Q = 0.5; tabular spacing =; 0.1 Continued

6.0

6.1

6.2

6. 3

6.4

6.5

6.6

6.7

6.8

6. 9

7.0

7.1

7.2

7.3

7.4

7.5

7.6

7.7

7.8

7.9

8.0

8.1

8.2

8.3

8.4

8.5

8.6

8.7

8.8

8.9

9.0

9.1

9.2

9.3

9.4

9.5 

9.6 

9.7 

9.8 

9.9

10.0

0. 3500 8749

. 3509 8941

. 3518 6253

. 3527 0820

. 3535 2769

. 3543 2220

. 3550 9284

. 3558 4067

. 3565 6668

. 3572 7182

. 3579 5696

. 3586 2295

. 3592 7057

. 3599 0057

. 3605 1366

. 3611 1051

. 3616 9175

. 3622 5799

. 3628 0979

. 3633 4771

. 3638 7226

. 3643 8392

. 3648 8318

. 3653 7046

. 3658 4621

.3663 1082

. 3667 6467

. 3672 0814

. 3676 4158

. 3680 6532

. 3684 7969

. 3688 8499

. 3692 8151

. 3696 6954

. 3700 4936

. 3704 2120 

. 3707 8533 

.3711 4198 

. 3714 9138 

. 3718 3374

. 3721 6929

-3024

-2880

-2745

-2618

-2499

-2387

-2281

-2182

-2088

-1999

-1916

-1837

-1762

-1691

-1624

-1560

-1500

-1443

-1389

-1337

-1288

-1241

-1197

-1154

-1114

-1075

-1038

-1003

-969

-938

-907

-878

-849

-822

-796

-772 

-748 

-725 

-703 

-682

144

135

127

119

112

106

99

94

88

84

79

75

71

67

64

60

57

55

52

49

47

45

43

40

38

37

35

34

32

31

29

28

27

26

25

24 

23 

22 

21 

20

-10

 Q<y

_ O

__ Q

-7

-6

-6

-6

-6

-4

-5

-4

-4

-4

-4

-4

-3

_ 3

-3

-3

-2

_ 2

_ 2

-2

_ 2

 1

_ 2

-1

-2

-1

-1

__ I

__ 1

-1

-1

-1 

-1

-1 

_ i

-1

Q= 0.5; tabular spacing =0.2

10.0

10.2

10.4 

10.6 

19. 8

11.0 

11.2

11.4

11.6

11. 8

12.0

12.2

12.4

12.6

12.8

13.0

0. 3721 6929

. 3728 2070

. 3734 4714 

. 3740 5002 

. 3746 3063

. 3751 9020 

. 3757 2983

. 3762 5059

. 3767 5343

. 3772 3927

. 3777 0895

. 3781 6328

. 3786 0297

. 3790 2874

. 3794 4124

. 3798 4107

-2649

-2497

-2357 

-2226 

-2105

-1993 

-1888

-1791

-1700

-1616

-1536

-1462

-1393

-1328

-1267

152

140

131 

121 

112

104 

97

91

85

79

74

70

65

61

58

-13

-12

-10

-10 

_ g

-8

_ «

-7

-6

-6

_ PJ

-4

-4

-4

-3
72 APPARENT RESISTIVITY OP A SINGLE UNIFORM OVERBURDEN

TABLE 38. Modified potential Continued

a W AI, Ai, Ai,

Q= 0.5; tabular spacing =0.5

13.0 

13.5 

14.0

14.5

15.0 

15.5 

16.0 

16.5

17.0

17.5 

18.0 

18.5 

19.0

19.5

20.0 

20.5 

21.0 

21.5

22. 0

22.5 

23.0 

23.5 

24.0

24. 5

25.0 

25.5 

26.0 

26.5

27.0

27.5 

28.0 

28. 5 

29.0

29.5

30.0

0. 3798 4107 

. 3807 8891 

. 3816 6917

. 3824 8882

. 3832 5392 

. 3839 6973 

. 3846 4086 

. 3852 7137

. 3858 6484

.3864 2443 

. 3869 5296 

. 3874 5296 

. 3879 2667

.3883 7610

. 3888 0308 

. 3892 0926 

. 3895 9610 

.3899 6497

. 3903 1708

. 3906 5355 

.3909 7540 

. 3912 8357 

. 3915 7890

. 3918 6219

. 3921 3415 

. 3923 9545 

. 3926 4670 

. 3928 8849

.3931 2132

. 3933 4568 

. 3935 6204 

. 3937 7080 

. 3939 7238

.3941 6712

. 3943 5537

-7567 

-6758 

-6061

-5456

-4929 

-4468 

-4062 

-3704

-3388

-3105 

-2854 

-2629 

-2427

-2245

-2081 

-1933 

-1798 

-1676

-1564

-1462 

-1369 

-1283 

-1205

-1132

-1066 

-1005

-948 

-895

-846

-801 

-759 

-720 

-683

-649

809 

697 

605

527

461 

405 

358 

317

282

251 

225 

202 

182

164

148 

135 

122 

112

102

93

85 

78

72

66

62

57 

52 

49

46

42 

39 

37 

34

32

-134 

-111 

-93

-78

-66 

-56 

-48 

-41

-35

-31 

-26 

-23 

-20

-18

-16 

-14 

-13 

-11

-10

-9 

-8

-7 

-6

-6

-5

-5 

-4

-3

-3 

_ 3

-3

_ 2

_ 2

Q = 0.4; tabular spacing =0.02

0.00 

.02 

.04

.06

.08

. 10

. 12 

. 14

. 16

.18

.20

. 22

.24

. 26

.28

.30

.32

.34

.36

.38

.40

.42

0. 0000 0000 

. 0000 7638 

. 0003 0525 

. 0006 8575

. 0012 1648

. 0018 9549

. 0027 2030 

. 0036 8795

. 0047 9498

. 0060 3753

. 0074 1130

.0089 1165

. 0105 3360

. 0122 7189

.0141 2103

. 0160 7534

.0181 2895

. 0202 7592

. 0225 1020

. 0248 2574

. 0272 1647

. 0296 7634

1 5277 

1 5248 

1 5163 

1 5023

1 4828

1 4580

1 4283 

1 3939

1 3551

1 3122

1 2657

1 2160

1 1635

1 1085

1 0516

9931

9335

8732

8125

7519

6915

6319

-29 

-85 

-140 

-195

-247

2Q7

-344 

-388

-428

-465

-497

-526

-550

-569

-585

-596

-603

-607

-607

-603

-596

-587

-57 

-56 

-56 

-54

-52

Cfl ou

-47 

-44

-40

-36

-32

-29

-24

-20

-16

-11

 7

-3

0

+ 3

7

9

TABLE 38. Modified potential Continued

a W ^

AL, Ai,

Q = -0.4; tabular spacing=0.02 Continued

0. 44 

.46

.48

.50

.52

.54 

.56

.58

. 60

.62 

.64 

.66 

.68

.70

.72 

.74 

.76 

.78

.80

.82 

.84 

.86 

.88

.90

.92 

.94 

.96 

.98

1.00

1.02 

1.04 

1.06 

1.08

1. 10

1. 12 

1. 14

1. 16

1. 18

L22 

1. 24 

1.26 

1.28

1.30

0. 0321 9941 

. 0347 7979 

.0374 1174

. 0400 8964

. 0428 0805 

.0455 6170 

. 0483 4550 

. 0511 5458

. 0539 8428

. 0568 3015 

. 0596 8797 

.0625 5374 

. 0654 2368

. 0682 9425

. 0711 6212 

. 0740 2418 

. 0768 7753 

. 0797 1950

. 0825 4758

. 0853 5949 

. 0881 5313 

. 0909 2658 

. 0936 7807

. 0964 0602

.0991 0900 

. 1017 8571 

. 1044 3502 

. 1070 5591

. 1096 4747

.1122 0894 

. 1147 3964 

. 1172 3901 

.1197 0656

. 1221 4192

. 1245 4478 

. 1269 1490

. 1292 5212

. 1315 5635

1 OO Q O'TC A . looo .£/o4 

. 1360 6572 

. 1382 7094 

. 1404 4331 

. 1425 8298

. 1446 9014

5732 

5156 

4596

4051

3524 

3016 

2528 

2062

L617

1195 

795 

418 

+ 63

-270

-581 

-870 

-1139 

-1388

-1617

-1827 

-2020 

-2195 

-2354

-2497

-2626 

-2741 

-2842 

-2932

-3010

-3077 

-3134 

-3181 

-3220

-3250

-3273 

-3290

-3300

-3303

-3296 

-3285 

-3270 

-3251

-575 

-561 

-545

-527

-508

-488 

-466 

-445

-422

-400 

-377 

-355 

-333

-311

-289 

-269 

-249 

-229

-211

-192 

-175 

-159 

-144

-129

-115 

-102 

-89 

-78

-67

-57 

-47 

-39 

-30

-23

-16 

-10

-4

+ 1

11 

15 

19 

22

12 

14 

16

18

19 

21 

21 

22

22

23 

22 

22 

22

22

22 

21 

20 

20

18

18 

17 

16 

15

15

14 

13 

12 

12

11

10 

10 

9

8

7

7 

7

6

5

5 

4 

4 

3 

3

Q = 0.4 ; tabular spacing =0.05

1.30

1.35

1.40

1.45

1.50

1.55

1. 60

1.65

1.70

1.75

1. 80

1.85

1. 90

1.95

0. 1446 9014

. 1498 1733

. 1547 4698

. 1594 8419

. 1640 3474

. 1684 0484

. 1726 0101

. 1766 2996

. 1804 9844

. 1842 1320

. 1877 8090

.1912 0806

. 1945 0104

. 1976 6601

-2 0174

-1 9754

-1 9243

-1 8667

-1 8045

-1 7392

-1 6723

-1 6047

-1 5372

-1 4706

-1 4054

-1 3418

-1 2801

-1 2207

420

510

576

622

652

670

676

674

666

653

636

616

595

572

119

90

66

46

30

17

+ 6

-2

-8

-13

-17

-20

-22

-23
NUMERICAL VALUES OP THE MODIFIED POTENTIAL 73

TABLE 38. Modified potential Continued

a w Aii Ai, Ai,

Q= 0.4; tabular spacing=0.05 Continued

2.00

2.05

2. 10

2. 15

2. 20

0. 2007 0891

. 2036 3546

. 2064 5115

. 2091 6121

. 2117 7064

-1 1635

-1 1087

-1 0562

-1 0062

548

524

500

478

-24

-24

-24

-22

Q = 0.4; tabular spacing=0.1

2.2

2.3

2. 4

2.5

2.6

2.7

2.8 

2.9

3.0 

3.1 

3.2

3.3 

3.4

3.5 

3.6 

3.7

3.8 

3. 9

4.0

4. 1

4. 2

4.3

4. 4

4.5

4.6

4.7

4.8

4.9

5.0

5. 1

5.2

5.3

5. 4

5.5

5.6

5.7

5.8

K Q o. y

6.0

6. 1

6. 2

6.3

6.4

6. 5

6.6

6.7

6. 8

6.9

7.0

7.1

7.2

7.3

7.4

0. 2117 7064

. 2167 0651 

. 2212 9416

. 2255 6556 

. 2295 4945 

. 2332 7164

. 2367 5529 

. 2400 2118

. 2430 8795 

. 2459 7234 

. 2486 8941

. 2512 5270 

. 2536 7440

. 2559 6553 

. 2581 3604 

. 2601 9493

. 2621 5037 

. 2640 0978

. 2657 7991

. 2674 6690

. 2690 7636

. 2706 1343

. 2720 8280

. 2734 8877

. 2748 3532

. 2761 2607

. 2773 6439

. 2785 5336

. 2796 9585

. 2807 9450

. 2818 5177

. 2828 6993

. 2838 5108

. 2847 9720

. 2857 1011

. 2865 9152

. 2874 4303

OQOO GG 1 1 . ZooZ 6611

. 2890 6215

. 2898 3247

. 2905 7828

. 2913 0072

. 2920 0088

. 2926 7975

. 2933 3830

. 2939 7741

. 2945 9794

. 2952 0068

. 2957 8637

. 2963 5573

. 2969 0943

. 2974 4810

. 2979 7235

-3 8363

-3 4821 

-3 1626

-2 8751 

-2 6170 

-2 3854

-2 1776 

-1 9912

-1 8238 

-1 6733 

-1 5378

-1 4158 

-1 3057

-1 2062 

-1 1162 

-1 0345

-9603

-8928

-8314

-7753

-7240

-6770

-6339

-5943

-5579

-5244

-4935

-4648

-4384

-4139

-3911

-3700

-3504

-3321

-3150

-2991

-2842

97HQ £/l Uo

-2573

-2451

-2336

-2229

-2128

-2033

-1943

-1859

-1779

-1704

-1633

-1566

-1503

-1442

-1386

3542

3195

2874

2581 

2316 

2078

1865

1674

1505 

1354 

1220

1101 

995

901 

817 

742

675 

615

561

513

470

431

396

364

335

309

286

265

245

228

211

196

183

171

159

148

139

1OU i 3fi

122

114

108

101

95

90

84

79

75

71

67

63

60

57

54

-369

-347 

-321

-293 

-265 -238

-213 

-190

-169 

-151 

-134

-119 

-106

-94

-84

-75

-67 

-60

-53

-48

-43

-39

-35

-32

-30

-26

-23

-21

-20

-17

-16

-15

-13

-12

-12

-11

-9

-9

-8

-8

_ IT

-7

-5

-5

-5

-5

-4

-4

-4

-4

-3

_ g

_ g

TABLE 38. Modified potential Continued

a w Ai, Ai, Als

Q= 0.4; tabular spacing=0.1 Continued

7.5

7.6

7.7

7.8

7.9

8.0 ai

8. 2

& Qo

8.4

8.5 

8. 6

8.7 

8.8 

8. 9

9.0 

9. 1

9.2 

9. 3 

9. 4

9. 5 

9.6

9.7 

9.8 

9. 9

10.0

0. 2984 8274

. 2989 7982

. 2994 6410

. 2999 3606

. 3003 9617

. 3008 4486

. 3012 8256

. 3017 0967

onoi OCKC . o021 26ob

. 3025 3359

. 3029 3111 

. 3033 1944

. 3036 9891 

. 3040 6980 

. 3044 3242

. 3047 8702 

. 3051 3388

. 3054 7324 

. 3058 0534 

. 3061 3042

. 3064 4868 

. 3067 6036

. 3070 6564 

. 3073 6472 

. 3076 5779

. 3079 4502

-1331

-1280

-1232

-1186

-1141

-1099

-1060

-1022

-986 

-951

-918 

-887

-857 

-828 

-801

-775 

-750

-726 

-703 

-681

-660 

-639

-620 

-601 

-583

51

49

46

44

42

40

38

36

34 

33

31 

30

29 

27 

26

25

24

23 

22

21

20 

20

19

18 

17

-3

-3

-2

-2

-2

_ o

-2

-2

0

&

-1

-2 

-1

_ 1

-1 

_ i

-1 

-1

-1 

-1

-1

_ 1 

_ 1

-1

-1 

_ i

Q= 0.4; tabular spacing = 0.2

10.0

10.2

10. 4

10. 6

10.8

11.0

11. 2

11.4

11.6

11. 8

12. 0

12. 2

12.4

12. 6

12. 8

13. 0

0. 3079 4502

. 3085 0268

. 3090 3898

. 3095 5514

. 3100 5226

. 3105 3137

. 3109 9344

. 3114 3935

.3118 6994

. 3122 8599

. 3126 8822

. 3130 7730

. 3134 5386

. 3138 1851

. 3141 7180

. 3145 1425

-2265

-2135

-2015

-1904

-1800

-1705

-1615

-1532

-1454

-1382

-1315

-1251

-1192

-1136

-1084

130

120

111

103

96

89

83

78

72

68

64

59

55

52

49

-10

-10

_ g

-8

_ IT

-7

-6

-6

-6

-5

-4

_ 4

-4

-3

-3

Q = 0.4; tabular spacing =0.5

13.0

13. 5

14.0

14. 5

15.0

15.5

16. 0

16. 5

17.0

17.5

18. 0

18. 5

19.0

19. 5

0. 3145 1425

. 3153 2609

. 3160 8010

. 3167 8222

. 3174 3764

. 3180 5085

. 3186 2582

. 3191 6599

. 3196 7445

. 3201 5390

. 3206 0674

. 3210 3515

. 3214 4104

. 3218 2614

-6476

-5784

-5188

-4671

-4220

-3825

-3478

-3172

-2901

-2660

-2444

-2252

-2079

-1923

692

596

517

451

395

347

306

271

242

215

192

173

156

140

-115

-95

-79

-66

-56

-48

-41

-35

-30

-26

-23

-19

-17

-15
74 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

TABLE 38. Modified potential Continued

a W Ai, Ai, Ai,

<J = 0.4; tabular spacing =0.5 Continued

20.0

20.5

21.0

21. 5

22.0

22. 5

23.0

23.5

24.0

24. 5

25.0

25.5

26.0

26. 5

27.0

27.5

28.0

28. 5

29.0

29.5

30.0

0. 3221 9200

. 3225 4004

. 3228 7153

. 3231 8761

. 3234 8934

. 3237 7767

. 3240 5348

. 3243 1 756

. 3245 7064

. 3248 1341

. 3250 4647

. 3252 7040

. 3254 8572

. 3256 9292

. 3258 9245

. 3260 8473

. 3262 7015

. 3264 4907

. 3266 2182

. 3267 8871

. 3269 5005

-1783

-1656

-1540

-1435

-1340

-1253

-1173

-1099

-1032

-970

-913

-861

-812

-767

-725

-686

-650

-617

-585

-556

127

115

105

95

87

80

73

67

62

57

53

48

45

42

39

36

34

31

29

27

-13

-12

-10

-10

-8

-7

-7

-6

-6

-5

-4

-4

-3

_ Q

_ 0

_ 0

-2

-2

-2

Q=-0.3; tabular spacing =0.02

0.00

.02

.04

.06

.08

. 10 

. 12

. 14

. 16

. 18

.20

.22

. 24

. 26

. 28

.30 

.32

.34

. 36 

.38

.40

.42

.44

. 46

.48

.50

.52

. 54

.56

.58

.60

.62

.64

.66

.68

.70

. 72

0. 0000 0000

. 0000 5791

. 0002 3143 

. 0005 1991

.0009 2231

. 0014 3714 

. 0020 6257

. 0027 9634

. 0036 3586 

. 0045 7823

. 0056 2021

. 0067 5830

. 0079 8877

. 0093 0766 

. 0107 1082

. 0121 9399 

. 0137 5275

. 0153 8262

. 0170 7905 

. 0188 3748

. 0206 5334

. 0225 2208

. 0244 3921

. 0264 0030

. 0284 0102

. 0304 3713

. 0325 0453

. 0345 9923

. 0367 1741

. 0388 5539

. 0410 0963

.0431 7678

. 0453 5365

. 0475 3722

. 0497 2462

. 0519 1317

. 0541 0035

1 1582

1 1561

1 1497 

1 1391

1 1244

1 1058 

1 0835

1 0576

1 0284 

9961

9612

9237

8842

8428 

8000

7560 

7111

6656

6200 

5743

5288

4839

4396

3962

3539

3128

2731

2347

1979

1627

1291

972

669

384

+ 115

-137

-373

-21

-64

-106 -146

-186

-224 

-259

-292

-350

-374

-396

-414

-428 -440

-448 

-454

-457

-457 -454

-450

-442

-434

-423

-411

-398

-384

-368

-352

-336

-319

-302

-286

-269

-252

-236

-220

-43

-43

-42 

-41

-40

00 

DO 

OK 

DO

-33

-30

-27

-24

 22

-18

-15 

-12

-8 

-6

-2

0

+ 2

5

7

9

11

12

13

14

16

16

16

17

17

17

17

17

16

16

TABLE 38. Modified potential Continued

a w Ai, Al, Ai,

Q=-0.3; tabular spacing =0.02 Continued

0. 74

. 76

.78

.80

.82

.84

.86

.88

.90

.92

.94

.96

.98

1.00

1.02

1.04

1. 06

1.08

1. 10

0. 0562 8380

. 0584 6132

. 0606 3088

. 0627 9058

. 0649 3867

. 0670 7356

.0691 9378

. 0712 9798

. 0733 8495

. 0754 5359

. 0775 0292

. 0795 3204

. 0815 4018

. 0835 2662

. 0854 9079

. 0874 3214

. 0893 5023

. 0912 4468

.0931 1518

-593

-797

-986

-1160

-1320

-1467

-1601

-1723

-1833

-1932

-2020

-2099

-2168

-2229

-2281

-2326

-2364

-2395

-204

-189

-174

-160

-147

-134

-122

-110

-99

-88

-79

-69

-60

-53

-45

-38

-31

-25

16

15

15

14

13

13

12

12

11

10

10

10

9

8

8

7

6

6

Q= 0.3; tabular spacing=0.05

1.10

1. 15 

1. 20

1.25 

1.30

1.35

1.40 

1.45

1.50

1.55 

1.60

1.65 

1.70

1. 75

1.80 

1.85

1.90

1.95

2.00

2.05 

2. 10

0. 0931 1518

. 0976 8516 

. 1021 0161

. 1063 6390 

. 1104 7290

. 1144 3061

.1182 3997 

.1219 0459

. 1254 2861

. 1288 1652 

. 1320 7308

. 1352 0315 

. 1382 1170

. 1411 0368

. 1438 8399 

. 1465 5744

. 1491 2874

. 1516 0246

. 1539 8301

. 1562 7465 

. 1584 8144

1 ^Jl OQ

-1 5353 

-1 5415

-1 5330 

-1 5128

-1 4835

-1 4473 

-1 4060

-1 3610

-1 3136 

-1 2648

-1 2152 

-1 1657

-1 1167

-1 0685 

-1 0215

-9758

-9317

-8892

-8484

_ 044

-62

+ 85

202 

293

362

413 

450

474

489 

495

495 

490

482

470

457

442

425

408

391

005

182 

147

117 

91

69

51 

37

24

15

+ 6

0 -5

__ Q

-12

-14

-15

-16

-17

-17

Q= 0.3; tabular spacing = 0.1

2. 1

2. 2

2.3

2.4

2.5

2.6

2. 7

2.8

2.9

3.0

3. 1

3.2

3.3

3. 4

0. 1584 8144

. 1626 5601

. 1665 3596

. 1701 4786

. 1735 1575

. 1766 6132

. 1796 0411

. 1823 6167

. 1849 4980

. 1873 8274

. 1896 7328

. 1918 3295

. 1938 7215

. 1958 0026

-3 2386

-2 9462

-2 6805

-2 4401

-2 2232

-2 0279

-1 8522

-1 6942

-1 5520

-1 4240

-1 3087

-1 2047

-1 1108

-1 0259

2925

2657

2404

2169

1953

1757

1580

1422

1280

1153

1040

939

849

769

-273

-268

-254

-235

-216

-196

-176

-158

-142

-126

-113

-101

-90

-80
NUMERICAL VALUES OF THE MODIFIED POTENTIAL 75

TABLE 38. Modified potential Continued

a w All Ai, Ai,

Q = 0.3; tabular spacing =0.1 Continued

3.5

3.6

3.7

3.8

3.9

4.0

4. 1

4. 2

4.3

4. 4

4.5

4. 6

4.7

4.8

4. 9

5.0

5. 1

5.2 

5.3

5.4

5.5

5. 6

5.7

5.8

5. 9

6.0

6. 1

6.2

6.3

6.4

6.5

6. 6

6. 7

6.8

6. 9

7.0

7. 1

7.2

7.3

7.4

7.5

7. 6

7.7

7.8

7.9

8.0

8. 1

8.2

8.3

8.4

8.5

8.6

8.7

8.8

8.9

9.0

9. 1

9.2

9. 3

9.4

0. 1976 2579

. 1993 5643

. 2009 9914

. 2025 6027

. 2040 4558

. 2054 6031

. 2068 0928

. 2080 9685

. 2093 2704

. 2105 0352

. 2116 2969

. 2127 0863

. 2137 4322

. 2147 3610

. 2156 8970

. 2166 0627

. 2174 8791

. 2183 3654 

. 2191 5398

. 2199 4189

. 2207 0182

. 2214 3523

. 2221 4347

. 2228 2780

. 2234 8939

. 2241 2936

. 2247 4874

. 2253 4851

. 2259 2956

. 2264 9277

. 2270 3892

. 2275 6878

. 2280 8308

. 2285 8247

. 2290 6759

. 2295 3904

. 2299 9739

. 2304 4317

. 2308 7690

. 2312 9904

. 2317 1006

. 2321 1040

. 2325 0045

. 2328 8061

. 2332 5125

. 2336 1272

. 2339 6535

. 2343 0947

. 2346 4537

. 2349 7336

. 2352 9370

. 2356 0665

. 2359 1247

. 2362 1140

. 2365 0367

. 2367 8949

. 2370 6909

. 2373 4265

.2376 1038

. 2378 7245

-9490

-8792

-8158

-7582

-7057

-6577

-6139

-5738

-5370

-5032

-4722

-4436

-4172

-3928

-3702

-3494

-3300

-3120 

-2953

-2798

-2652

-2517

-2391

-2273

-2163

-2059

-1962

-1871

-1785

-1705

-1629

-1557

 1490

-1427

-1367

-1310

-1256

-1206

-1158

-1112

-1069

-1028 -989

-952

-917

-884

-852

-821

-792

-765

-738

-713

-689

-666

-644

-623

-603

-584

-565

-548

698

634

576

525

480

438

401

368

338

311

286

264

244

225

209

194

180

167 

155

145

135

126

118

110

104

97

91

85

80

76

72

67

63

60

57

54

50

48

46

43

41

39

37

35

33

32

31

29

28

26

25

24

23

22

21

20

19

18

17

17

-72

-64

-58

-51

-45

-42

-37

-33

-30

-27

 24

-22

-20

-18

-16

-15

-14

-13 

-12

-10

-10

-9

-8

-8

-6

-6

-6

-6

~"~ O

-4

-4

-4

-4

_ 3

-3

-3

__ Q

-2

-2

-2

__ Q

-2

_ 2

-2

_ o

-2

_ i

-2

_ 2

-1

 1

 1

-1

_ j

_ j

 1

_ i

-1

-1

0

TABLE 38. Modified potential Continued

a W Ai, Ai, Ai,

Q= 0.3; tabular spacing=0.1 Continued

9. 5

9. 6

9.7

9.8

9.9

10. 0

10. 1

10.2

10.3

10.4

10. 5

10.6

10.7

10. 8

10. 9

11.0

0. 2381 2904

. 2383 8032

. 2386 2646

. 2388 6762

. 2391 0393

. 2393 3554

. 2395 6260

. 2397 8524

. 2400 0358

. 2402 1774

. 2404 2785

. 2406 3402

. 2408 3635

. 2410 3496

. 2412 2995

. 2414 2140

-531

-514

-499

-484

-470

-456

-442

-430

-417

-406

-394

-383

-373

-362

-353

17

16

15

14

14

13

12

12

12

11

11

10

10

10

10

0

 1

 1

0

0

-1

0

0

0

_ i

0 -1

0

0

0

Q= 0.3; tabolar spacing =0.5

11.0

11. 5

12. 0

12. 5

13.0

13. 5

14.0

14.5

15.0

15.5

16. 0

16. 5

17.0

17.5

18.0

18.5

19.0

19. 5

20.0

20. 5

21.0

21. 5

22.0

22. 5

23.0

23. 5

24.0

24. 5

25.0

25. 5

26.0

26. 5

27. 0

27.5

28.0

28. 5

29.0

29.5

30.0

0. 2414 2140

. 2423 2900

.2431 6132

. 2439 2733

. 2446 3465

. 2452 8976

. 2458 9824

. 2464 6488

. 2469 9386

. 2474 8881

. 2479 5289

. 2483 8892

. 2487 9936

. 2491 8639

. 2495 5197

. 2498 9782

. 2502 2550

. 2505 3640

. 2508 3179

. 2511 1279

. 2513 8042

. 2516 3563

. 2518 7925

.2521 1206

. 2523 3475

. 2525 4798

. 2527 5234

. 2529 4837

. 2531 3656

. 2533 1738

. 2534 9125

. 2536 5856

. 2538 1969

. 2539 7496

. 2541 2468

. 2542 6916

. 2544 0867

. 2545 4344

. 2546 7373

-8596

-7528

-6630

-5870

-5220

-4664

-4183

-3767

-3404

-3086

-2806

-2559

-2340

-2146

-1972

-1817

-1678

-1552

-1439

-1336

-1243

-1158

-1081

-1011 -946

-887

-833

-783

-738

-695

-656

-619

-585

-554

-525

-498

-473

-449

1068

898

761

649

557

480

417

363

318

280

247

218

194

174

155

140

126

113

103

93

85

77

70

65

59

54

50

46

43

39

37

34

31

29

27

25

24

22

-211

-170

-138

-112

-92

-76

-64

-54

-45

-38

-33

-29

-24

-21

-18

-16

-14

-12

-11

-10

__ 0

__ 0

-6

-6

-6

-5

-5

-4

. _ o

-4

_ 2

-3

_ 2

-2

-2

-2

-1

-2
76 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

TABLE 38. Modified potential Continued

a w Ai, Ai, Ai,

Q=-0.2; tabular spacing*^ 0.02

0.00

.02

.04

.06

.08

. 10

. 12

. 14

. 16

. 18

.20

. 22

. 24 

.26

.28

.30

.32

.34

.36

.38

.40

. 42

.44

.46

. 48

.50

.52

.54

.56

.58

.60

.62

. 64

. 66

.68

.70

0. 0000 0000

. 0000 3904

. 0001 5603

. 0003 5053

. 0006 2184

. 0009 6898

. 0013 9070

. 0018 8551

. 0024 5167

. 0030 8724

. 0037 9006

. 0045 5779

. 0053 8792 

. 0062 7780

. 0072 2468

. 0082 2569

. 0092 7787

. 0103 7822

. 0115 2372

. 0127 1128

. 0139 3787

. 0152 0042

. 0164 9593

. 0178 2144

. 0191 7404

. 0205 5089

. 0219 4924

. 0233 6642

. 0247 9985

. 0262 4707

. 0277 0571

. 0291 7351

. 0306 4833

. 0321 2813

. 0336 1099

. 0350 9511

7809

7794

7752

7681

7583

7458

7309

7136

6940

6725

6491

6240

rrvTfi o97o 

5699

5412

5118

4818

4514

4208

3901

3597

3296

3000

2709

2425

2150

1883

1626

1378

1142

916

702

498

306

125

-14

-43

-71

-98

-124

-149

-173

-195

-216

-234

-250

-264

-277 

-287

-295

-300

-304

-306

-306

-304

-301

-296

-290

-284

-276

-267

-257

-247

-237

-226

-214

-203

-192

-181

-170

-29

-28

-28

-28

-26

-25

-24

-22

-20

-18

-16

-14

 19\.£i

-10

-8

 o

_ 4.

-2

0

+ 2

3

5

6

7

8

9

10

10

11

11

11

11

11

11

11

Q=-0.2; tabular spacing =0.05

0.70

.75

. 80

. 85

.90

.95

1. 00

1. 05

1. 10

1. 15

1. 20

1. 25

1.30

1.35

1.40

1. 45

1.50

1.55

1.60

1. 65

1. 70

0. 0350 9511

. 0387 9999

. 0424 7876

.0461 1183

. 0496 8322

.0531 8027

. 0565 9320

.0599 1475

. 0631 3978

. 0662 6494

. 0692 8842

. 0722 0959

. 0750 2883

. 0777 4732

. 0803 6684

. 0828 8968

.0853 1844

. 0876 5600

. 0899 0543

. 0920 6987

. 0941 5255

-250

-2610

-4572

-6167

-7434

-8412

-9139

 9652

-9986

-1 0169

-1 0231

-1 0193

-1 0075

-9896

-9669

-9407

-9120

-8814

-8498

-8177

-7854

-2360

-1962

-1595

-1267

-978

-727

-513

-333

-184

-61

+ 38

117

179

227

262

288

305

316

321

322

320

425

399

366

329

289

250

214

180

149

122

100

79

62

48

35

26

18

11

5

+ 1

-2

TABLE 38. Modified potential Continued

a w *, Ai, Ala

Q=-0.2; tabular spacing =0.05 Continued

1. 75

1.80

1. 85

1.90

1. 95

2.00

2.05

2. 10

2. 15

2.20

2.25

2.30

0.0961 5668

. 0980 8547

. 0999 4207

. 1017 2956

. 1034 5095

. 1051 0914

. 1067 0695

. 1082 4706

. 1097 3206

. 1111 6445

. 1125 4658

. 1138 8073

-7534

-7219

-6911

-6610

-6320

-6039

-5769

-5510

 5262

-5025

-4799

315

308

300

291

280

270

259

248

237

226

216

-5

_ y

-8

-10

-10

-10 -11 -11

 11 -11

-10

Q- 0.2; tabular spacing=0.1

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3.0

3. 1

3.2

3.3

3.4

3.5

3.6

3. 7

3.8

3.9

4.0

4. 1

4.2

4.3

4.4

4. 5 

4. 6

4.7

4.8

4.9

5.0

5. 1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9

6.0

6. 1

6.2

6.3

6.4

6.5

6.6

6.7

6. 8

6.9

0. 1138 8073

. 1164 1359

. 1187 7907

. 1209 9171

. 1230 6462

. 1250 0964

. 1268 3743

. 1285 5762

. 1301 7891

. 1317 0914

. 1331 5540

. 1345 2411

. 1358 2109

. 1370 5163

. 1382 2052

. 1393 3214

. 1403 9045

. 1413 9910

. 1423 6141

. 1432 8042

. 1441 5892

. 1449 9948

. 1458 0444 

. 1465 7598

. 1473 1610

. 1480 2664

. 1487 0932

. 1493 6572

. 1499 9729

. 1506 0541

. 1511 9134

. 1517 5626

. 1523 0126

. 1528 2736

. 1533 3553

. 1538 2665

. 1543 0155

. 1547 6102

. 1552 0580

. 1556 3657

. 1560 5397

. 1564 5861

. 1568 5107

. 1572 3188

. 1576 0155

. 1579 6056

. 1583 0935

-1 8342

-1 6737

-1 5285

-1 3973

-1 2790

-1 1722

-1 0760

-9890

-9106

-8397

-7755

-7173

-6644

-6165

-5728

-5330

-4966

-4634

-4330

-4051

-3795

-3559

-3342 

-3142

-2957

-2786

-2628

-2482

-2346

 2219

-2101

-1992

-1890

-1794

-1705

 1621

-1543

-1470

-1401

-1336

-1276

-1218

-1165

-1114

-1066

-1021

-979

1605

1452

1312

1183

1067

963

869

784

709

642.

582

528

480

437

398

364

332

304

279

256

236

217

200

185

171

158

146

136

127

118

110

102

96

89

83

78

74

69

64

61

57

54

51

48

45

42

40

-164

-153

-140

-128

-116

-104

-94

-85

-75

-67

-60

-54

-48

-43

-38

-35

-31

-28

-25

-23

-20

-19

-17 

-15

-14

-13

-12

-10

-10

-9

-8

-7

-7

-6

-6

-5

-4

-5

_ 4

_ 4

-4

-4

_ __ 0

_ 3

_ 3

-3

-2
NUMERICAL VALUES OF THE MODIFIED POTENTIAL 77

TABLE 38. Modified potential Continued

a w Ai, Ai, At,

|Q = -0.2; tabular spacing=0.1 Continued

7.0

7. 1

7.2

7.3

7.4

7.5

7. 6

7.7

7.8

7.9

8.0

8. 1

8.2

8.3

8.4

8.5

8.6

8.7

8. 8

8.9

9.0

9. 1

9. 2

9. 3

9.4

9.5

9. 6

9.7

9.8

9.9

10.0

10. 1

10.2

10.3

10.4

10.5

10. 6

10.7

10.8

10.9

11.0

0. 1586 4836

. 1589 7799

. 1592 9862

. 1596 1062

. 1599 1431

. 1602 1004

. 1604 9810

. 1607 7879

. 1610 5239

. 1613 1915

. 1615 7934

. 1618 3320

. 1620 8094

. 1623 2278

. 1625 5895

. 1627 8962

. 1630 1499

. 1632 3524

. 1634 5053

. 1636 6105

. 1638 6693

. 1640 6834

. 1642 6542

. 1644 5830

. 1646 4711

. 1648 3199

. 1650 1306

.1651 9042

. 1653 6420

. 1655 3450

. 1657 0142

. 1658 6506

. 1660 2552

. 1661 8289

. 1663 3726

. 1664 8871

. 1666 3733

. 1667 8319

. 1669 2637

. 1670 6694

. 1672 0497

-938

-900

-864

-830

-797

-766

-737

-709

-683

-658

-634

-611

-589

-569

-549

-530

-512

-495

-478

-463

-448

-433

-420

-406

-394

-382

-370

-359

-348

-338

-328

-318

-309

-300

-292

-283

-276

-268

-261

-254

38

36

34

33

31

29

28

26

25

24

22

22

21

20

19

18

17

17

16

15

14

14

13

13

12

12

11

11

10

10

9

9

9

9

8

8

8

7

7

6

_ 2

-2

-2

-1

-2

 1

_ 2

 1

_ j

-1

-2

 1

_ j

_ j

-1

_ j

-1

-1

-1

_ j

0

_ j

0

0

_ 1

0

0

0

0

0

 1

0

0

0

0

-1

0

0

0

_ t

Q= 0.2; tabular sparing =0.5

11.0

11. 5

12.0

12. 5

13.0

13. 5

14.0

14. 5

15. 0

15. 5

16.0

16. 5

17. 0

17.5

18.0

18. 5

0. 1672 0497

. 1678 5936

. 1684 5955

. 1690 1200

. 1695 2218

. 1699 9475

. 1704 3371

. 1708 4253

. 1712 2419

. 1715 8133

. 1719 1622

. 1722 3087

. 1725 2707

. 1728 0640

. 1730 7025

. 1733 1987

-6186

-5420

-4774

-4228

-3761

-3360

-3015

-2715

-2453

 2224

-2023

-1845

-1688

-1548

-1423

-1311

767

645

547

467

400

346

300

262

229

201

178

158

140

125

112

101

-151

 122

-98

-80

-66

-55

-46

-38

-33

-28

-23

-20

-18

-15

-13

-11

TABLE 38. Modified potential Continued

a W Ai Ai, Ai,

Q= 0.2; tabular spacing =0.5 Continued

19.0

19. 5

20.0

20. 5

21.0

21. 5

22.0

22. 5

23.0

23.5

24.0

24. 5

25.0

25. 5

26.0

26.5

27.0

27.5

28.0

28. 5

29.0

29.5

30. 0

0. 1735 5639

. 1737 8080

. 1739 9402

. 1741 9686

. 1743 9005

. 1745 7428

. 1747 5015

. 1749 1822

. 1750 7899

. 1752 3293

. 1753 8046

. 1755 2199

. 1756 5786

. 1757 8840

. 1759 1394

. 1760 3474

. 1761 5107

. 1762 6318

. 1763 7129

. 1764 7561

. 1765 7633

. 1766 7364

. 1767 6772

-1210

-1120

-1038

-964

-897

-836

-780

-730

-683

-640

-601

-565

-532

-501

-473

-447

-422

-400

-379

-360

-341

-324

90

82

74

67

61

56

51

46

43

39

36

33

31

28

26

24

23

21

20

18

17

16

-10

-9

-8

-7

-6

-6

__ Q

-4

-4

_ 4.

-3

-3

-3

-2

-2

-2

_ 2

-2

-1

-1

_ i

__ i

Q= 0.1; tabular spacing =0.02

0.00

.02

.04

.06

.08

. 10

.12

. 14

.16

. 18

.20

. 22

. 24

.26

.28

.30

.32 

.34

.36

.38

.40

.42

.44

.46

.48

.50

. 52

. 54

.56

.58

.60

. 62

.64

.66

.68

0. 0000 0000

. 0000 1975

. 0000 7893

. 0001 7733

. 0003 1459

. 0004 9022

. 0007 0360

. 0009 5397

. 0012 4047

. 0015 6211

. 0019 1783

. 0023 0643

. 0027 2667

. 0031 7722

. 0036 5669

. 0041 6364

. 0046 9659 

. 0052 5404

. 0058 3445

. 0064 3630

. 0070 5804

. 0076 9815

. 0083 5510

. 0090 2743

. 0097 1365

. 0104 1234

. 0111 2213

.0118 4166

. 0125 6965

. 0133 0484

. 0140 4605

. 0147 9214

. 0155 4202

. 0162 9467

. 0170 4912

3950

3943

3922

3886

3837

3774

3700

3613

3515

3407

3289

3164

3031

2892

2748

2600

2449 

2297

2143

1989

1836

1685

1536

1390

1247

1109

975

845

721

602

488

380

277

180

89

_ ijr

-21

-35

-49

-62

-75

-87

-98

-108

-117

-126

-133

-139

-144

-148

-151

-152 

-154

-154

-153

-151

-149

-146

-143

-138

-134

-130

-124

-119

-114

-108

-102

-97

-92

-86

-14

-14

-14

-14

-13

-12

-12

-12

-10

-9

-8

-7

-6

-5

-4

-3

_ 2 

-1

0

+ 1

2

2

3

4

4

4

4

5

5

5

6

6

6

6

6

547123 6C
78 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

TABLE 38. Modified potential Continued

a w Aii Ai, Ai,

Q= 0.1; tabular spacing =0.02 Continued

0.70 

.72

.74 

.76 

.78

.80

.82

.84 

.86 

.88

.90 

.92

.94 

.96 

.98

1.00 

1.02

1. 04 

1.06 

1.08

1. 10 

1. 12

1. 14 

1. 16 

1.18

1. 20 

1. 22

1. 24 

1. 26

1.28

1.30

1.32

1.34

1.36

1.38

1.40

1. 42

1. 44

1.46

1.48

1.50

1.52

1.54

1.56

1.58

1.60

1.62

1.64

1.66

1.68

1.70

1.72

1.74

1.76

1.78

1.80

1.82

1.84

1.86

1.88

1.90

0. 0178 0446 

.0185 5983

.0193 1442 

. 0200 6749 

. 0208 1833

. 0215 6630 

. 0223 1079

. 0230 5126 

. 0237 8720 

. 0245 1814

. 0252 4367 

. 0259 6339

. 0266 7697 

. 0273 8408 

. 0280 8446

. 0287 7785 

. 0294 6404

.0301 4283 

.0308 1407 

. 0314 7761

. 0321 3332 

. 0327 8112

. 0334 2093 

. 0340 5268 

. 0346 7634

. 0352 9186 

. 0358 9924

. 0364 9848 

. 0370 8958

. 0376 7256

. 0382 4746

.0388 1431

.0393 7316

. 0399 2406

. 0404 6708

. 0410 0228

. 0415 2973

. 0420 4951

.0425 6170

. 0430 6639

. 0435 6366

. 0440 5360

.0445 3631

.0450 1187

. 0454 8039

. 0459 4197

. 0463 9670

. 0468 4467

. 0472 8600

. 0477 2078

.0481 4912

.0485 7110

. 0489 8685

. 0493 9644

. 0497 9999

.0501 9759

.0505 8934

. 0509 7534

.0513 5568

. 0517 3046

. 0520 9979

+ 3 -78

-152 

-222 

-287

-347 

-402

-453 

-500 

-542

-580 

-615

-646 

-674 

-698

-720 

-740

-756 

-770 

-782

-792 

-799

-805 

-810 

-813

-814 -814

-814 

-812

-808

-805

-800

-795

-789

-782

-775

-767

-759

-750

-742

-733

-724

-714

-704

-694

-685

-675

-665

-655

-644

-635

-625

-614

-605

-595

-585

-575

-566

-556

-546

-80 

-75

-70 

-65 

-60

-55 

-51

-46 

-42 

-38

-35 

-31

-28 

-25 

-22

-19 

-16

-14 

-12 

-10

-8 

-6

-5 

-3

-1

0

+ 1

2 

3

4

5

6

6

7

7

8

8

8

9

9

9

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

10

5 

6

5 

5 

5

5

4

4 

4 

4

4 

3

4 

3 

3

3 

3

2 

2 

3

2 

2

1 

2 

2

1 

1

1 

1

0

1

1

0

1

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

TABLE 38. Modified potential Continued

a w Ali Ai, Ai,

Q 0.1; tabular spacing = 0.1

1.9

2.0 

2. 1 

2.2

2.3

2.4

2.5 

2.6 

2.7

2.8 

2.9

3.0 

3. 1 

3. 2

3.3

3.4

3.5 

3.6 

3.7

3.8 

3.9

4.0 

4. 1

4.2

4.3 

4.4

4.5

4.6 

4.7

4.8

4. 9

5.0

5. 1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9

6.0

6.1

6.2

6.3

6.4

6.5

6.6

6.7

6.8

6.9

7.0

7. 1

7.2

7.3

7.4

7.5

7.6

7.7

7.8

7.9

0. 0520 9979

. 0538 6765 

.0555 1246 

. 0570 4467

. 0584 7393 

. 0598 0905

. 0610 5806 

. 0622 2826 

.0633 2622

. 0643 5792 

. 0653 2876

. 0662 4362 

. 0671 0692 

. 0679 2267

. 0686 9448 

.0694 2564

.0701 1914 

. 0707 7770 

. 0714 0380

.0719 9970 

. 0725 6744

.0731 0893 

. 0736 2588

.0741 1988

. 0745 9239 

. 0750 4474

. 0754 7817

. 0758 9381 

. 0762 9271

. 0766 7585

. 0770 4411

.0773 9834

. 0777 3931

. 0780 6773

. 0783 8428

. 0786 8958

. 0789 8420

. 0792 6870

. 0795 4357

. 0798 0930

. 0800 6631

. 0803 1504

. 0805 5587

. 0807 8917

. 0810 1527

. 0812 3451

.0814 4719

. 0816 5360

. 0818 5400

. 0820 4866

. 0822 3781

. 0824 2168

. 0826 0050

. 0827 7445

. 0829 4375

.0831 0856

. 0832 6907

.0834 2544

. 0835 7783

. 0837 2639

. 0838 7125

-1 3433

-1 2306 

-1 1259 

-1 0295

-9414 

-8611

-7882 

-7223 

-6626

-6086 

-5598

-5156 

-4756 

-4394

-4065 

-3766

-3494 

-3246 

-3021

-2815 

-2626

-2453 

-2295

-2149

-2016 

-1893

-1779

-1674 

-1577

-1487

-1404

-1326

-1254

-1188

-1125

-1067

-1013

-962

-915

-870

-829

-790

-753

-719

-687

-656

-627

-600

-575

-551

-528

-506

-486

-466

-448

-430

-414

-398

-383

-369

-356

1127

1047 

964 

882

803

728

660 

597 

540

488 

442

400 

362 

329

299 

272

248 

226 

206

189 

173

158 

146

134

123 

114

105

97 

90

83

78

72

67

62

58

54

51

47

44

42

39

36

34

32

30

29

27

25

24

23

22

20

20

18

18

17

16

15

14

13

13

-68

-80

-82 

-82

-79 

-74

-69 

-63 

-57

-52 

-46

-42 

-38 

-33

-30

-27

-24 

-22 

-20

-17 

-16

-15 

-13

-12

-11

_ Q {/

-9

-8

-7

-6

-6

-6

-5

-4

_ 4

-4

-4

-3

__ Q

-3

-2

-3

-2

-2

-2

_ j

 2

_ 2

-1

_ j

-1

 1

-1

_ j

-1

_ 1

. _ I

-1

-1

-1

i
NUMERICAL VALUES OF THE MODIFIED POTENTIAL 79

TABLE 38. Modified potential Continued

a w All A!, A!,

Q= 0.1; tabular spacing =0.1 Continued

8.0 

8. 1 

8.2 

8. 3 

8.4

8.5 

& 6 

8. 7 

8.8 

8. 9

9.0 

9. 1 

9. 2 

9. 3 

9.4

9.5 

9.6 

9. 7 

9.8 

9.9

10.0

0.0840 1256 

. 0841 5043 

. 0842 8500 

. 0844 1639 

. 0845 4470

. 0846 7003 

. 0847 9250 

. 0849 1219 

. 0850 2920 

.0851 4362

. 0852 5554 

. 0853 6503 

. 0854 7217 

. 0855 7703 

. 0856 7970

. 0857 8023 

. 0858 7869 

.0859 7515 

. 0860 6966 

. 0861 6228

. 0862 5308

-343 

-330 

-319 

-308 -297

-287 

-277 

-268 

-259 

-251

-242 

-235 

-228 

-220 

-213

-207 

-201 

-195 

-189 

-183

12 

12 

11 

11 

10

10 

9 

9

8 

8

8

7 

7 

7 

7

6 

6 

6 

6 

5

0 -1 -1

0

_ J

0 -1

0 -1

0

_ J

0 

0 

0 

0

0 

0 

0 

0 

0

Q= 0.1; tabular spacing =0.5

10.0

10. 5 

11.0 

11.5 

12. 0

12. 5

13.0

13.5

140

145

15.0 

15.5 

16.0 

16.5 

17.0

17.5

18.0

18. 5

19.0

19.5

20.0 

20. 5 

21.0 

21. 5 

22. 0

22. 5

23.0

23. 5

24. 0

245

25.0

25.5 

26.0 

26. 5 

27.0

27. 5

28.0

28. 5

29.0

29.5

30.0

0. 0862 5308

. 0866 8137 

. 0870 7112 

. 0874 2727 

. 0877 5398

. 0880 5476

. 0883 3256

. 0885 8991

. 0888 2900

. 0890 5168

. 0892 5959 

. 0894 5416 

. 0896 3662 

. 0898 0806 

. 0899 6947

.0901 2168

. 0902 6547

. 0904 0152

. 0905 3042

. 0906 5274

. 0907 6895 

. 0908 7951 

. 0909 8483 

. 0910 8525 

.0911 8112

. 0912 7274

. 0913 6039

. 0914 4431

. 0915 2475

. 0916 0190

. 0916 7598

. 0917 4716 

.0918 1560 

. 0918 8146 

. 0919 4489

. 0920 0602

. 0920 6497

. 0921 2185

.0921 7677

.0922 2983

.0922 8113

-4455

-3856 

-3359 

-2944 

-2594

-2298

-2044

-1827

-1640

-1477

 

1 OO C Ioo5 

-1210 

-1101 

-1004 

-919

-843

-775

-714

-659

-610

-566 

-525 

-489 

-455 

-425

-398

-372

-349

-328

-308

-290

-273

-258 

-244 

-230

-218

-207

-196

-186

-177

599

497 

415 

350 

296

253

217

188

163

142

124 

110 

96 

86 

76

68

61

55

49

44

40 

36 

33 

30 

28

25

23

21

20

18

16

15 

14 

13 

12

12

11

10

9

9

-130

-103 

-81 

-66 

-53

-43

-36

-30

-25

-21

-17 

-15 

-13 

 11

_ g

-8

-7

-6

-6

-5

-4 

-4 

-3 

-3 

-2

-2

-2

-2

-2

_ o

_ 2

 1 

 1 

 1 

 1

 1

-1

_ j

-1

-1

TABLE 38. Modified potential Continued

a

0.00 

.02 

.04 

.06 

.08

. 10 

. 12 

. 14 

. 16 

. 18

.20 

. 22 

. 24 

.26

.28

. 30 

.32 

. 34 

. 36 

. 38

.40 

.42

.44

.46

.48

.50 

.52 

.54 

. 56

.58

. 60

-w Ai, Ali Ai,

Q= -1-0.1; tabular spacing =0.02

0. 0000 0000

. 0000 2025 

. 0000 8093 

. 0001 8183 

. 0003 2259

. 0005 0271 

. 0007 2157 

. 0009 7841 

. 0012 7236 

. 0016 0242

. 0019 6752 

.0023 6646 

. 0027 9799 

. 0032 6077 

. 0037 5340

. 0042 7442 

. 0048 2235 

. 0053 9566 

. 0059 9282 

.0066 1228

. 0072 5248 

. 0079 1189

AAO K OOH£ . OOoo soyo

. 0092 8220

. 0099 9013

.0107 1130 

. 0114 4432 

. 0121 8782 

. 0129 4049

. 0137 0107

. 0144 6834

4050 

4043 

4022 

3986 

3936

3874 

3798 

3711 

3612 

3503

3385 

3258 

3125 

2985 

2840

2691 

2538 

2385 

2230 

2074

1920 

1767

1 C1 *7 Ibl7

1469

1325

1184 

1048 

917 

790

670

-7 

-21 

-36 

-50 

-63

-75

-87 

-99 

-109 

-118

-126 

-134 

-140 

-145 

-149

-152 

-154 

-155 

-156 

-154

-153 

-151

-148

-144

-140

-136 

-131 

-126 

-121

-115

-15 

-14 

-14 

-14 

-13

-13 

-12 

-12 

-10 

-9

-8 

-7 

-6 

o -4

-3 

_ 2

-1 

-1

+ 1

1 

2

4

4

4 

5 

5 

6

6

Q=+0.1; tabular spacing =0.05

0.60 

.65 

. 70

.75

.80

.85

.90

.95

1.00 

1.05 

1. 10 

1. 15 

1. 20

1. 25

1.30

1.35

1.40

1.45

1.50 

1.55 

1.60 

1.65

1.70

1. 75

1.80

1.85

1.90

0. 0144 6834 

. 0164 0837 

. 0183 6663

. 0203 2876

. 0222 8250

.0242 1761

.0261 2568

. 0279 9994

. 0298 3508 

. 0316 2707 

.0333 7296 

. 0350 7071 

.0367 1905

. 0383 1735

. 0398 6548

. 0413 6375

.0428 1278

.0442 1345

. 0455 6685 

. 0468 7421 

.0481 3687 

.0493 5622

.0505 3372

.0516 7085

. 0527 6907

. 0538 2985

. 0548 5464

3478 

1824

+ 387

-838

-1863

-2704

-3381

-3912

-4315 

-4610 

-4814 

-4941 

-5004

-5016

-4987

-4924

-4836

-4727

-4604 

-4471 

-4330 

-4185

-4038

-3890

-3744

-3599

-1654 

-1437 

-1225

-1025

-841

-676

-531

-404

-295 

-204 

-127 

-63 

-12

+ 30

63

88

108

123

133 

141 

145 

147

148

147

144

142

215 

217 

212

200

184

165

146

127

108 

91 

77 

64 

51

42

33

26

20

14

10

8 

4

+ 2

0

_ i

-2

-3
80 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

TABLK 38. Modified potential Continued

a -w All All Ats

Q=+0.1; tabular epacing=0.1

1.9

2.0

2. 1

2.2

2.3

2. 4

2.5

2.6

2.7

2.8

2. 9

3.0

3. 1

3.2

3.3

3.4

3. 5

3.6

3.7

3. 8

3.9

4.0

4. 1

4. 2

4.3

4. 4

4.5

4.6

4.7

4.8

4.9

5.0

5. 1

5. 2

5.3

5. 4

5.5

5.6

5. 7

5. 8

5.9

6.0

6. 1

6. 2

6.3

6. 4

6. 5

6.6

6.7

6.8

6.9

7.0

7. 1

7.2

7.3

7.4

7.5

7.6

7.7

7. 8

7.9

0. 0548 5464

. 0568 0188

. 0586 2165

.0603 2415

.0619 1881

.0634 1436

.0648 1874

. 0661 3925

. 0673 8253

. 0685 5461

.0696 6099

. 0707 0666

.0716 9617

. 0726 3367

.0735 2293

. 0743 6738

. 0751 7017

. 0759 3416

. 0766 6198

. 0773 5603

. 0780 1853

. 0786 5149

. 0792 5679

. 0798 3613

.0803 9109

. 0809 2314

. 0814 3364

.0819 2381

.0823 9482

. 0828 4776

. 0832 8360

. 0837 0328

. 0841 0766

. 0844 9754

. 0848 7368

. 0852 3676

.0855 8746

.0859 2637

. 0862 5407

.0865 7110

. 0868 7796

.0871 7513

. 0874 6305

.0877 4213

.0880 1278

.0882 7537

. 0885 3024

. 0887 7772

. 0890 1813

. 0892 5176

. 0894 7889

.0896 9978

. 0899 1469

.0901 2385

. 0903 2749

.0905 2582

. 0907 1904

. 0909 0735

. 0910 9093

. 0912 6995

. 0914 4458

-1 3834

-1 2747

-1 1728

-1 0783

-9912

-9115

-8388

-7724

-7120

-6570

-6070

-5616

-5202

-4824

-4480

-4166

-3880

-3617

-3376

-3156

-2953

-2767

-2596

-2437

-2291

-2156

-2032

-1916

-1808

-1709

-1616

-1530

-1450

-1375

-1305

-1239

-1178

-1121

-1067

-1017

-970

-925

-883

-844

-807

-772

-739

-707

-678

-650

-624

-599

-575

-552

-531

-511

-492

-473

-456

-439

-423

1087

1019

944

870

797

728

664

604

549

500

455

414

377

344

314

287

263

240

221

203

186

171

158

146

135

124

116

108

100

92

86

81

75

70

66

61

57

54

50

47

45

42

40

37

35

33

31

30

28

26

25

24

22

21

20

19

18

18

16

16

15

-56

-68

-75

-74

-73

-69

-64

-60

-55

-50

-45

-41

-37

-33

-30

-27

-24

-22

-20

-18

-17

-15

-13

-12

-11

-10

-9

-8

-8

-7

-6

-6

-6

-5

-4

-5

-4

-3

-4

-3

-2

_ g

-2

-3

-2

_ 2

-2

_ 2

-2

-2

-1

-1

-1

-1

-1

_ j

-1

-1

-1

 1

-1

TABLE 38. Modified potential Continued

a -W Al, Al, 4i,

Q=+0.1; tabular spacing =0.1 Continued

8.0

8.1

8.2

8.3

8.4

8.5

8.6

8.7

8.8

8. 9

9.0

9. 1

9.2

9. 3

9.4

9.5

9.6

9.7

9. 8

9. 9

10.0

0. 0916 1497

. 0917 8128

. 0919 4366

. 0921 0223

. 0922 5713

. 0924 0849

. 0925 5642

. 0927 0103

. 0928 4245

. 0929 8076

.0931 1608

. 0932 4849

. 0933 7808

. 0935 0496

. 0936 2920

. 0937 5088

. 0938 7007

. 0939 8686

. 0941 0132

. 0942 1352

. 0943 2351

-408

-394

-380

-367

-355

-342

-331

-320

-310

-300

-290

-281

-272

-264

-256

-248

-241

-233

-227

-220

14

14

13

13

12

11

11

11

10

10

9

9

9

8

8

8

7

7

7

6

-1

-1

_ i

0

-1

_ j

0

0 -1

0

0

0

0

_ J

0

0

0 -1

0

0

Q=+O.I; tabular spacing=0.5

10.0

10.5

11.0

11. 5

12.0

12. 5

13.0

13. 5

14.0

14. 5

15.0

15.5

16.0

16. 5

17.0

17. 5

18.0

18. 5

19.0

19. 5

20.0

20. 5

21.0

21. 5

22.0

22. 5

23.0

23.5

24.0

24. 5

25.0

25. 5

26.0

26. 5

27.0

0. 0943 2351

. 0948 4262

. 0953 1535

. 0957 4761

.0961 4437

. 0965 0981

. 0968 4748

. 0971 6042

. 0974 5125

. 0977 2221

. 0979 7528

. 0982 1216

. 0984 3436

. 0986 4319

. 0988 3983

. 0990 2530

. 0992 0054

. 0993 6636

. 0995 2351

. 0996 7264

. 0998 1435

. 0999 4918

. 1000 7762

. 1002 0012

. 1003 1707

. 1004 2884

. 1005 3578

. 1006 3818

. 1007 3633

. 1008 3048

. 1009 2089

.1010 0776

. 1010 9129

. 1011 7169

. 1012 4911

-5352

-4639

-4046

-3550

-3132

-2777

-2473

-2212

-1986

-1790

-1619

-1468

-1336

-1220

-1116

-1024

-941

-868

-802

-742

-688

-639

-595

-554

-518

-484

-453

-425

-399

-375

-353

- 333

-314

-297

-281

713

592

496

418

356

304

261

226

196

171

150

132

117

104

92

82

74

66

60

54

49

44

40

37

34

31

28

26

24

22

20

19

18

16

15

-153

-120

-97

-78

-63

-52

-43

-35

-30

-25

-21

-18

-15

-13

-11

-10

-9

-7

-6

-6

-5

-4

-4

_ 4

-3

-3

-3

-3

_ 2

_ 2

-2

-1

-1

-2

-1
NUMERICAL VALUES OF THE MODIFIED POTENTIAL 81

TABLE 38. Modified potential Continued

a -w Ai, Ai, Ai,

Q=+0.1; tabular spacing =0.5 Continued

27.5

28.0

28. 5

29.0

29.5

30.0

0. 1013 2373

. 1013 9569

. 1014 6513

. 1015 3218

.1015 9697

.1016 5960

-266

-252

-239

-227

-215

14

13

12

11

11

_ i

-1

-1

-1

_ i

Q=+0.2; tabular spacing =0.02

0.00

.02

.04

.06

.08

. 10

. 12

.14

.16

. 18

.20

. 22

 94. ^i^t

. 26 

.28

.30

.32 

.34 

.36 

.38

.40

.42 

. 44 

. 46 

.48

.50

.52 

.54 

.56 

.58

.60

.62 

.64 

.66 

.68

.70

0. 0000 0000

. 0000 4105

. 0001 6407

. 0003 6861

.0006 5396

. 0010 1914

. 0014 6287

. 0019 8364

. 0025 7971

. 0032 4908

. 0039 8958

. 0047 9884

. 0056 7430 

. 0066 1328 

. 0076 1299

. 0086 7050

. 0097 8282 

. 0109 4690 

.0121 5964 

. 0134 1793

. 0147 1866

. 0160 5871 

. 0174 3501 

. 0188 4454 

. 0202 8432

. 0217 5144

. 0232 4310 

. 0247 5653 

. 0262 8911 

. 0278 3828

. 0294 0161

. 0309 7677 

.0325 6153 

.0341 5379 

. 0357 5154

. 0373 5289

8210

8196

8153

8081

7982

7856

7704

7529

7331

7112

6875

6621

/JO KO OOOO

6072 

5781

5481

5176 

4866 

4555 

4243

3933

3625 

3322 

3025 

2735

2452

2178 

1914 

1660 

1416

1183

960 

749 

550 

361

-14

-43

-72

-99

-126

-152

-175

-198

-219

-237

-254

-268

 9R1 AoL

-291 

-300

-306

-309 

-312 

-312 

-311

-307

-303 

-297 

-290 

-283

-274

-264 

-254 

-244 

-233

-222

-211 

-199 

-189 

-178

-28

-29

-28

-28

-27

-26

-24

-23

-21

-18

-17

-14

_ 19x ^ 

-10 

-8

-6

_ 4. 

_ 2 

0

+ 1

3

4 

6

7 

8

9

10 

10 

10 

11

11

11 

12 

11 

11

Q=+0.2; tabular spacing =0.05

0.70

.75

.80

.85

.90

.95

1.00

1.05

1. 10

1. 15

1. 20

0. 0373 5289

. 0413 6069

. 0453 5542

. 0493 1628

. 0532 2611

. 0570 7109

. 0608 4030

. 0645 2537

.0681 2013

. 0716 2023

. 0750 2286

+ 1175

-1306

-3388

-5102

-6485

-7577

-8414

-9032

-9466

-9746

-9897

-2482

-2082

-1714

-1383

-1091

-837

-618

-434

-280

-151

-47

424

400

368

331

292

254

218

184

154

129

104

TABLE 38. Modified potential Continued

a -W Ai, Ai, Ai,

Q =4-0.2; tabular spacing=0.05 Continued

1.25

1.30

1.35

1.40

1.45

1.50

1. 55

1.60

1.65 

1.70

1.75

1. 80

1.85

1.90

1.95

2. 00

2.05

2. 10

2. 15

2.20

0. 0783 2653

. 0815 3075

. 0846 3591

. 0876 4305

. 0905 5376

. 0933 7002

. 0960 9414

. 0987 2863

. 1012 7617 

. 1037 3953

. 1061 2155

. 1084 2508

. 1106 5295

.1128 0798

.1148 9291

. 1169 1044

. 1188 6321

. 1207 5375

. 1225 8452

. 1243 5789

-9944

-9907

-9802

-9643

-9444

-9214

-8963

-8695

-8418 

-8134

-7850

-7566

-7285

-7009

-6739

-6477

-6223

-5977

-5740

+ 38

105

158

199

230

252

268

278

283 

285

284

281

276

270

262

254

246

237

229

84

68

53

41

31

22

16

10

6 

+ 1

-1

-3

-5

-6

_ 7

-8

-9

-8

_ o

Q=+0.2; tabular spacing= 0.1

2.2 

2. 3

2. 4

2. 5

2.6 

2.7 

2.8

2. 9

3.0 

3. 1 

3. 2 

3. 3

3.4

3. 5 

3.6 

3.7 

3.8

3. 9

4.0 

4. 1 

4.2 

4.3

4. 4

A K

T:. O

4. 6 

4.7

4.8

4.9

5.0

5. 1

5.2

5.3

5. 4

5.5

5. 6

5.7

5.8

5.9

0. 1243 5789 

. 1277 4149

. 1309 2177

. 1339 1454 

. 1367 3432 

. 1393 9436 

.1419 0676

. 1442 8259

. 1465 3191 

. 1486 6393 

. 1506 8704 

. 1526 0892

. 1544 3657

. 1561 7644 

. 1578 3438 

. 1594 1580 

. 1609 2565

. 1623 6850

. 1637 4852 

. 1650 6960 

. 1663 3528 

. 1675 4888

. 1687 1343

. 1698 3176

. 1709 0648 

.1719 4002

. 1729 3464

. 1738 9244

. 1748 1537

. 1757 0526

. 1765 6379

. 1773 9258

. 1781 9309

. 1789 6672

. 1797 1477

. 1804 3846

.1811 3895

. 1818 1730

-2 2054 

-2 0333

-1 8750

-1 7300 

-1 5974 

-1 4763 

-1 3658

-1 2650

-1 1730 

-1 0891 

-1 0124 

-9422

-8780

-8192 

-7652 

-7157 

-6701

-6282

-5895 

-5538 

-5209 

-4904

-4622

-4361

-4118 

-3892

-3682

-3487

-3305

-3135

-2976

-2827

-2688

-2558

-2436

-2321

-2214

-2112

1721 

1583

1450

1326 

1211 

1105 

1008

920

839 

767 

702 

642

588

539 

496 

456 

419

386

357 

329 

305

282

262

94 Q ^SrtG

226 

210

195

182

170

159

149

139

130

122

115

107

101

95

-142 

-138

-132

-124 

-115 

-106 

-97

-88

-80 

-72 

-66 

-60

-54

-49 

-44 

-40 

-37

-33

-30 

-27 

-25 

-22

-21

1O *7

-17 

-16

-14

-13

-12

-11

-10

-10

-8

-8

-8

_ rj

-6

-6
82 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

TABLE 38. Modified potential Continued

a -w Al, Al, Ai,

Q=+0.2; tabular spacing =0.1 Continued

6.0

6. 1

6. 2

6. 3

6. 4

6.5

6.6

6.7

6.8

6.9

7.0

7.1

7.2

7.3

7.4

7. 5

7.6

7.7

7.8

7. 9

8.0

8. 1

8. 2

8. 3

8.4

8.5

8.6

8.7

8.8

8.9

9.0

9. 1

9.2

9. 3

9.4

9.5

9. 6

9.7

9.8

Q Q t7. t7

10.0 

10.1 

10.2 

10. 3 

10.4

10. 5 

10.6 

10.7 

10. 8 

10.9

11.0 

11. 1 

11.2 

11.3

11.4

11. 5

0. 1824 7453

. 1831 1159

. 1837 2939

. 1843 2876

. 1849 1050

. 1854 7537

. 1860 2408

. 1865 5729

. 1870 7566

. 1875 7978

. 1880 7022

. 1885 4752

. 1890 1220

. 1894 6474

. 1899 0561

. 1903 3524

. 1907 5406

.1911 6246

. 1915 6083

.1919 4952

. 1923 2888

. 1926 9923

. 1930 6088

. 1934 1415

. 1937 5931

. 1940 9664

. 1944 2640

. 1947 4883

. 1950 6418

. 1953 7267

. 1956 7452

. 1959 6995

. 1962 5916

. 1965 4233

. 1968 1965

. 1970 9130

. 1973 5746

. 1976 1827

. 1978 7391

. 1981 2453

. 1983 7026 

. 1986 1126 

. 1988 4764 

. 1990 7955 

. 1993 0711

. 1995 3044 

. 1997 4966 

. 1999 6487 

. 2001 761 8 

. 2003 8371

. 2005 8754 

. 2007 8779 

. 2009 8453 

. 2011 7786 

. 2013 6787

. 2015 5464

-2017

-1927

-1842

-1762

-1687

-1616

-1549

-1485

-1425

-1368

-1314

-1262

-1214

-1168

-1123

-1082

-1042

-1004

-968

-933

-901

-869

-839

-811

-783

-757

-732

-708

-686

-664

-643

-622

-603

-585

-567

-550

-534

-518

-503

 4oo AQQ

-474 

-461 

-448 

-435 

-423

-412 

-400 

-389 

-379 

-369

-360 

-350 

-341 

-332 

-324

90

85

80

75

71

68

64

60

57

54

52

49

46

44

42

40

38

36

34

33

32

30

28

27

26

25

24

23

22

21

20

19

18

18

17

16

16

15

15

14

13 

13 

13

12 

11

11 

11 

10 

10 

10

10 

9 

9

8 

8

-6

-5

-5

" ~ o

_4

-3

-4

-4

-4

-2

_ o

-3

_ g

-2

-3

-2

_ i

-2

-2

_ i

-1

-2

-2

_ i

-1

-1

_ i

-1

-1

_ i

0

-1

_ -i

_ j

0

-1-1

-1

0

1

-1

0 

0 -1

0

0 

0 

0 -1

0

0 

0 

0 

0 

0

Q= +0.2; tabular spacing -0.5

11.5

12.0

0. 2015 5464

. 2024 4270

-7904

-6978

926

787

-170

-139

TABLE 38. Modified potential Continued

a -w Al, Ai, At,

Q=+0.2; tabular spacing =0.5 Continued

12. 5

13.0

13. 5

14.0

14. 5

15.0

15.5

16.0

16.5

17.0

17. 5

18.0

18.5

19.0

19. 5

20.0

20.5

21.0

21.5

22.0

22. 5

23.0

23. 5

24.0

24. 5

25.0

25. 5

26.0

26.5

27.0

27.5

28.0

2&5

29.0

29.5

30.0

0. 2032 6097

. 2040 1733

. 2047 1852

. 2053 7034

. 2059 7780

. 2065 4527

. 2070 7656

. 2075 7500

. 2080 4355

. 2084 8480

. 2089 0107

. 2092 9442

. 2096 6668

. 2100 1950

. 2103 5436

. 2106 7259

. 2109 7541

.2112 6390

.2115 3905

. 2118 0177

. 2120 5287

. 2122 9312

. 2125 2320

. 2127 4373

. 2129 5531

. 2131 5847

. 2133 5369

.2135 4143

. 2137 2212

. 2138 9614

. 2140 6386

. 2142 2561

. 2143 8170

. 2145 3243

. 2146 7807

. 2148 1886

-6191

-5517

-4937

-4436

-3999

-3618

-3284

-2990

-2729

-2498

-2292

-2109

-1944

-1796

-1663

-1542

-1433

-1334

-1243

-1161

-1086

-1017

-954

-896

-842

-793

-748

-706

-6P.7

-630

-597

-566

-536

-509

-484

674

580

502

436

381

334

294

260

231

206

184

165

148

133

121

109

99

90

82

75

69

63

58

54

49

45

42

39

36

33

32

29

27

25

24

-114

-94

-78

-66

-55

-47

-40

-34

-29

-26

-22

-19

-17

-15

-12

-11

-10

-9

-8

-7

-6

-6

_ g

_ 4.

-4

-4

-3

_ 3

_ o

-3

-2

-2

-2

_ 2

-2

Q= +0.3; tabular spacing =0.02

0.00 

.02 

.04 

.06 

.08

. 10

.12 

. 14 

.16 

.18

. 20 

. 22

.24 

.26

.28

. 30 

. 3204. . OTC

.36

.38

.40

.42

0. 0000 0000 

. 0000 6246 

. 0002 4963 

. 0005 6085 

. 0009 9505

. 0015 5073 

. 0022 2600 

. 0030 1858 

. 0039 2582 

. 0049 4474

. 0060 7206 

. 0073 0419 

. 0086 3733 

. 0100 6742 

.0115 9023

. 0132 0138 

. 0148 9635

. 0166 7055

. 0185 1931 

. 0204 3794

. 0224 2174

. 0244 6603

1 2492 

1 2470 

1 2406 

1 2298 

1 2148

1 1959 

1 1731 

1 1466 

1 1169 

1 0839

1 0482 

1 0100 

9696 

9272 

8834

8383

7922

7456

6987 

6517

6049

5585

-22 

-65 

-108 

-150 

-190

-228 

-264 

-298 

-329 

-358

-382 

-404 

-423 

-438 

-451

-460 

-466

-470

-470 

-468

-463

-457

-44 

-43 

-43

-42 

-40

-39 

-36 

-33 

-32 

-28

-24 

-22 

-19 

-15 

-13

_ g

-6

_ oo

-1

+ 2

4

6
NUMERICAL VALUES OF THE MODIFIED POTENTIAL 83

TABLE 38. Modified potential Continued

a -w Ai, Al, Ai,

Q =4-0.3; tabular apacing=0.02 Continued

0.44

.46

.48

. 50

.52 

.54

. 56 

. 58

. 60

.62 

. 64 

. 66 

. 68

.70

. 72

. 74 

. 76 

. 78

.80

. 82 

.84 

. 86 

.88

.90

.92 

.94 

. 96 

.98

1.00

1.02 

1.04 

1.06 

1.08

1. 10

1. 12 

1. 14 

1. 16 

1. 18

1. 20

0. 0265 6617

. 0287 1759

.0309 1582

. 0331 5645

. 0354 3524 

. 03*77 4804

. 0400 9086 

. 0424 5985

. 0448 5132

. 0472 6175 

. 0496 8777 

.0521 2618 

. 0545 7396

. 0570 2825

. 0594 8634 

. 0619 4571 

. 0644 0399 

. 0668 5895

. 0693 0854

. 0717 5084 

.0741 8407 

. 0766 0659 

. 0790 1689

.0814 1359

. 0837 9542 

. 0861 6124 

. 0885 0997 

. 0908 4069

. 0931 5254

. 0954 4476 

.0977 1666 

. 0999 6765 

. 1021 9721

. 1044 0486

. 1065 9023 

. 1087 5298 

. 1108 9282 

.1130 0954

. 1151 0295

5128

4680

4242

3815

3401

orjrjo OUUZ

2617 

2248

1895

1559 

1240 

937 

650

381

+ 128 

-109 

-331 

-538

-729

-907 

-1071 

-1222 

-1360

-1487

-1602 

-1707 

-1802 

-1887

-1963

-2031 

-2091 

-2144 

-2190

-2229

-2262 

-2290 

-2313 

-2331

-448

-438

-427

-414

-400

 

OQ C ooo 

-369 

-353

-336

-320 

-303 

-286 

-270

-253

-237 

-222 

-207 

-192

-178

-164 

-151 

-138

-127

-116

-105 

-95 

-85 

-76

-68

-60 

-53 

-46 

-39

-33

-28 

-23 

-18 

-13

9

10

12

13

14 i 10 ^

16 

16

16

17 

17 

16 

17

16

16 

15 

15 

15

14

14 

13 

13 

12

11

11 

10 

10 

8

9

8

7 

7 

7

6

5 

5 

5 

5

Q +0.3; tabular spacing =0.06

1.20

1. 25

1.30

1.35

1.40

1.45

1.50

1.55

1.60

1. 65

1.70

1.75

1.80

1.85

1. 90

1.95

2.00

2.05

0. 1151 0295

. 1202 3372

. 1252 1702

. 1300 5306

. 1347 4308

. 1392 8917

. 1436 9402

. 1479 6082

. 1520 9309

. 1560 9464

. 1599 6940

. 1637 2146

. 1673 5490

. 1708 7383

. 1742 8232

. 1775 8439

. 1807 8396

. 1838 8486

 1 4640

-1 4748

-1 4726

-1 4601

-1 4394

-1 4124

-1 3806

-1 3452

-1 3073

-1 2678

-1 2272

-1 1861

-1 1451

 1 1044

-1 0643

-1 0250

-9866

-9494

-107

+ 22

125

207

270

318

353

379

396

406

410

410

407

401

393

383

372

361

159

129

103

82

63

48

35

26

16

10

+ 4

0

-3

-6

 oQ 

-9

-11

-12

TABLE 38. Modified potential Continued

a -w Ai, Ai, Ai»

Q= 4-0.3; tabular spacing =0.05 Continued

2. 10

2. 15

2.20

0. 1868 9082

. 1898 0546

. 1926 3225

-9133

-8784

350

336

-12

-13

Q=4-0.3; tabular spacing=0.1!

2.2 

2.3

2. 4

2. 5 

2.6 

2. 7 

2. 8

2.9

3.0 

3. 1 

3.2 

3.3

3.4

3.5 

3. 6 

3.7 

3.8

3.9

4.0 

4. 1 

4. 2 

4.3

4. 4

4. 5 

4. 6 

4. 7 

4. 8

4. 9

5.0 

5. 1 

5.2 

5.3

5.4

5

K

. O

5.6

5. 7

5.8

5. 9

6.0

6. 1

6.2

6.3

6.4

6.5

6.6

6. 7

6.8

6.9

7.0

7. 1

7.2

7.3

7.4

7.5

7.6

0. 1926 3225 

. 1980 3568

. 2031 2652

. 2079 2828 

.2124 6254 

. 2167 4910 

. 2208 0610

. 2246 5016

. 2282 9648 

. 2317 5896 

. 2350 5034 

. 2381 8231

. 2411 6555

. 2440 0990 

. 2467 2436 

.2493 1726 

. 2517 9621

. 2541 6827

. 2564 3993 

. 2586 1720 

. 2607 0562 

. 2627 1034

. 2646 3611

. 2664 8737 

. 2682 6821 

. 2699 8246 

. 2716 3367

. 2732 2517

. 2747 6004 

. 2762 4117 

. 2776 7126 

. 2790 5284

. 2803 8827

901 ft 7Q77 . £tfj ±\J 1 J7 I 1

. 2829 2942

.2841 3918

. 2853 1089

. 2864 4628

. 2875 4696

. 2886 1448

. 2896 5028

. 2906 5573

. 2916 3211

. 2925 8065

. 2935 0250

. 2943 9874

. 2952 7041

. 2961 1849

. 2969 4391

. 2977 4755

. 2985 3025

. 2992 9280

. 3000 3595

. 3007 6043

.3014 6692

-3 3801 

-3 1257

-2 8909

-2 6750 

-2 4770 

-2 2955 

-2 1294

-1 9775

-1 8383 

-1 7110 

-1 5942 

-1 4872

-1 3890

-1 2987 

-1 2158 

-1 1394 

-1 0689

-1 0040

-9440 

-8884 

-8370 

-7894

-7452

-7041 

-6659 

-6304 

-5972

-5663

-5374 

-5104 

-4851 

-4615

-4393

-4185

-3989

-3805

-3633

-3470

-3316

-3172

-3035

-2906

-2785

-2669

-2560

-2457

-2359

-2266

-2178

-2094

-2015

-1939

-1868

-1799

-1734

2544 

2348

2159

1980 

1815 

1661 

1520

1391

1274 

1167 

1070 

982

902

830 

764 

704 

650

600

555

514 

476 

442

411

382 

356 

332 

309

289

270 

253 

236 

222

208

IQfi j. w

184

173

163

154

145

136

129

122

115

109

103

98

93

88

84

80

76

72

69

65

62

 199 

-195

-190

-179 

-165 

-154 

-142

-128

-117 

-107 

-97 

-88

-80

-73 

-66 

-60 

-55

-49

-45 

-41 

-38 

-34

-31

-29 

-27 

 24 

-23

-20

-19 

-18 

-16 

-14

-13

i lo^

-12

 11

-10

-9

__ Q u

-8

-7

-7

-6

-6

-6

 ""O

-5

 . fj

-5

-4

-4

-4

__ Q

 4

-3
84 APPARENT RESISTIVITY OP A SINGLE UNIFORM OVERBURDEN

TABLE 38. Modified potential Continued

a -w Ai, Ai, At,

Q=40.3; tabular spacing=0.1 Continued

7.7

7.8

7.9

8.0

8. 1

8. 2

8.3

8.4

8.5

8.6

8.7

8.8

8.9

9.0

9. 1

9.2

9.3

9.4

9. 5

9.6

9.7

9.8

9.9

10.0

0. 3021 5607

. 3028 2851

. 3034 8482

. 3041 2557

. 3047 5130

. 3053 6253

. 3059 5975

. 3065 4342

. 3071 1400

. 3076 7192

. 3082 1758

. 3087 5139

. 3092 7372

. 3097 8493

. 3102 8537

. 3107 7537

. 3112 5526

. 3117 2533

.3121 8589

. 3126 3721

. 3130 7958

.3135 1324

. 3139 3845

O1 40 KK4R 

. OlTiO OOIO

-1672

-1612

-1556

-1502

-1450

-1402

-1354

-1309

-1267

-1225

-1186

-1148

-1112

-1077

-1044

-1012

-981

-952

-924

-896

-870

-845

-820

59

57

54

51

49

47

45

42

41

40

38

36

35

33

32

31

29

28

28

26

25

24

23

-2

-3

-3

-2

-2

-2

_ o

-2

-1

-1

-2

_ o

-1

-1

-1

_ 2

-1

-1

_ i

-1

_ j

-1

_ i

Q=+0.3; tabular spacing =0.2

10.0

10.2

10. 4

10.6

10.8

11.0

11.2

11.4

11.6

11.8

12.0

12. 2

12.4

12. 6

12. 8

13.0

0. 3143 5546

.3151 6578

. 3159 4596

. 3166 9764

. 3174 2233

.3181 2146

. 3187 9634

. 3194 4819

. 3200 7818

. 3206 8737

. 3212 7677

. 3218 4732

. 3223 9990

. 3229 3535

. 3234 5445

. 3239 5792

-3190

-3014

-2850

-2698

-2557

-2425

-2302

-2187

-2080

-1979

-1885

-1796

-1714

-1636

-1562

176

164

152

142

132

123

115

108

100

94

88

83

78

73

69

-14

-12

-12

-10

-10

_ g

-8

-7

-7

-6

-6

"" 0

 * o

~"~ O

-4

Q=+0.3; tabular spacing =0.5

13.0

13. 5

14.0

14. 5

15.0

15.5

16.0

16. 5

17.0

17.5

18.0

18. 5

19.0

19.5

0. 3239 5792

. 3251 5268

. 3262 6377

. 3272 9962

. 3282 6759

.3291 7412

. 3300 2484

. 3308 2473

. 3315 7820

. 3322 8917

. 3329 6111

. 3335 9714

. 3342 0006

. 3347 7238

-9343

-8368

-7523

-6788

-6145

-5580

-5083

-4642

-4251

-3902

-3591

-3311

-3060

-2834

975

845

735

643

565

498

441

391

348

312

280

251

226

205

-157

 130

-110

-92

-78

-67

-57

-49

-43

-37

-32

-28

-25

-22

TABLE 38. Modified potential Continued

a -w Aii Aii Ml

Q=+0.3: tabular spacing=0.5 Continued

20.0

20.5

21.0

21.5

22.0

22.5

23.0

23. 5

24.0

24. 5

25.0

25. 5

26.0

26. 5

27.0

27.5

28.0

28.5

29.0

29. 5

30.0

0. 3353 1637

. 3358 3406

. 3363 2733

. 3367 9784

. 3372 4715

. 3376 7664

. 3380. 8759

. 3384 8119

. 3388 5849

. 3392 2050

. 3395 6812

. 3399 0219

. 3402 2348

. 3405 3272

. 3408 3057

. 3411 1764

. 3413 9451

. 3416 6172

. 3419 1976

.3421 6909

. 3424 1014

-2629

-2443

-2275

-2121

-1981

-1853

-1736

-1629

-1530

-1439

-1355

-1278

-1206

-1139

-1077

-1020

-967

-917

-871

-826

186

168

154

140

128

117

108

99

91

84

77

72

67

62

57

53

50

46

43

40

-19

-17

-15

-14

-12

-11

-10

-8

-8

__ "7

-6

-6

-5

_ g

-4

-4

_ 4.

-4

-3

-3

Q=+0.4; tabular spacing 0.02

0.00 

.02 

.04

.06

.08

. 10

. 12

. 14

. 16

. 18

.20

.22

.24

.26

.28

.30

.32

.34

.36

.38

.40 

.42

.44

.46

.48

.50

.52

.54

.56

.58

.60

.62

.64

.66

.68

. 70

.72

0. 0000 0000 

. 0000 8455 

. 0003 3791

. 0007 5922

. 0013 4702

. 0020 9932

.0030 1358

. 0040 8674

. 0053 1529

. 0066 9522

. 0082 2214

. 0098 9125

. 0116 9745

. 0136 3532

. 0156 9919

. 0178 8320

. 0201 8130

. 0225 8732

. 0250 9503

. 0276 9813

. 0303 9033 

.0331 6534

. 0360 1696

. 0389 3906

. 0419 2562

. 0449 7075

. 0480 6873

.0512 1401

. 0544 0122

. 0576 2520

. 0608 8097

.0641 6379

. 0674 6914

. 0707 9271

.0741 3041

. 0774 7840

. 0808 3303

1 6910 

1 6881 

1 6794

1 6650

1 6450

1 6196

1 5891

1 5537

1 5139

1 4698

1 4220

1 3708

1 3167

1 2601

1 2013

1 1409

1 0793

1 0168

9539

8909

QOOO OiiOii

7660

7048

6446

5858

5285

4730

4193

3676

3180

2705

2252

1822

1414

1028

664

322

 on£i\)

-87 

-144

-200

-254

-305

-354

-399

-440

-478

-512

-541

-566

-588

-604

-616

-625

-629

-630

-627

 

coo Q2Z

-613

-602

-588

-573

-555

-537

-517

-496

-475

-453

-430

-408

-386

-364

-342

-321

 DoC O

-58

-57

-56

-54

-51

-48

-45

-42

-38

-34

-30

-25

-21

-16

-12

-8

-4

0

+ 2

9

11

14

16

17

19

20

21

22

22

22

22

22

22

22

21
NUMERICAL VALUES OF THE MODIFIED POTENTIAL 85

TABLE 38. Modified potential Continued

a -w Ait Ai, A!,

Q=+0.4; tabular spacing =0.02 Continued

0. 74 

.76

.78

.80

.82 

.84 

.86 

.88

.90

.92

.94 

.96 

.98

1.00

1.02

1. 04

1.06 

1.08

1. 10

1. 12

1. 14

1. 16 

1. 18

1.20

1.22 

1.24

1. 26

1.28

1.30

1.32

1.34

1.36

1.38

1.40

0. 0841 9088 

. 0875 4875 

.0909 0364

. 0942 5274

. 0975 9347 

. 1009 2340 

. 1042 4031 

. 1075 4214

. 1108 2700

. 1140 9316

. 1173 3904 

. 1205 6320 

. 1237 6433

. 1269 4125

. 1300 9292

. 1332 1838

. 1363 1679 

. 1393 8741

. 1424 2960

. 1454 4278

. 1484 2649

. 1513 8031 

. 1543 0390

. 1571 9699

. 1600 5936 

. 1628 9086

. 1656 9136

. 1684 6082

. 1711 9920

. 1739 0653

. 1765 8286

. 1792 2828

. 1818 4290

. 1844 2687

+ 2 

-298 -578

-838

-1079 

-1302 

-1508 

-1697

-1870

-2028

-2172 

-2303 

-2420

-2526

-2621

-2705

-2779 

-2844

-2900

-2948

-2989

-3023 

-3050

-3072

-3088 

-3099

-3105

-3107

-3105

-3100

-3091

-3080

-3065

-300 

-280 -260

-241

-223 

-206 

-189 -173

-158

-144

-130 

-118 -106

-94

-84

-74

-65 

-56

-48

-41

-34

-27 

-22

-16

-11 

-6

-2

5

9

12

14

17

21 

20 

19

19

18 

17 

17 

16

15

14

14 

13 

12

11

11

10

9

8

8

7

7

6 

6

5

5 

5

4

4

4

3

3

2

3

Q=40.4; tabular spacing =0.05

1.40

1.45

1.50

1.55

1.60

1.65

1.70

1.75

1.80

1.85

1.90

1.95

2.00

2.05

2. 10

2. 15

2.20

2. 25

2.30

2.35

2.40

2.45

2.50

0. 1844 2687

. 1907 5385

. 1968 9355

. 2028 4984

. 2086 2707

. 2142 2998

. 2196 6353

. 2249 3286

. 2300 4318

. 2349 9973

. 2398 0772

. 2444 7231

. 2489 9856

. 2533 9143

. 2576 5574

. 2617 9618

. 2658 1729

. 2697 2346

. 2735 1894

. 2772 0779

. 2807 9395

. 2842 8120

. 2876 7315

-1 9046

- 1 8728

-1 8342

-1 7905

-1 7433

 1 6935

-1 6422

-1 5901

-1 5377

-1 4856

-1 4340

-1 3834

-1 3338

-1 2856

-1 2387

-1 1933

-1 1494

-1 1070

-1 0662

-1 0269

-9892

-9529

319

386

436

472

498

513

521

524

521

515

506

496

483

469

454

439

424

408

393

377

363

348

87

67

50

36

25

16

8

+ 2

-2

-6

-9

-11

-13

-14

-14

-15

-16

-16

-15

-16

-15

-14

TABLE 38. Modified potential Continued

a -w Al, Al, Ai,

Q= +0.4; tabular spacing=0.1

2.5 

2.6 

2.7 

2.8

2.9

3.0 

3.1 

3.2 

3.3

3.4

3.5 

3.6 

3.7 

3.8

3.9

4.0

4. 1 

4.2 

4.3

4.4

4.5

4.6

4.7

4.8

4.9

5.0

5. 1

5.2 

5.3

5.4

5.5

5.6

5.7 

5.8

5.9 

6f\. 0 

6. 1

6.2

6.3

6.4

6.5

6.6

6.7

6.8

6.9

7.0

7. 1

7. 2

7.3

7.4

7.5

7.6

7.7

7.8

7. 9

8.0

8. 1

8.2

8.3

8.4

8.5

8.6

0. 2876 7315 

. 2941 8498 

. 3003 5563 

. 3062 0916

.3117 6767

. 3170 5146 

. 3220 7917 

. 3268 6791 

. 3314 3340

. 3357 9011

. 3399 5134 

. 3439 2937 

. 3477 3551 

. 3513 8018

. 3548 7305

. 3582 2303

. 3614 3839 

.3645 2680 

. 3674 9535

. 3703 5068

. 3730 9891

. 3757 4578 

. 3782 9661

. 3807 5639

. 3831 2976

. 3854 2106

. 3876 3437

. 3897 7348 

. 3918 4197

. 3938 4320

. 3957 8030

. 3976 5623

. 3994 7377 

. 4012 3555

.4029 4403

. 4046 0153 

. 4062 1026

. 4077 7230

. 4092 8959

. 4107 6399

.4121 9727

. 4135 9108

. 4149 4699

. 4162 6651

. 4175 5105

. 4188 0195

. 4200 2050

. 4212 0790

. 4223 6533

. 4234 9388

. 4245 9459

. 4256 6847

. 4267 1647

. 4277 3950

. 4287 3842

. 4297 1407

. 4306 6724

. 4315 9868

. 4325 0911

.4333 9923

. 4342 6970

. 4351 2115

-3 6737 

-3 4118 

-3 1712 

-2 9502

 2 7472

-2 5608 

-2 3897 

-2 2324 

-2 0878

-1 9547

-1 8321 

-1 7190 

 1 6146 

-1 5181

-1 4288

-1 3462

-1 2696 

-1 1984 

-1 1323

-1 0709

-1 0137

-9603 

-9106

-8641

-8206

-7800

-7419

-7062 

-6727

-6412

-6117

-5839

-5576 

-5330

-5097

4878 T:Ol O

-4670

-4474

-4289

-4113

-3947

-3789

-3640

-3498

-3363

-3236

-3114

-2998

-2888

-2783

-2683

 2588

-2497

-2410

-2327

-2248

-2173

-2100

-2031

-1965

-1902

-1841

2619 

2407 

2210 

2029

1864

1711 

1573 

1446 

1331

1227

1131 

1044 

965 

893

826

766

712 

661 

614

572

533

498 

465

434

406

381

357

335 

314

295

278

262

247 

232

220

196

185

176

166

158

149

142

135

128

122

116

110

105

100

95

91

87

83

79

75

73

69

66

64

61

58

-226 

-212 

-197 

-181

-166

-152 

-139 

-126 -115

-104

-96

-88 

-79 

-72

-66

-60

-55 

-50 

-47

-42

-39

-36 

-33

-30

-28

-26

-23

-22 

-21

-19

-17

-16

-15 

-15

-12

-12

-11

-9

-10

-9

-8

-8-

-7

-7

-6

-fr

-6

-5

-5

_ g

-4

-4

 4

 4

-2'

-3

_ 4.

_ 2"

-3

3-
86 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

TABLE 38. Modified potential Continued

a -w Ai, Ai, Ai,

Q=+0.4; tabular spacing=0.1 Continued

8.7

8.8

8.9

9.0

9.1

9.2

9.3

9.4

9.5

9.6

9.7

9.8

9.9

10.0

0. 4359 5419

. 4367 6940

. 4375 6735

. 4383 4856

.4391 1355

. 4398 6281

. 4405 9682

.4413 1603

. 4420 2088

.4427 1179

. 4433 8916

. 4440 5339

. 4447 0484

.4453 4388

-1783

-1727

-1673

-1622

-1573

-1526

-1480

-1436

-1394

-1354

-1315

-1277

-1241

56

54

52

49

47

46

44

42

40

39

37

36

35

-2

_ 2

-2

-2

-2

_ 2

-2

-2

-2

-2

_ 2

_ j

-1

Q=+0.4; tabular spacing=0.2

10.0 

10.2

10.4 

10.6 

10.8

11.0 

11.2 

11.4 

11.6 

11.8

12.0 

12.2 

12.4 

12.6 

12.8

13.0 

13.2 

13.4 

13. 6 

13.8

14.0

0. 4453 4388 

. 4465 8610 

. 4477 8266 

. 4489 3603 

. 4500 4847

.4511 2209 

. 4521 5888 

.4531 6068 

.4541 2921 

. 4550 6609

. 4559 7283 

. 4568 5084 

. 4577 0147 

. 4585 2596 

. 4593 2548

. 4601 0115 

. 4608 5401 

. 4615 8504 

. 4622 9517 

. 4629 8528

. 4636 5619

-4827 

-4565 

-4320 

-4093 

-3881

-3684 

-3499 

-3327 

-3165 

-3014

-2872 

-2739 

-2614 

-2496 

-2386

-2281 

-2183 

-2090 

-2002 

-1920

263 

244 

227 

212 

198

184 

172 

162 

151 

142

133 

125 

118 

111 

104

98 

93 

88 

83 

78

(

-21 

-18 

-17 

-16 

-14

-13 

-12 

-11 

-10 

-9

-9 

-8

-7

_ 7 

-6

-6

-6 

_ g

-5

-4

Q=-\-OA; tabular spacing =0.5

14.0

14.5

15.0

15.5

16.0

16.5

17.0

17.5

18.0

18.5

19.0

19.5

20.0

20.5

21.0

21.5

22.0

22.5

23.0

0. 4636 5619

. 4652 5455

. 4667 4889

. 4681 4898

. 4694 6343

. 4706 9980

. 4718 6483

. 4729 6447

. 4740 0407

. 4749 8837

. 4759 2167

. 4768 0781

. 4776 5026

. 4784 5216

. 4792 1636

. 4799 4545

. 4806 4178

. 4813 0751

. 4819 4460

-1 1518

-1 0402

-9424

-8565

-7807

-7135

-6538

-6005

-5529

-5101

-4716

-4369

-4055

-3770

-3511

-3275

-3060

-2864

-2683

1116

977

860

758

672

598

533

476

428

385

347

314

285

259

236

215

197

180

165

-164

-139

-118

-102

-86

-74

-65

-56

-49

-43

-38

-33

-30

-26

-23

-21

-18

-16

-15

TABLE 38. Modified potential Continued

a -W Ai, Ai, Ai,

Q= 4-0.4; tabular spacing = 0.5 Continued

23.5

24.0

24.5

25.0

25.5

26.0

26.5

27.0

27.5

28.0

28.5

29.0

29.5

30.0

0. 4825 5487

.4831 3995

. 4837 0138

. 4842 4056

. 4847 5877

. 4852 5723

. 4857 3702

.4861 9919

. 4866 4468

. 4870 7437

. 4874 8910

. 4878 8962

. 4882 7665

. 4886 5086

-2518

-2366

-2225

-2096

-1976

-1866

-1763

-1668

-1579

-1497

-1420

-1349

-1282

Q=+0.5; tabular spacing =

0.00 

.02 

.04 

.06 

.08

. 10 

. 12 

. 14 

.16 

.18

.20 

.22 

.24 

.26

.28

.30 

.32 

.34 

.36 

.38

.40

.42

.44

.46 

.48

.50

.52

.54

.56

.58

.60

.62

. 64

. 66

.68

.70

.72

.74

.76

.78

.80

.82

.84

.86

.88

0. 0000 0000 

. 0001 0741 

. 0004 2928 

. 0009 6452 

. 0017 1131

. 0026 6716 

. 0038 2886 

. 0051 9261 

. 0067 5396 

. 0085 0791

. 0104 4892 

. 0125 7102 

. 0148 6776 

. 0173 3235 

. 0199 5768

. 0227 3638 

. 0256 6087 

. 0287 2340 

. 0319 1613 

. 0352 3115

. 0386 6056 

. 0421 9645

rvicQ Qlfl^ . U^tOo OJ.UO

. 0495 5659 

. 0533 6554

. 0572 5050

. 0612 0426

. 0652 1981

. 0692 9041

. 0734 0952

. 0775 7091

. 0817 6858

. 0859 9681

. 0902 5018

. 0945 2351

. 0988 1194

. 1031 1085

. 1074 1591

.1117 2306

. 1160 2849

. 1203 2865

. 1246 2023

. 1289 0015

. 1331 6558

. 1374 1387

2 1482 

2 1446 

2 1336 

2 1156 

2 0905

2 0586 

2 0204 

1 9760 

1 9260 

1 8707

1 8107 

1 7465 

1 6785 

1 6074 

1 5337

1 4578 

1 3805 

1 3020 

1 2229 

1 1438

1 0650 

9868

9097 *7V/*7 I

8340 

7600

6880

6180

5504

4852

4227

3628

3057

2513

1997

1509

1049

615

+ 209

-172

-527

-858

-1165

-1450

-1713

-1955

152

140

129

120

111

102

95

88

82

77

72

67

63

-13

-12

-11

-10

-9

-8

-7

_ 7

-6

-6

_ g

_ g

-4

=0.02

-36 

-110 

-181 

-251 

-318

-383 

-444 

-500 

-553 

-600

-642 

-680 

-711

-737 

-758

-774 

-785 

-790 

-791 

-788

-781 

-771

 *7C*7 /o/

-740 

-721

-700

-676

-651

-625

-599

-571

-544

-516

-488

-460

-433

-406

-381

-355

-331

-307

-285

-263

-242

-222

-73 

-73 

-71 

-70 

-68

-64 

-61

-57 

-52

-47

-42 

-37 

-32 

-26 

-21

-15 

-11

-6 

_ 1

+ 3

7 

10

14

17 

19

21

23

25

26

26

28

28

28

28

28

27

27

26

26

24

24

23

22

21

20
NUMERICAL VALUES OF THE MODIFIED POTENTIAL 87

TABLE 38. Modified potential Continued

a -w Ai, Ai, Al,

Q=+0,5; tabular spacing =0.02 Continued

0.90

.92

.94

.96

.98

1.00

1. 02

1.04

1.06

1.08

1. 10

1. 12

1. 14

1. 16

1. 18

1.20

1.22

1. 24

1. 26

1.28

1.30

1.32

1.34

1. 36

1. 38

1.40

1.42

1.44

1.46

1.48

1.50

1.52

1. 54

1.56

1.58

1.60

0. 1416 4263

. 1458 4961

. 1500 3280

. 1541 9032

. 1583 2051

. 1624 2183

. 1664 9291

. 1705 3250

. 1745 3952

. 1785 1298

. 1824 5202

. 1863 5588

. 1902 2392

. 1940 5557

. 1978 5036

. 2016 0791

. 2053 2790

. 2090 1008

. 2126 5427

. 2162 6034

. 2198 2824

. 2233 5793

. 2268 4945

. 2303 0287

. 2337 1829

. 2370 9585

. 2404 3572

. 2437 3812

. 2470 0326

. 2502 3140

. 2534 2280

. 2565 7777

. 2596 9660

. 2627 7963

. 2658 2718

. 2688 3960

-2177

-2380

-2566

-2734

-2887

-3024

-3148

-3258

-3356

-3442

-3517

-3583

-3639

-3686

-3725

-3756

-3781

-3799

-3811

-3818

-3820

-3817

-3810

-3800

-3786

-3768

-3748

-3725

-3700

-3673

-3644

-3613

-3581

-3547

-3513

-204

-185

-168

-153

-138

-124

-110

-98

-86

-75

-65

-56

-47

-39

-32

-25

-18

-12

-7

-2

+ 3

7

11

14

17

20

23

25

27

29

31

32

34

34

36

19

18

17

16

15

14

13

12

12

11

10

9

9

8

7

7

6

6

5

5

5

4

4

4

3

3

3

2

2

2

2

2

1

1

2

Q=+0.5; tabular spacing=0.05

1.60

1.65

1. 70

1.75

1.80

1.85

1.90

1.95

2.00

2.05

2. 10

2. 15

2. 20

2.25

2.30

2.35

2.40

2.45

2.50

2.55

2.60

2.65

2.70

0. 2688 3960

. 2762 1934

. 2833 8759

. 2903 5037

. 2971 1386

. 3036 8425

. 3100 6777

. 3162 7058

. 3222 9875

. 3281 5826

. 3338 5491

.3393 9439

. 3447 8220

. 3500 2367

. 3551 2397

. 3600 8806

. 3649 2076

. 3696 2669

. 3742 1030

. 3786 7587

. 3830 2752

. 3872 6920

. 3914 0470

-2 1729

-2 1149

-2 0546

-1 9930

-1 9309

-1 8688

-1 8071

-1 7463

-1 6867

-1 6285

-1 5718

-1 5167

-1 4634

-1 4118

-1 3620

-1 3140

-1 2677

-1 2232

-1 1804

-1 1392

-1 0997

-1 0618

-1 0253

580

603

616

622

621

616

608

596

582

567

551

533

516

498

480

463

445

428

412

395

380

364

350

34

23

13

+ 6

0

-5

-9

-12

-14

-15

-16

-17

-17

-18

-18

-18

-18

-17

-17

-16

-16

-15

-14

TABLE 38. Modified potential Continued

a -W Al, 4. Al,

Q=+0.5; tabular spacing =0.05 Continued

2.75

2.80

2.85

2.90

2.95

3.00

0. 3954 3768

. 3993 7162

. 4032 0989

. 4069 5571

. 4106 1217

.4141 8224

-9903

-9568

-9245

-8936

-8639

336

322

309

297

285

-14

-13

-13

-13

-12

Q=+0.5; tabular spacing =0.1

3.0

3.1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4.0

4. 1

4. 2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

5.0

5. 1

5.2

5.3

5.4

5. 5

5.6

5.7

5.8

5.9

6.0

6. 1

6.2

6.3

6.4

6.5

6.6

6.7

6.8

6.9

7.0

7. 1

7.2

7. 3

7.4

7. 5

7.6

7. 7

7.8

7.9

8.0

.8. 1

0. 4141 8224

. 4210 7449

. 4276 5391

. 4339 4028

. 4399 5184

. 4457 0541

.4512 1657

. 4564 9969

. 4615 6810

. 4664 3414

.4711 0926

. 4756 0408

. 4799 2849

. 4840 9169

. 4881 0224

. 4919. 6813

. 4956 9680

. 4992 9523

. 5027 6992

.5061 2697

. 5093 7208

.5125 1062

.5155 4762

. 5184 8778

. 5213 3556

. 5240 9512

. 5267 7042

.5293 6514 

.5318 8280

. 5343 2670

. 5366 9994

. 5390 0550

. 5412 4615

. 5434 2454

. 5455 4316

. 5476 0440

. 5496 1050

. 5515 6361

.5534 6575

.5553 1884

. 5571 2472

. 5588 8515

. 5606 0177

. 5622 7618

. 5639 0988

. 5655 0430

. 5670 6083

. 5685 8077

. 5700 6538

. 5715 1585

. 5729 3332

. 5743 1888

-3 3428

-3 1283

-2 9306

-2 7482

-2 5798

-2 4242

-2 2803

-2 1471

-2 0237

-1 9092

-1 8029

-1 7041

-1 6121

-1 5265

-1 4466

-1 3721

-1 3025

-1 2374

-1 1764

-1 1193

-1 0658

 1 0155

-9683

-9239

-8821

-8427

-8057

-7707 

-7376

-7064

-6770

-6490

-6226

-5976

-5739

-5513

-5300

-5097

-4904

-4721

-4546

-4380

-4222

-4071

-3927

-3790

-3659

-3534

-3414

-3300

-3190

-3086

2145

1977

1824

1684

1556

1439

1332

1234

1145

1063

988

920

856

798

745

697

651

609

571

536

503

472

444

418

394

371

350

330 

312

295

279

265

250

237

226

214

202

193

184

175

166

158

151

144

137

131

125

120

114

109

105

100

-184

-168

-153

-140

-128

-117

-107

-97

-89

-82

-75

-68

-63

-58

-53

-49

-46

-42

-38

-35

-33

-31

-28

-26

 24

-23

-21

-20 

-18

-17

-16

-15

-15

-13

-12

-12

-11

-10

_ g

-9

_ g

 8

-7

_ 7

_ 7

-6

-5

-6

-5

-5

* o

_ g
88 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

TABLE 38. Modified potential Continued

a -w Ai, Al, Al,

Q=+0.5: tabular spacing=0.1 Continued

8.2

8.3

8.4

8. 5

8.6

8.7

8. 8

8.9

9.0

9. 1

9. 2

9.3

9.4

9. 5

9.6

9. 7

9. 8

9.9

10.0

0. 5756 7359

. 5769 9844

. 5782 9440

. 5795 6238

. 5808 0327

. 5820 1790

. 5832 0710

. 5843 7163

. 5855 1225

. 5866 2967

. 5877 2458

. 5887 9765

. 5898 4951

. 5908 8078

.5918 9204

. 5928 8386

. 5938 5680

. 5948 1137

. 5957 4809

-2986

-2890

-2798

-2710

-2625

-2544

-2466

-2392

-2320

-2251

-2184

-2121

-2060

-2001

-1944

-1889

-1836

-1786

96

92

88

84

81

78

75

72

69

67

64

61

59

57

55

53

51

49

-4

-4

_ 4.

-4

_ 3

-4

-3

-3

-3

-2

-3

-2

_ 2

-2

-2

-2

-2

-2

Q=+0.5; tabular spacing =0.2

10.0

10. 2

10. 4

10.6

10. 8

11.0

11. 2

11.4

11. 6

11. 8

12.0

12. 2

12.4

12. 6

12. 8

13.0

13.2

13.4

13. 6

13. 8

14.0

14. 2

14.4

14. 6

14. 8

15.0

15.2

15. 4

15. 6

15. 8

16.0

0. 5957 4809

. 5975 6990

. 5993 2593

. 6010 1965

. 6026 5426

. 6042 3276

. 6057 5797

. 6072 3250

. 6086 5880

. 6100 3919

. 6113 7581

. 6126 7069

. 6139 2574

. 6151 4274

. 6163 2338

. 6174 6925

. 6185 8185

. 6196 6259

. 6207 1281

. 6217 3376

. 6227 2666

. 6236 9261

. 6246 3270

. 6255 4794

. 6264 3930

. 6273 0769

. 6281 5398

. 6289 7900

. 6297 8353

. 6305 6832

. 6313 3408

-6948

-6577

-6232

-5910

-5610

-5330

-5068

-4822

-4592

-4376

-4174

-3984

-3804

-3636

-3477

-3327

-3186

-3052

-2926

-2807

-2694

-2586

-2485

-2388

-2297

-2210

-2127

-2049

-1974

-1903

371

345

322

300

280

262

245

230

216

202

190

180

169

159

150

141

134

126

120

113

107

102

96

92

87

82

78

75

71

68

-28

-25

-24

-22

-20

-18

-17

-15

-14

-13

-12

-11

-10

-10

-9

-8

-8

_ >Y

-6

-6

-6

-6

-5

-5

-4

-4

-4

-4

-4

-4

Q=+0.5; tabular spacing =0.5

16.0

16. 5

17.0

17.5

18.0

0. 6313 3408

. 6331 6960

. 6349 0011

. 6365 3433

. 6380 8000

-1 1478

-1 0500

-9630

-8854

-8157

977

870

777

696

625

-124

-108

-93

-80

-71

TABLE 38. Modified potential Continued

a -w Al, Ai, Ai,

Q=+0.5; tabular spacing^ 0.5 Continued

18.5

19.0

19.5

20.0

20. 5

21.0

21. 5

22.0

22. 5

23.0

23. 5

24.0

24. 5

25.0

25. 5

26.0

26. 5

27.0

27. 5

OR Zo. nU

28. 5

29.0

29. 5

30.0

0. 6395 4410

. 6409 3288

. 6422 5198

. 6435 0647

. 6447 0096

. 6458 3964

. 6469 2632

. 6479 6445

. 6489 5720

. 6499 0748

. 6508 1794

. 6516 9101

. 6525 2896

. 6533 3384

. 6541 0757

. 6548 5191

. 6555 6851

. 6562 5888

. 6569 2444

. 6575 6648

. 6581 8623

. 6587 8484

. 6593 6335

. 6599 2276

-7532

-6969

-6460

-5999

-5581

-5201

-4854

-4538

-4248

-3982

-3738

-3513

-3306

-3115

-2938

-2775

-2623

-2482

 ^oD 1

OOOQ £t£i£i\J

-2115

-2009

-1910

563

509

461

418

380

347

317

290

265

244

225

207

191

177

164

152

141

132

122

114

106

99

93

-62

-54

-48

-43

-38

-34

-30

-27

-25

-21

-19

-18

-16

-14

-13

-12

-11

-9

-10

_ 7

-7

-6

Q=+0.6; tabular spacing =0.02

0.00

.02

.04

.06

.08

. 10

. 12

. 14

. 16

. 18

.20

. 22

. 24

.26

. 28

.30

.32

.34

. 36

.38

.40

. 42

.44

.46

.48

.50

CO . u£

. 54

. 56

.58

.60

.62

.64

.66

.68

0. 0000 0000

. 0001 3116

. 0005 2422

.0011 7784

. 0020 8985

. 0032 5724

. 0046 7616

. 0063 4201

. 0082 4945

. 0103 9244

. 0127 6432

. 0153 5788

. 0181 6536

.0211 7858

. 0243 8897

. 0277 8764

. 0313 6544

.0351 1304

. 0390 2098

. 0430 7973

. 0472 7973

. 0516 1147

. 0560 6552

. 0606 3256

. 0653 0346

. 0700 6928

. A7AQ u/'iy91^n ^iou

. 0798 5106

. 0848 5039

. 0899 1140

. 0950 2651

. 1001 8848

. 1053 9037

. 1106 2560

. 1158 8789

2 6233

2 6189

2 6057

2 5839

2. 5537

2. 5154

2. 4693

2 4158

2 3556

2 2890

2 2167

2 1393

2 0574

1 9717

1. 8828

1 7914

1 6980

1 6034

1 5080

1 4126

1 3174

1 2231

1 1300

1 0386

9492

8620

7774

6956

6168

5411

4685

3993

3333

2707

2114

-44

-132

-218

-302

-384

-461

-534

-603

-666

-723

-774

-819

-857

-889

-914

-933

-946

-954

-955

-952

-943

-931

-914

-894

-871

-846

 olo 81 8 

-788

-758

-725

-693

-660

-626

-593

-560

-88

-88

-86

-84

-82

-77

-74

-68

-63

-57

-51

-45

-38

-32

-26

-19

-13

-7

_ i

+ 3

8

12

17

20

23

26

98Zo

30

31

32

33

33

34

33

33
NUMERICAL VALUES OF THE MODIFIED POTENTIAL 89

TABLE 38. Modified potential - Continued

a -w A!. Al, Ai,

Q=+0.6; tabular spacing =0.02 Continued

0. 70

.72

.74

.76

.78

.80

.82

. 84

.86

.88

.90

. 92

. 94

.96

.98

1.00

1.02

1. 04

1.06

1.08

1. 10

1. 12

1. 14

1. 16

1. 18

1.20

1.22

1.24

1.26

1. 28

1. 30

1. 32

1.34

1. 36

1. 38

1. 40

1. 42

1. 44

1. 46

1.48

1.50

1.52

1.54

1.56

1.58

1.60

0. 1211 7132

. 1264 7030

. 1317 7955

. 1370 9412

. 1424 0937

. 1477 2096

. 1530 2486

. 1583 1729

. 1635 9478

. 1688 5410

. 1740 9228

. 1793 0657

. 1844 9447

. 1896 5368

. 1947 8210

. 1998 7784

. 2049 3918

. 2099 6457

. 2149 5261

. 2199 0207

. 2248 1185

. 2296 8098

. 2345 0861

. 2392 9401

. 2440 3655

. 2487 3571

. 2533 9104

. 2580 0220

. 2625 6891

. 2670 9096

. 2715 6821

. 2760 0060

. 2803 8809

. 2847 3072

. 2890 2857

. 2932 8176

. 2974 9044

. 3016 5482

. 3057 7510

. 3098 5157

. 3138 8449

. 3178 7417

. 3218 2094

. 3257 2515

. 3295 8716

. 3334 0734

1554

1027

532

+ 68

-366

-770

-1146

-1494

-1817

-2114

-2388

-2639

-2869

-3078

-3268

-3440

-3595

-3734

-3858

-3968

-4065

-4150

-4223

-4286

-4339

-4382

 4418

 4445

-4466

-4479

-4487

-4489

-4486

-4478

-4466

-4450

-4431

-4408

-4383

-4354

-4324

-4291

-4256

 4220

-4182

-527

-495

-464

-434

-404

-376

-349

-323

-297

-274

-251

-230

-209

-190

-172

-155

-139

-124

-110

-97

-85

-73

-63

-53

-44

-35

-28

-20

-14

-8

-2

+ 3

8

12

16

19

23

26

28

31

33

35

36

38

39

32

32

31

31

29

28

27

26

25

24

23

22

20

19

18

17

16

15

14

13

12

12

10

10

10

8

7

7

7

6

6

5

5

5

4

4

4

3

3

2

2

2

2

1

2

Q=+0.6; tabular spacing =0.05

1. 60

1. 65

1.70

1.75

1.80

1.85

1.90

1. 95

2.00

2.05

2. 10

2. 15

2.20

0. 3334 0734

. 3427 7741

. 3518 9496

. 3607 6670

. 3693 9951

. 3778 0043

. 3859 7648

. 3939 3468

. 4016 8196

. 4092 2516

. 4165 7096

. 4237 2589

. 4306 9631

-2 5892

-2 5252

-2 4582

-2 3891

-2 3190

-2 2486

-2 1785

-2 1091

-2 0409

-1 9740

-1 9087

-1 8451

-1 7834

640

671

690

701

704

701

694

683

669

653

636

617

598

46

31

20

11

+ 3

-3

-7

-11

-14

-15

-17

 1 loo 

-19

TABLE 38. Modified potential Continued

a -W Al, Al, Ai,

Q=+0.6; tabular spacing=0.05 Continued

2. 25

2.30

2.35

2. 40

2.45

2.50

2.55

2.60

2.65

2.70

2.75

2.80

2.85

2.90

2.95

3.00

0. 4374 8839

. 4441 0813

. 4505 6130

. 4568 5351

. 4629 9015

. 4689 7643

.4748 1736

.4805 1778

. 4860 8234

.4915 1550

. 4968 2157

. 5020 0468

. 5070 6882

. 5120 1781

. 5168 5532

. 5215 8488

-1 7235

-1 6656

-1 6097

-1 5557

-1 5036

-1 4534

-1 4052

-1 3586

-1 3139

-1 2709

-1 2295

-1 1898

-1 1515

-1 1148

-1 0794

579

559

540

521

502

483

465

447

430

414

398

382

368

353

340

-19

-20

-20

-19

-19

-19

-18

-18

-17

-16

-16

-15

-15

-14

-14

Q=+0.6; tabular spacirg=0.1

3.0

3. 1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4.0

4. 1

4.2

4. 3

4.4

4.5

4. 6

4.7

4.8

4.9

5.0

5. 1

5. 2

5.3

5.4

5.5

5. 6

5. 7

5.8

5.9

6.0

6.1

6.2

6.3

6.4

6.5

6. 6

6.7

6.8

6.9

7.0

7. 1

0. 5215 8488

. 5307 3364

. 5394 8972

. 5478 7682

.5559 1690

. 5636 3027

. 5710 3575

. 5781 5082

. 5849 9168

.5915 7341

.5979 1003

. 6040 1458

. 6098 9925

. 6155 7537

. 6210 5355

. 6263 4371

. 6314 5512

. 6363 9646

. 6411 7588

. 6458 0102

. 6502 7906

.6546 1674

. 6588 2041

. 6628 9605

. 6668 4929

. 6706 8544

. 6744 0951

. 6780 2624

. 6815 4011

. 6849 5534

. 6882 7595

. 6915 0573

. 6946 4829

. 6977 0703

. 7006 8521

. 7035 8589

. 7064 1200

. 7091 6632

.7118 5151

. 7144 7008

. 7170 2443

.7195 1685

-4 1832

-3 9268

-3 6897

-3 4703

-3 2671

-3 0788

-2 9042

-2 7420

-2 5914

-2 4512

-2 3206

-2 1989

-2 0854

-1 9794

-1 8802

-1 7875

-1 7007

-1 6192

-1 5428

-1 4710

-1 4036

-1 3401

-1 2803

-1 2240

-1 1709

-1 1208

-1 0734

-1 0287

-9863

-9462

-9082

-8723

-8381

-8057

-7750

-7457

-7179

-6914

-6662

 6422

-6193

-5975

2564

2371

2194

2032

1883

1746

1622

1507

1402

1306

1216

1135

1061

991

927

868

815

765

717

674

635

598

563

531

501

474

448

424

401

380

360

342

324

308

293

278

265

252

240

229

218

209

-210

-193

-177

-162

-149

-136

-125

-115

-105

-96

-90

-82

-74

-70

-64

-59

-53

-50

-47

-43

-39

-37

-35

-32

-30

-28

-26

 24

-22

-21

-20

-18

-17

-16

-15

-14

-13

-13

-12

-11

-11

-9
90 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

TABLE 38. Modified potential Continued

a -w Al, Al, A!S

Q=-f-0.6; tabular spacing =0.1 Continued

7.2

7.3

7.4

7.5

7.6

7.7 

7.8

7.9

8.0

8. 1

8.2

8.3

8.4

8.5

8.6

8.7

8.8

8.9

9.0

9.1

9.2

9.3

9.4

9.5

9.6

9.7

9.8

9.9

10.0

0. 7219 4952

. 7243 2453

. 7266 4386

. 7289 0941

.7311 2300

. 7332 8635 

. 7354 0114

. 7374 6894

. 7394 9127

. 7414 6960

. 7434 0532

. 7452 9976

. 7471 5422

. 7489 6990

. 7507 4802

. 7524 8969

. 7541 9600

. 7558 6801

. 7575 0671

. 7591 1309

. 7606 8806

. 7622 3253

. 7637 4736

. 7652 3338

. 7666 9138

. 7681 2216

. 7695 2643

. 7709 0494

. 7722 5835

-5766

-5568

-5378

-5196

-5023

-4857 

-4698

-4546

-4401

-4261

-4128

-3999

-3876

-3758

-3644

-3536

-3431

-3330

-3233

-3140

-3050

-2964

-2881

-2801

-2724

-2649

-2578

-2508

199

190

182

174

166

159 

152

145

140

134

128

123

118

113

109

105

101

97

93

90

86

83

80

77

74

72

69

67

-10

-9

-8

-8

-8

-7

_ 17 _ fj

-6

-6

-6

-5

 o

-5

-4

_4

_ 4

-4

-4

-4

-3

_ q

_ q

_ 3

-3

-2

-3

-2

Q=+0.6; tabular spacing= 0.2

10.0

10.2

10.4

10.6

10.8

11.0 

11.2

11.4

11.6

11.8

12.0

12.2

12.4

12.6

12. 8

13.0

13.2

13.4

13.6

13.8

14.0

14.2

14.4

14.6

14.8

15.0

15. 2

0. 7722 5835

. 7748 9259

. 7774 3420

. 7798 8792

. 7822 5817

. 7845 4907 

. 7867 6448

. 7889 0802

. 7909 8308

. 7929 9286

. 7949 4034

. 7968 2833

. 7986 5948

. 8004 3630

. 8021 6113

. 8038 3619

. 8054 6358

. 8070 4527

. 8085 8315

. 8100 7899

. 8115 3447

. 8129 5118

. 8143 3064

. 8156 7427

. 8169 8345

. 8182 5945

. 8195 0352

-9768

-9262

-8789

-8348

-7935

-7549 

-7187

-6848

-6529

-6230

-5948

-5684

-5434

-5199

-4977

-4767

-4569

-4382

-4204

-4036

-3877

-3726

-3582

-3446

-3317

-3194

-3076

507

473

442

413

386

362 

339

319

299

281

265

250

235

222

210

198

188

177

168

159

151

144

136

129

123

117

111

-38

-34

-31

-29

-26

-24 

-22

-21

-19

-18

-16

-15

-14

-14

-12

-11

-11

-10

_ g

_ g

-8

-8

__ rj _ >7

-6

-6

-6

TABLE 38. Modified potential Continued

a -w Ai, Ai, Ai,

Q=+0.6; tabular spacing** 0.2 Continued

15.4

15.6

15.8

16.0

0. 8207 1682

. 8219 0048

. 8230 5555

. 8241 8304

-2965

-2859

-2757

106

101

97

-5

_ 5

-5

Q=+0.6; tabular spacing =0.5

16.0

16.5

17.0

17.5

18.0

18.5

19.0

19.5

20.0

20.5

21.0

21.5

22.0

22.5

23.0

23.5

24.0

24.5

25.0

25.5

26.0

26.5

27.0

27.5

28.0

28.5

29.0

29.5

30.0

0. 8241 8304

. 8268 8737

. 8294 3922

. 8318 5099

. 8341 3383

. 8362 9771

. 8383 5164

. 8403 0370

. 8421 6123

. 8439 3090

. 8456 1874

. 8472 3027

. 8487 7051

. 8502 4406

. 8516 5512

. 8530 0758

. 8543 0498

. 8555 5059

. 8567 4745

. 8578 9833

. 8590 0584

. 8600 7236

. 8611 0011

. 8620 9116

. 8630 4744

. 8639 7072

. 8648 6269

. 8657 2490

. 8665 5881

-1 6642

-1 5249

-1 4006

-1 2894

-1 1895

-1 0996

-1 0186

-9452

-8787

-8182

-7632

-7129

-6669

-6248

-5862

-5506

-5178

-4876

-4597

-4338

-4099

-3877

-3670

-3478

-3299

-3132

-2976

-2830

1393

1242

1112

999

899

810

734

665

605

551

503

460

421

386

356

328

302

279

259

239

222

206

192

179

167

156

146

137

-173

-151

-130

-113

-101

-88

-77

-68

-61

-54

-48

-43

-39

-35

-31

-28

-26

-22

-21

-19

-17

-16

-14

-14

-12

-11

-10

__ QJ7

Q=-t0.7; tabular spacing =0.02

0.00

.02

.04

.06

.08

.10

.12

. 14

.16

. 18

.20

.22

.24

.26

.28

.30

.32

.34

.36

.38

0. 0000 0000

. 0001 5597

. 0006 2336

. 0014 0064

. 0024 8525

. 0038 7365

. 0055 6135

. 0075 4296

. 0098 1221

. 0123 6204

.0151 8466

. 0182 7159

.0216 1377

. 0252 0159

. 0290 2502

. 0330 7362

. 0373 3670

. 0418 0329

. 0464 6230

. 0513 0256

3 1194

3 1142

3 0988

3 0733

3 0379

2 9930

2 9390

2 8765

2 8058

2 7278

2 6431

2 5524

2 4565

2 3560

2 2518

2 1446

2 0352

1 9242

1 8124

1 7003

-52

 154

-255

-354

-449

-540

-626

-706

-780

-847

-907

-960

-1004

-1042

-1072

-1094

-1110

-1118

-1121

-1116

-103

-102

-101

-99

-95

-91

-86

-80

-74

-67

-60

-53

-45

-38

-30

-23

-15

-9

_ 3

+ 4
NUMERICAL VALUES OF THE MODIFIED POTENTIAL 91

TABLE 38. Modified potential Continued

a

0.40 

.42

.44 

.46

.48

.50 

.52 

.54 

.56

.58

.60 

.62 

.64 

.66

.68

.70

.72

.74 

.76

.78

QA

. oU

.82 

.84

.86 

.88

.90 

.92 

.94

.96 

.98

1.00 

1.02 

1.04

1.06 

1.08

1.10 

1.12 

1. 14

1. 16 

1.18

1.20 

1.22 

1.24

1.26 

1.28

1.30

1.32 

1.34

1.36 

1.38

1.40 

1.42 

1.44

1.46 

1.48

1.50 

1.52 

1.54

1.56 

1.58

1

f»f\ 

. GO

-w Al, Ai, Ai,

Q-+0.7; tabular spacing =0.02 Continued

0.0563 1284 

. 0614 8200 

. 0667 9895 

. 0722 5276

. 0778 3269

. 0835 2825 

. 0893 2918 

. 0952 2555 

. 1012 0773

. 1072 6643

. 1133 9275 

. 1195 7814 

. 1258 1441 

. 1320 9381

. 1384 0894

. 1447 5279 

. 1511 1874

. 1575 0056 

. 1638 9238

. 1702 8869

1 *7CC Q>4Q£i . 1760 o4oo 

. 1830 7457 

. 1894 5484

. 1958 2103 

. 2021 6928

. 2084 9602 

. 2147 9799 

. 2210 7215

. 2273 1575 

. 2335 2625

. 2397 0136 

. 2458 3898 

. 2519 3722

. 2579 9438 

. 2640 0892

. 2699 7948 

. 2759 0486 

. 2817 8398

. 2876 1591 

. 2933 9985

. 2991 3509 

. 3048 2106 

. 3104 5728

. 3160 4334 

.3215 7893

. 3270 6383

. 3324 9788 

. 3378 8100

. 3432 1313 

. 3484 9432

. 3537 2464 

. 3589 0422 

. 3640 3322

.3691 1184 

.3741 4033

.3791 1896 

. 3840 4803 

. 3889 2787

. 3937 5883 

. 3985 4128

. 4032 7562

1 5887 

1 4780 

1 3686 

1 2613

1 1562

1 0538 

9543 

8581 

7653

6761

5906 

5090 

4312 

3573

2872

2210 

1587

1000 

+ 450

cc DO

-546 

-993 

-1409

-1794 

-2150

-2478 

-2780 

-3056

-3310 

-3540

-3749 

-3938 

-4109

-4261 

-4398

-4519 

-4626 

-4719

-4800 

-4869

-4927 

-4976 

-5015

-5046 

-5069

-5085

-5094 

-5097

-5095 

-5087

-5074 

-5058 

-5037

-5013 

-4986

-4956 

-4923 

-4888

-4851 

-4812

-1107 

-1093 

-1074 

-1050

-1024

-995 

-962 

-928 -892

-855

-817 

-778 

-739 -701

-662

-624 -587

-550 -515

4.RO

-448 

-416 

-385

-356 -328

-302 

-277 

-253

-230 -209

-189 

-171 -153

-136 -121

-107 -93 

-81

-70 

-58

-48 

-40 

-31

-23 

-16

-9

-3

+ 2

8 

12

16 

21 

24

27 

30

33 

35 

37

39 

41

9 

14 

20 

23

26

30 

33 

34 

36

37

38 

39 

39 

38

39

38 

37

36 

35

OK DO

33 

32 

31

29 

28

27 

25 

24

23 

21

20 

19 

18

16 

15

14 

14 

12

11 

11

10 

9

8

8 

7

6

6 

5

6

4

4 

4 

3

3 

3

3 

2 

2

2 

2

TABLE 38. Modified potential Continued

a -w Al, Ai, Ai,

Q=+0.7; tabular spacing=0.05

1.60 

1.65

1.70

1.75

1.80 

1.85 

1.90 

1. 95

2.00

2.05 

2. 10 

2. 15 

2.20

2.25

2.30 

2.35

2.40

0. 4032 7562 

. 4149 0365 

. 4262 4026

. 4372 9259

. 4480 6804 

.4585 7420 

.4688 1871 

. 4788 0923

. 4885 5337

. 4980 5865 

. 5073 3244 

. 5163 8200 

.5252 1437

. 5338 3644

. 5422 5489 

. 5504 7618

. 5585 0660

-2 9815 

-2 9142 

-2 8428

-2 7687

-2 6930 

-2 6165 

-2 5399 

-2 4638

-2 3887

-2 3148 

-2 2424 

-2 1718 

-2 1030

-2 0363

-1 9715 

-1 9088

673

714 

741

758

765 

766 

761 

751

739

724 

706 

687 

668

648

627 

606

58 

41 

27

16

+ 8 

0 

-5 

-10

-13

-15 

-17 

-19 

-20

-20

-21 

-21

Q=+0.7; tabular spacing =0.1

2.4

2.5

2. 6

2.7 

2.8 

2.9

3.0 

3. 1

3.2 

3.3 

3.4

3.5 

3.6

3.7 

3.8 

3.9

4.0 

4. 1

4.2 

4.3

4.4

4.5 

4.6

4.7 

4.8

4.9

5.0 

5. 1

5.2 

5.3 

5.4

5. 5 

5.6

5.7 

5.8 

5.9

6.0 

6. 1

5.2

6.3

6. 4

0. 5585 0660

. 5740 1887 

. 5888 3779

. 6030 0626 

. 6165 6417 

. 6295 4850

. 6419 9358 

. 6539 3124

. 6653 9098 

. 6764 0019 

. 6869 8431

. 6971 6694 

. 7069 7004

. 7164 1405 

. 7255 1798 

. 7342 9957

. 7427 7537 

. 7509 6082

. 7588 7037 

. 7665 1755 

. 7739 1500

. 7810 7458 

. 7880 0744

. 7947 2403 

. 8012 3418

. 8075 4714

. 8136 7161 

.8196 1580

. 8253 8744 

. 8309 9386 

. 8364 4193

. 8417 3820 

. 8468 8884

.8518 9968 

. 8567 7628 

. 8615 2388

. 8661 4746 

. 8706 5176

. 8750 4127

. 8793 2026

. 8834 9280

-7 3944

-6 9336 

-6 5044

-6 1057 

-5 7357 

-5 3925

-5 0742 

-4 7791

-4 5053 

-4 2510 

-4 0149

-3 7953 

-3 5909

-3 4008 

-3 2234 

-3 0579

-2 9035 

-2 7590

-2 6238 

-2 4973 

-2 3786

 2 2673 

-2 1627

-2 0644 

-1 9719

-1 8849

-1.8028 

-1 7254

-1 6524 

-1 5834 

-1 5181

-1 4563 

-1 3979

-1 3425 

-1 2900 

-1 2402

-1 1928 

-1 1479

-1 1052

-1 0646

-1 0259

4609

4292 

3987

3700 

3432 

3183

2951 

2739

2543 

2361 

2196

2043 

1902

1774 

1655 

1545

1445 

1352

1265 

1186 

1114

1046 

983

925 

870

821

774 

730

690 

653 

618

584 

554

525 

498 

473

450

427

406

387

369

-327

-317 -304

-287 

-268 -249

-232 -212

-196 

-181 -165

-153 -141

-128 

-119 -110

-100 -93

-87 

-79 

-73

-68 

-63

-58 

-54

-50

-47 

-44

-40 

-37 

-35

-33 

-31

-29 

-27 

-25

-24 

-23

-21

-19

-18
92 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

TABLE 38. Modified potential Continued

a -w Ail Ai, A!,

Q=+0.7; tabular spacing+0.1 Continued

6.5

6.6

6.7

6.8

6.9

7.0

7.1

7.2

7.3

7.4

7.5

7.6

7.7

7.8

7.9

8.0

8. 1

8.2

8.3

8.4

8.5

8. 6

8.7

8.8

8.9

9.0

9. 1

9.2

9.3

9.4

9.5

9.6

9.7

9.8

9.9

10.0

0. 8875 6275

. 8915 3380

. 8954 0946

. 8991 9308

. 9028 8785

. 9064 9682

. 9100 2291

. 9134 6890

. 9168 3744

. 9201 3108

. 9233 5225

. 9265 0329

. 9295 8643

. 9326 0380

. 9355 5746

. 9384 4938

. 9412 8144

. 9440 5546

. 9467 7316

. 9494 3624

. 9520 4628

. 9546 0485

. 9571 1342

. 9595 7341

. 9619 8622

. 9643 5316

. 9666 7551

. 9689 5450

. 9711 9131

. 9733 8709

. 9755 4293

. 9776 5991

. 9797 3904

. 9817 8132

. 9837 8771

. 9857 5912

-9890

-9539

-9204

-8885

-8580

-8289

-8010

-7744

-7490

-7247

-7013

-6790

-6576

-6371

-6174

-5986

-5805

-5631

-5463

-5302

-5148

-5000

-4857

-4719

-4587

-4459

-4336

-4218

-4104

-3993

-3887

-3784

-3685

-3590

-3497

351

334

320

305

291

278

266

254

243

234

223

213

205

197

189

181

174

168

161

154

148

143

138

132

127

124

118

114

110

107

102

99

96

93

89

-18

-16

-15

-15

-14

-12

-12

-12

 11

-10

-10

-10

-8

-8

-8

-8

_ 7

-7

-7

_ tj

-6

-5

-6

-6

-5

-4

-5

_ 4.

-3

_ 4.

-4

-4

-3

_ 3

-3

Q=-j 0.7; tabular spacing=0.2

10.0

10.2

10. 4

10. 6

10.8

11.0

11.2

11.4

11.6

11.8

12.0

12.2

12. 4

12. 6

12.8

13.0

13.2

13.4

13. 6

13.8

14.0

14.2

0. 9857 5912

. 9896 0057

. 9933 1246

. 9969 0115

1. 0003 7257

1. 0037 3231

1. 0069 8559

1. 0101 3731

1. 0131 9210

1. 0161 5429

1. 0190 2797

1. 0218 1700

1. 0245 2500

1. 0271 5541

1.0297 1147

1. 0321 9625

1.0346 1264

1. 0369 6339

1. 0392 5110

1. 0414 7825

1. 0436 4716

1. 0457 6008

-1 3635

-1 2956

-1 2321

-1 1726

-1 1169

 1 0646

-1 0155

-9694

-9260

-8851

-8466

-8102

-7759

-7435

-7129

-6838

-6564

-6304

-6057

-5823

-5600

-5389

679

635

595

557

523

491

461

434

409

385

363

343

324

306

290

274

260

247

234

222

211

201

-48

-44

-40

-37

-34

-32

-30

-27

-26

-23

-22

-20

-19

-18

-16

-16

-14

-14

-12

-12

-11

-10

TABLE 38. Modified potential Continued

a -W Ai, A!, Ai,

Q=+0.7; tabular spacing=0.2 Continued

14.4

14. 6

14.8

15.0

15.2

15.4

15. 6

15.8

16.0

16.2

16. 4

16. 6

16.8

17.0

1.0478 1910

1. 0498 2625

1. 0517 8342

1. 0536 9244

1. 0555 5504

1. 0573 7287

1. 0591 4751

1. 0608 8045

1. 0625 7314

1. 0642 2694

1. 0658 4316

1. 0674 2305

1. 0689 6781

1. 0704 7858

-5188

-4997

-4815

-4642

-4477

-4319

-4169

-4026

-3889

-3758

-3633

-3513

-3399

191

182

173

165

157

150

143

137

131

125

120

114

110

-1C

_ c

-S

-*

-*

ft -e

-e

_ i

-i

_ i

_ i

Q-4-0.7; tabular spacing=0.5

17.0

17.5

18.0

18.5

19.0

19.5

20.0

20.5

21.0

21.5

22.0

22.5

23.0

23.5

24. 0

24. 5

25.0

25. 5

26.0

26.5

27.0

27.5

28.0

28.5

29.0

29.5

30.0

1. 0704 7858

1.0741 1389

1. 0775 5941

1. 0808 2947

1. 0839 3699

1. 0868 9368

1. 0897 1015

1. 0923 9604

1. 0949 6013

1.0974 1046

1. 0997 5435

1. 1019 9853

1. 1041 4918

1. 1062 1196

1. 1081 9212

1. 1100 9448

1. 1119 2350

1. 1136 8330

1. 1153 7771

1. 1170 1028

1. 1185 8430

1. 1201 0286

1. 1215 6881

1. 1229 8482

1. 1243 5337

1. 1256 7681

1. 1269 5731

-2 0570

-1 8979

-1 7546

-1 6253

-1 5084

-1 4022

-1 3058

-1 2179

-1 1377

-1 0643

-9971

-9354

-8786

-8263

-7780

-7334

-6922

-6539

-6184

 5854

-5547

-5261

-4994

-4745

-4512

-4294

1591

1432

1293

1170

1061

965

879

802

734

672

617

568

523

483

446

413

382

355

330

307

286

267

249

233

218

204

-181

-159

-139

-123

-109

-96

-86

-77

-68

-61

-55

-50

-45

-40

-37

-33

-30

-28

-25

-23

-21

-20

-17

-16

-15

-14

Q=+0.8; tabular spacing=0.02

0.00

.02

.04

.06

.08

.10

. 12

. 14

. 16

. 18

.20

.22

.24

.26

.28

0. 0000 0000

. 0001 8207

. 0007 2770

. 0016 3510

. 0029 0137

. 0045 2243

. 0064 9313

. 0088 0728

. 0114 5770

. 0144 3629

. 0177 3408

. 0213 4136

. 0252 4771

. 0294 4213

.0339 1307

3 6414

3 6355

3 6178

3 5886

3 5480

3 4964

3 4345

3 3627

3 2816

3 1921

3 0949

2 9908

2 8806

2 7652

2 6456

-59

-177

-293

-406

-515

-620

-718

-810

-895

-972

-1041

-1102

-1154

-1197

-1231

-119

-117

-116

-113

-109

 104

-99

-92

-85

-77

-69

-60

-52

-43

-34
NUMERICAL VALUES OF THE MODIFIED POTENTIAL 93

TABLE 38. Modified potential Continued

a -w A!. A!, Ai,

Q=-t 0.8; tabular spacing =0.02 Continued

0.30

.32

.34

.36

.38

.40

. 4.9 *i£

.44

.46

.48

.50

.52

.54

.56

.58

.60

.62

.64

.66

.68

.70

. 72

.74

.76

.78

.80

.82

.84

.86

.88

.90

.92

.94

.96

.98

1.00

1.02

1.04

1,06

1.08

1. 10

1. 12

1. 14

1. 16

1. 18

1. 20

1.22

1.24

1. 26

1.28

1.30

1.32

1.34

1.36

1.38

1.40

1.42

1.44

1.46

1.48

0. 0386 4856

. 0436 3630

. 0488 6371

.0543 1804

. 0599 8644

. 0658 5602

O71Q IQQK . u/ iy loyo 

. 0781 4749

. 0845 4406

. 0910 9130

. 0977 7712

. 1045 8972

. 1115 1764

. 1185 4980

. 1256 7547

. 1328 8439

. 1401 6666

. 1475 1287

. 1549 1401

. 1623 6154

. 1698 4734

. 1773 6375

. 1849 0352

. 1924 5986

. 2000 2635

. 2075 9701

. 2151 6624

. 2227 2881

. 2302 7987

.2378 1491

. 2453 2974

. 2528 2052

. 2602 8369

. 2677 1598

. 2751 1439

. 2824 7620

. 2897 9890

. 2970 8022

.3043 1813

.3115 1078

. 3186 5649

. 3257 5380

. 3328 0138

. 3397 9807

. 3467 4285

. 3536 3484

. 3604 7328

. 3672 5752

. 3739 8701

. 3806 6132

. 3872 8011

. 3938 4309

. 4003 5009

. 4068 0099

. 4131 9573

. 4195 3433

. 4258 1685

. 4320 4340

. 4382 1414

. 4443 2927

2 5224

2 3967

2 2692

2 1407

2 0118

1 8834

1

*7C£J 1

75ol

1 6303

1 5067

1 3858

1 2678

1 1532

1 0423

9353

8324

7336

6393

5494

4638

3828

3060

2337

1656

1016

+ 417

-143

-665

-1151

-1602

-2020

-2406

-2761

-3088

-3387

-3661

-3910

-4137

-4342

-4527

-4693

-4841

-4973

-5089

-5191

-5279

-5355

-5420

-5474

-5518

-5553

-5580

-5599

-5610

-5615

-5614

-5608

-5597

-5581

-5561

-5537

-1257

-1275

-1285

-1288

-1284

-1274

 19^7 l^Oi

 1 Izob OO£S

-1210

-1179

-1146

-1109

-1070

-1029

-987

-943

-899

-855

-811

_ 7«7 ( \)f

-724

-681

-640

-599

-560

-522

-486

-451

-418

-386

-356

-327

-299

-274

-249

-226

-205

-185

-166

-148

-132

-116

-102

-88

-76

-65

-54

-44

-35

-27

-19

-12

-5

+ 1

6

12

16

20

24

27

-26

-18

-10

-3

+ 4

101 C

lo 

91£t L

26

30

33

37

39

41

42

44

44

44

44

44

44

42

41

40

39

38

36

35

33

32

30

29

27

26

24

23

22

20

19

18

17

15

14

14

12

11

11

10

9

8

8

7

7

6

5

5

4

4

4

3

TABLE 38. Modified potential Continued

a w *i Al, Al,

Q=+0.8; tabular spaciHg=0.02 Continued

1.50

1.52

1.54

1.56

1.58

1.60

0. 4503 8904

. 4563 9372

. 4623 4361

. 4682 3905

. 4740 8039

. 4798 6802

-5509

-5479

-5445

-5409

-5371

30

33

36

38

40

3

3

3

2

3

Q=+0.8; tabular spacing =0.05

1.60

1.65

1.70

1.75

1.80

1.85

1.90

1.95

2.00

2.05

2. 10

2. 15

2. 20

2. 25

2.30

2.35 

2.40

2.45

2.50

0. 4798 6802

. 4941 0479

.5080 1516

. 5216 0639

. 5348 8611

. 5478 6218

. 5605 4259

. 5729 3541

. 5850 4866

. 5968 9034

. 6084 6831

. 6197 9029

. 6308 6386

. 6416 9642

. 6522 9515

. 6626 6706 

. 6728 1895

. 6827 5740

. 6924 8878

-3 3314

-3 2640

-3 1914

-3 1152

-3 0365

-2 9565

-2 8760

-2 7956

-2 7158

-2 6371

-2 5598

-2 4841

-2 4102

-2 3382

-2 2682

-2 2002 

-2 1344

-2 0706

674

726

762

787

800

805

804

798

787

773

757

739

720

700

680

658 

638

617

72

52

37

24

13

+ 5

-1

-6

-11

-14

-16

-18

-19

-20

-20

-21 

-21

-21

Q=+0.8; tabular spacing=0.1

2.5

2.6

2. 7

2.8

2.9

3.0

3. 1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4.^0

4. 1

4.2

4.3

4. 4

4.5

4.6

4.7

4.8

4.9

5.0

5.1

5.2

5.3

5.4

0. 6924 8878

. 7113 5482

. 7294 6397

. 7468 5999

. 7635 8361

. 7796 7277

. 7951 6274

. 8100 8636

. 8244 7420

. 8383 5473

. 8517 5447

. 8646 9816

. 8772 0890

. 8893 0824

. 9010 1636

. 9123 5211

. 9233 3318

. 9339 7614

. 9442 9653

. 9543 0897

. 9640 2718

. 9734 6408

. 9826 3184

. 9915 4193

1. 0002 0517

1. 0086 3178

1. 0168 3141

1. 0248 1320

1. 0325 8578

1. 0401 5734

-8 0379

-7 5689

-7 1314

-6 7239

-6 3447

-5 9918

-5 6635

-5 3578

-5 0731

-4 8079

-4 5605

-4 3296

-4 1139

-3 9123

-3 7236

-3 5469

-3 3812

-3 2256

-3 0796

-2 9423

-2 8131

-2 6914

-2 5767

-2 4685

-2 3663

-2 2698

-2 1785

-2 0920

-2 0102

-1 9326

4691

4375

4074

3792

3529

3284

3057

2847

2653

2474

2309

2156

2016

1887

1767

1657

1555

1461

1373

1292

1217

1147

1082

1022

965

913

864

819

776

736

-327

-316

-301

-282

-264

-245

-227

-210

-194

-179

-165

-153

-140

-129

-120

-110

-102

-95

-88

-81

-75

-70

-65

-61

-56

-52

-49

-46

-43

-39
94 APPARENT RESISTIVITY OP A SINGLE UNIFORM OVERBURDEN

TABLE 38. Modified potential Continued

a -w Ai, Ai, Ai,

Q=+0.8; tabular spacing = 0.1 Continued

5.5

5.6

5.7

5.8

5.9

6.0

6. 1

6.2

6.3

6.4

6.5

6.6

6.7

6.8

6.9

7.0

7.1

7.2

7.3

7.4

7.5

7.6

7.7

7.8

7.9

8.0

8. 1

8. 2

8.3

8.4

8.5

8

C. . O

8.7

8.8

8.9

9.0

9. 1

9.2

9.3

9.4

9.5

9.6

9.7

9.8

9.9

10.0

1. 0475 3564

1. 0547 2804

1. 0617 4154

1. 0685 8277

1. 0752 5803

1. 0817 7333

1. 0881 3438

1. 0943 4663

1. 1004 1525

1. 1063 4519

1. 1121 4117

1. 1178 0770

1. 1233 4907

1. 1287 6939

1. 1340 7261

1. 1392 6247

1. 1443 4259

1. 1493 1640

1. 1541 8722

1. 1589 5821

1. 1636 3241

1. 1682 1273

1. 1727 0198

1. 1771 0283

1. 1814 1787

1. 1856 4958

1. 1898 0036

1. 1938 7249

1. 1978 6818

1. 2017 8955

1. 2056 3866

1. 2094 1747

1 1. 01 -ilol qi £ O7QA i yu

1. 2167 7177

1. 2203 5085

1. 2238 6685

1. 2273 2140

1. 2307 1610

1. 2340 5249

1. 2373 3204

1. 2405 5618

1. 2437 2630

1. 2468 4374

1. 2499 0980

1. 2529 2572

1. 2558 9272

-1 8590

-1 7890

-1 7227

-1 6596

-1 5996

-1 5425

-1 4881

-1 4362

-1 3868

-1 3396

-1 2946

-1 2516

-1 2104

-1 1711

-1 1335

-1 0975

-1 0630

-1 0300

-9983

-9679

-9387

-9108

-8839

-8581

-8333

-8094

-7864

-7644

-7432

-7226

-7029

 

COO A uooy 

-6656

-6479

-6309

-6144

-5985

-5831

-5684

-5541

-5402

-5268

-5139

-5013

-4892

699

663

630

600

571

544

519

494

472

450

430

411

393

376

360

345

330

317

304

292

280

268

258

248

239

230

220

213

205

197

190 t oq loo

170

164

159

154

148

143

139

134

129

125

121

117

-37

-36

-33

-30

-29

-27

-26

-24

-23

-21

-20

-19

-18

-16

-16

-16

-15

-13

-13

-12

-12

-12

-10

-10

-10

__ Q

-9

-8

-8

-8

_ f

-7

-7

-6

-6

-5

-5

-6

-5

-4

 o

-4

-4

_4

_ 4

Q=+0.8; tabular spacing =0.2

10.0

10.2

10.4

10.6

10.8

11.0

11.2

11.4

11.6

11.8

1. 2558 9272

1. 2616 8460

1. 2672 9438

1. 2727 3046

1. 2780 0072

1. 2831 1258

1. 2880 7300

1. 2928 8857

1. 2975 6548

1. 3021 0959

-1 9104

-1 8210

-1 7370

-1 6582

-1 5841

-1 5143

-1 4486

-1 3866

-1 3281

-1 2728

894

840

788

741

698

657

620

585

553

522

-59

-55

-51

-47

-43

-41

-37

-35

-32

-30

TABLE 38. Modified potential Continued

a -w Al, Aii Ai,

Q-+0.8; tabular spacing=0.2 Continued

12.0

12.2

12.4

12.6

12.8

13.0

13.2

13.4

13.6

13.8

140

14.2

14.4

14.6

14.8

15.0

15.2

15.4

15. 6

15.8

16.0

16.2

16.4

16.6

16.8

17.0

17.2

17.4

17. 6

17.8

18.0

1. 3065 2642

1. 3108 2120

1. 3149 9885

1. 3190 6408

1. 3230 2130

1. 3268 7473

1. 3306 2834

1. 3342 8593

1. 3378 5110

1. 3413 2727

1. 3447 1769

1. 3480 2546

1. 3512 5354

1. 3544 0474

1. 3574 8174

1. 3604 8710

1. 3634 2326

1. 3662 9257

1. 3690 9725

1. 3718 3943

1. 3745 2116

1.3771 4439

1. 3797 1099

1. 3822 2275

1. 3846 8140

1. 3870 8857

1. 3894 4585

1. 3917 5476

1. 3940 1674

1. 3962 3321

1. 3984 0550

-1 2206

-1 1712

-1 1244

-1 0800

-1 0380

-9981

-9602

-9242

-8900

-8575

-8265

-7970

-7688

-7420

-7164

-6920

-6686

-6463

-6250

-6045

-5850

-5663

-5484

-5312

-5147

-4989

-4838

-4692

-4552

-4418

494

468

443

420

399

379

360

342

326

310

295

282

268

256

244

234

223

213

204

196

187

179

172

165

158

152

146

140

134

129

-28

-26

-25

-23

-22

-20

-19

-18

-16

-16

-15

-14

-13

-12

-12

-11

-10

-10

_ Q

-9

_ o

 oo 

-7

-7

__ >7

-6

-6

-6

-5

___ *y

Q=+0.8; tabular spacing =0.5

18.0

18.5

19.0

19.5

20.0

20.5

21.0

21.5

22.0

22.5

23.0

23.5

24. 0

24.5

25.0

25.5

26.0

26.5

27.0

27.5

28.0

28.5

29.0

29.5

30.0

1. 3984 0550

1. 4036 5129

1. 4086 4780

1. 4134 1215

1. 4179 5999

1. 4223 0558

1. 4264 6196

1. 4304 4107

1. 4342 5391

1.4379 1055

1. 4414 2030

1. 4447 9172

1. 4480 3276

1. 4511 5075

1. 4541 5249

1. 4570 4429

1. 4598 3202

1. 4625 2113

1.4651 1672

1. 4676 2352

1. 4700 4598

1. 4723 8823

1. 4746 5415

1. 4768 4738

1. 4789 7133

-2 6817

-2 4930

-2 3214

-2 1652

-2 0225

-1 8921

-1 7726

-1 6628

-1 5619

-1 4690

-1 3832

-1 3039

-1 2305

-1 1625

-1 0994

-1 0407

-9861

-9353

-8878

-8435

-8021

-7633

-7269

-6928

1887

1715

1563

1427

1304

1195

1098

1009

929

858

793

734

680

631

587

546

509

475

443

414

388

363

341

320

-194

-172

-152

-136

-122

-109

-98

-88

-80

-72

-65

-59

-54

-49

-45

-41

-37

-34

-32

-29

-26

-25

 22

91£i J.
NUMERICAL VALUES OF THE MODIFIED POTENTIAL 95

TABLE 38. Modified potential Continued

a -w Al, Al, A!*

Q= 4-0.9; tabular spacing =0.02

0.00

.02

.04

.06

.08

. 10

. 12

. 14

. 16

. 18

.20

.22

.24

.26

.28

.30

.32

. 34

.36

.38

.40

.42

. 44

.46

.48

. 50

.52

. 54

.56

.58

. 60

.62

.64

. 66

.68

.70

.72

.74

. 76

.78

.80

.82

.84

.86

.88

.90

.92

. 94

.96

.98

1.00

1. 02

1.04

1.06

1.08

1. 10

1. 12

1. 14

1. 16

1. 18

1. 20

1.22

0. 0000 0000

. 0002 0988

. 0008 3884

. 0018 8488

. 0033 4471

.0052 1373

. 0074 8612

.0101 5488

. 0132 1190

. 0166 4803

. 0204 5314

.0246 1625

. 0291 2559

. 0339 6870

. 0391 3255

. 0446 0360

. 0503 6793

. 0564 1131

. 0627 1933

. 0692 7743

. 0760 7104

. 0830 8561

. 0903 0672

. 0977 2012

. 1053 1181

. 1130 6807

. 1209 7553

. 1290 2118

. 1371 9243

. 1454 7712

. 1538 6355

. 1623 4051

. 1708 9725

. 1795 2354

. 1882 0963

. 1969 4627

. 2057 2471

. 2145 3667

. 2233 7438

. 2322 3049

. 2410 9815

. 2499 7092

. 2588 4280

. 2677 0818

. 2765 6188

. 2853 9906

. 2942 1524

. 3030 0632

. 3117 6847

. 3204 9820

. 3291 9232

. 3378 4788

. 3464 6221

. 3550 3289

. 3635 5771

. 3720 3468

. 3804 6202

. 3888 3812

. 3971 6157

. 4054 3108

. 4136 4556

. 4218 0404

4 1975

4 1909

4 1708

4 1378

4 0920

4 0337

3 9637

3 8826

3 7910

3 6899

3 5800

3 4623

3 3378

3 2074

3 0720

2 9328

2 7906

2 6463

2 5008

2 3550

2 2096

2 0654

1 9229

1 7829

1 6457

1 5120

1 3819

1 2560

1 1344

1 0175

9052

7979

6955

5980

5055

4180

3353

2574

1841

1154

+ 511

-89

-649

-1169

-1652

-2099

-2512

-2892

-3242

-3562

-3855

-4123

-4366

-4586

-4785

-4964

-5124

-5266

-5392

-5504

-5600

-5684

-67

-200

-331

-458

-582

-700

-811

-916

-1012

-1099

-1177

-1245

-1304

-1353

-1393

-1422

-1443

-1455

-1458

-1454

-1442

-1425

-1401

-1372

-1338

-1300

-1259

-1216

-1170

-1122

-1074

-1024

-974

-925

-875

-827

-779

-732

-687

-643

-601

-560

-520

-483

-447

-413

-380

-350

-321

-293

-267

-243

-220

-199

-179

-160

-142

-126

-111

-97

-84

-72

-133

-133

-131

-128

 124

-118

-111

-104

-96

-87

-78

-68

-59

-49

-40

-29

-21

-12

-4

+ 4

11

18

24

29

34

37

41

44

46

47

49

50

50

50

50

48

48

47

46

44

42

41

39

38

36

34

33

31

29

28

26

24

23

21

20

19

18

16

15

14

13

12

TABLE 38. Modified potential Continued

a -W Al, Al, Al,

Q=4-0.9; tabular spacing=0.02 Continued

1. 24

1. 26

1.28

1. 30

1.32

1. 34

1. 36

1.38

1.40

1.42

1. 44

1.46

1.48

1.50

1.52

1.54

1. 56

1.58

1.60

0. 4299 0567

. 4379 4974

. 4459 3564

. 4538 6287

. 4617 3103

. 4695 3980

. 4772 8894

.4849 7831

. 4926 0780

. 5001 7739

. 5076 8712

. 5151 3706

. 5225 2736

. 5298 5819

. 5371 2977

. 5443 4236

. 5514 9625

. 5585 9176

. 5656 2922

-5756

-5817

-5867

-5907

-5939

-5962

-5978

-5987

-5990

-5987

-5978

-5965

-5947

-5925

-5899

-5870

-5838

-5804

-60

-50

-40

-32

-23

-16

-9

-2

+ 3

9

14

18

22

26

29

32

35

37

11

10

10

9

8

8

7

6

6

6

5

4

4

4

3

3

3

2

Q=-4-0.9; tabular spar ing=0.05

1. 60

1. 65

1. 70

1. 75

1.80

1. 85

1.90

1.95

2.00

2.05

2. 10

2. 15

2. 20

2.25

2.30

2.35

2.40

0. 5656 2922

.5829 7148

. 5999 5975

. 6166 0105

. 6329 0284

. 6488 7294

. 6645 1936

. 6798 5022

. 6948 7369

. 7095 9790

. 7240 3091

. 7381 8069

. 7520 5506

. 7656 6166

. 7790 0798

. 7921 0131

. 8049 4874

-3 6034

-3 5398

-3 4698

-3 3950

-3 3170

-3 2368

-3 1556

-3 0740

-2 9926

-2 9119

-2 8323

-2 7542

-2 6776

-2 6028

-2 5299

-2 4590

636

701

748

780

801

812

816

814

807

796

782

766

748

729

709

689

86

65

47

32

21

11

+ 4

-2

-7

-11

-14

-16

-18

-19

-20

-20

Q=+0.9; tabular spacing=0.1

2.4

2. 5

2.6

2.7

2.8

2.9

3.0

3. 1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4.0

4. 1

0. 8049 4874

. 8299 3328

. 8540 1426

. 8772 4112

. 8996 6013

. 9213 1455

. 9422 4476

. 9624 8846

. 9820 8084

1. 0010 5475

1. 0194 4089

1. 0372 6795

1. 0545 6275

1.0713 5039

1. 0876 5439

1. 1034 9682

1. 1188 9836

1. 1338 7846

-9 5623

-9 0355

-8 5412

-8 0785

-7 6460

-7 2421

-6 8651

-6 5132

-6 1847

-5 8777

-5 5908

-5 3226

-5 0716

-4 8363

-4 6158

-4 4088

-4 2145

-4 0317

5268

4943

4627

4325

4039

3770

3518

3286

3070

2869

2682

2510

2352

2206

2070

1944

1827

1719

-326

-325

-316

-302

-286

-269

-251

-233

-216

-201

-186

-172

-158

-147

-136

-126

-116

-108
96 APPARENT RESISTIVITY OF A SINGLE UNIFORM OVERBURDEN

TABLE 38. Modified potential Continued

a -W Aii Ali Ai2

Q=+0.9; tabular spacing=0.1 Continued

4.2 

4.3

4.4

4.5 

4.6

4. 7 

4.8 

4. 9

5.0 

5. 1

5.2 

5. 3

5.4

5.5 

5.6

5.7 

5.8 

5.9

6.0 

6. 1

6.2 

6.3 

6. 4

6.5 

6.6

6.7 

6.8 

6.9

7.0

7. 1

7. 2 

7. 3 

7.4

7.5 

7.6

7. 7 

7.8 

7.9

8.0

8. 1

8.2 

8.3 

8.4

8. 5 

8.6

8. 7 

8.8

8.9

9.0

9. 1

9.2

9.3

9.4

9.5

9.6

9.7

9.8

9. 9

10.0

1. 1484 5539 

1. 1626 4633 

1. 1764 6748

1. 1899 3412 

1. 2030 6065

1. 2158 6067 

1. 2283 4703 

1. 2405 3189

1. 2524 2676 

1. 2640 4257

1. 2753 8966 

1. 2864 7782 

1. 2973 1640

1. 3079 1424 

1. 3182 7979

1. 3284 2105 

1. 3383 4568 

1. 3480 6094

1. 3575 7377 

1. 3668 9082

1. 3760 1839 

1. 3849 6254 

1. 3937 2905

1. 4023 2344 

1. 4107 5100

1.4190 1680 

1. 4271 2569 

1. 4350 8233

1. 4428 9118 

1. 4505 5652

1. 4580 8246 

1. 4654 7297 

1. 4727 3182

1. 4798 6267 

1. 4868 6903

1. 4937 5428 

1. 5005 2166 

1. 5071 7432

1. 5137 1526 

1. 5201 4740

1. 5264 7354 

1. 5326 9638 

1. 5388 1853

1. 5448 4252 

1. 5507 7077

1. 5566 0564 

1. 5623 4940

1. 5680 0423

1. 5735 7227

1. 5790 5558

1. 5844 5612

1. 5897 7583

1. 5950 1657

1. 6001 8014

1. 6052 6829

1. 6102 8271

1. 6152 2504

1. 6200 9687

1. 6248 9974

-3 8598 

-3 6979

 o D 40^

-3 4011 

-3 2651

-3 1366 

-3 0150 

-2 8998

-2 7907 

-2 6873

-2 5892 

-2 4959 

-2 4073

-2 3230 

-2 2428

-2 1664 

-2 0936 

-2 0242

-1 9580 

-1 8947

-1 8342 

-1 7764 

-1 7212

-1 6683 

-1 6176

-1 5691 

-1 5225 

-1 4779

-1 4351 

-1 3940

-1 3544 

-1 3165 

-1 2800

-1 2449 

-1 2111

-1 1786 

-1 1473 

-1 1171

-1 0881 

-1 0600

-1 0329 

-1 0068 

-9817

-9574 

-9338

-9111 

-8892

-8679

-8474

-8276

-8084

-7897

-7717

-7542

-7373

-7209

-7050

-6896

1619 

1527 

1441

1360 

1285

1216 

1152 

1091

1034 

981

932 

886 

843

802 

764

728 

694 

663

633 

604

578 

553 

529

506

486

466 

446

428

411 

396

379 

364 

351

338 

325

313 

302

291

280 

271

261 

252 

243

235 

227

220 

212

one ^uo

198

192

187

180

174

169

164

159

154

, 150

-100 

-92 

-86

-81

-75

-69 

-64 

-61

-57

COoo -49 

-46 

-43

-41 

-38

-36 

-33 

-32

-30

-28

-26 

-25

-24

-22

-21

-20

-19 

-18

-16 

-16

-16 

-15 

-13

-13 

-13

-12 

-12 

-11

-10 

-10

-10 

-9

-8

-8 

-8

-8

-7

-7

-7

-6

-6

-6

-6

-5

-5

_ g

-5

-4

TABLE 38. Modified potential Continued

a -w All Ai, Ai,

Q=+0.9; tabular spacing =0.2

10.0 

10. 2 

10.4

10. 6 

10. 8

11.0 

11.2 

11. 4

11. 6 

11. 8

12.0 

12. 2 

12.4

12.6 

12. 8

13.0 

13. 2 

13. 4

13. 6 

13. 8

14.0 

14.2 

144

14.6 

14.8

15. 0 

15. 2 

15. 4

15.6 

15. 8

16.0 

16. 2 

16.4

16. 6 

16. 8

17.0 

17.2 

17.4

17.6 

17. 8

18. 0 

18.2 

18. 4

18. 6 

18.8

19.0

1. 6248 9974 

1. 6343 0457 

1. 6434 5096

1. 6523 4970 

1. 6610 1098

1. 6694 4441 

1. 6776 5908 

1. 6856 6358

1. 6934 6605 

1. 7010 7420

1. 7084 9536 

1. 7157 3647 

1. 7228 0411

1. 7297 0457 

1. 7364 4380

1. 7430 2747 

1. 7494 6101 

1. 7557 4956

1. 7618 9803 

1. 7679 1112

1. 7737 9332 

1. 7795 4889 

1. 7851 8193

1. 7906 9635 

1. 7960 9591

1. 8013 8419 

1. 8065 6462 

1. 8116 4050

1. 8166 1500 

1. 8214 9113

1. 8262 7181 

1. 8309 5984 

1. 8355 5789

1. 8400 6856 

1. 8444 9431

1. 8488 3754 

1. 8531 0055 

1. 8572 8554

1. 8613 9465 

1. 8654 2992

1. 8693 9334 

1. 8732 8682 

1. 8771 1218

1. 8808 7121 

1. 8845 6562

1. 8881 9706

-2 6988 

-2 5843 

-2 4764

-2 3746 

-2 2785

-2 1876 

-2 1017 

-2 0203

-1 9432 

-1 8700

-1 8005 

-1 7346 

-1 6719

-1 6123 

-1 5555

-1 5014 

-1 4499 

-1 4007

-1 3538 

-1 3090

-1 2662 

-1 2253 

-1 1861

-1 1487 

-1 1128

-1 0784 

-1 0455 

-1 0139

-9836 

-9545

-9266 

-8997 

-8739

-8491 

-8252

-8023 

-7801

-7588

-7383

-7185

-6995 

-6811 

-6633

-6462 

-6297

1145 

1079 

1018

961 

909

860 

814 

771

732

694

659 

627 

596

568 

541

515 

492 

469

448 

428

409 

392 

374

359 

344

330 

316 

303

291 

279

269

258 

248

239 

230

221 

213 

205

198 

191

184 

178 

171

165 

160

-70 

-66 

-61

-57 

-52

-49 

-46 

-43

-39

-38

-35 

-32 

-31

_ on£t*J

-27

-26 

-24 

-23

-21 

-20

-19 

-18 

-17

-16 

-15

-14 

-14 

-13

-12 

-12

-11 

-11 

-10

-10

_ n t/

_ o

-8 

-8

-8

-7

-7 

-6 

-6

-6 

-6

Q=+0.9; tabular spacing =0.5

19. 0

19.5

20.0

20.5

21.0

21.5

22.0

22.5

23.0

23.5

1. 8881 9706

1. 8970 1054

1. 9054 6384

1.9135 7858

1. 9213 7469

1. 9288 7052

1. 9360 8306

1. 9430 2802

1. 9497 1997

1. 9561 7243

-3 8371

-3 6018

-3 3856

-3 1864

-3 0027

-2 8329

-2 6758

-2 5301

-2 3949

-2 2692

2352

2163

1992

1837

1698

1572

1457

1352

1257

1170

-212

-190

-171

-155

-139

-126

-115

-104

-95

-87
NUMERICAL VALUES OF THE MODIFIED POTENTIAL 97

TABLE 38. Modified potential Continued

a -w Ml Al, Ai,

Q=+0.9; tabular spacing =0.5 Continued

24.0

24.5

25.0

25.5

26.0

26.5

27.0

27.5

28.0

28.5

29.0

29.5

30.0

1. 9623 9796

1. 9684 0829

1. 9742 1431

1. 9798 2621

1. 9852 5351

1. 9905 0510

1. 9955 8934

2. 0005 1403

2. 0052 8651

2. 0099 1368

2. 0144 0202

2. 0187 5765

2. 0229 8630

-2 1521

-2 0430

-1 9412

-1 8460

-1 7570

-1 6736

-1 5954

-1 5221

-1 4531

-1 3883

-1 3272

-1 2697

1091

1018

952

890

834

782

734

690

649

610

575

542

-79

-73

-67

-62

-56

-52

-48

_ 4.4.

-41

-38

-35

-33

Q=4-1.0; tabular spacing=0.02

0.00

.02

.04

.06

.08

.10

. 12

. 14

. 16

. 18

.20

.22

.24

.26

.28

.30

.32

.34

.36

.38

.40

.42

.44

.46

. 48

.50

.52

.54

.56

.58

. 60

.62

.64

.66

.68

.70

. 72

.74

.76

.78

.80

.82

.84

.86

0. 0000 0000

. 0002 4035

. 0009 6065

. 0021 5868

. 0038 3074

. 0059 7171

. 0085 7511

. 0116 3311

.0151 3667

. 0190 7555

. 0234 3847

. 0282 1316

. 0333 8648

. 0389 4453

. 0448 7273

.0511 5598

. 0577 7870

. 0647 2502

. 0719 7881

. 0795 2380

. 0873 4370

. 0954 2222

. 1037 4324

. 1122 9080

. 1210 4923

. 1300 0316

. 1391 3760

. 1484 3797

. 1578 9014

. 1674 8046

. 1771 9582

. 1870 2358

. 1969 5169

. 2069 6860

. 2170 6336

. 2272 2553

. 2374 4524

. 2477 1315

. 2580 2045

. 2683 5884

. 2787 2054

. 2890 9824

. 2994 8513

. 3098 7483

4 8070

4 7995

4 7772

4 7403

4 6892

4 6242

4 5461

4 4555

4 3533

4 2404

4 1177

3 9863

3 8472

3 7016

3 5504

3 3948

3 2359

3 0747

2 9120

2 7490

2 5863

2 4249

2 2655

2 1086

1 9550

1 8051

1 6593

1 5180

1 3816

1 2502

1 1241

1 0034

8881

7784

6742

5754

4820

3939

3110

2331

1601

918

+ 281

-313

-74

-223

-369

-512

-650

-781

-906

-1022

-1129

-1227

-1314

-1391

-1456

-1512

-1556

-1589

-1613

-1626

-1630

-1626

-1614

-1594

-1568

-1536

-1499

-1458

-1413

-1364

-1314

-1261

-1207

-1153

-1097

-1042

-988

-934

-881

-829

-779

-730

-683

-637

-594

-552

-149

-149

-146

-143

-138

-132

-124

-116

-107

-97

-88

-77

-66

-55

-44

-33

-24

-14

-4

+ 4

12

20

26

32

37

41

45

49

50

52

54

54

56

55

54

54

52

52

50

48

48

46

43

42

TABLE 38. Modified potential Continued

a -w Ai, A!, Ai,

Q=+1.0; tabular spacing=0.02 Continued

0.88

.90

.92

.94

.96

.98

1.00

1.02

1. 04

1.06

1.08

1. 10

1. 12

1. 14

1

1 G. . lo 

1. 18

1.20

1.22

1. 24

1.26

1.28

1.30

1.32

1.34

1.36

1.38

1. 40

1.42

1.44

1.46

1.48

1.50

1.52

1.54

1.56

1.58

1.60

1.62

1. 64

1. 66

1.68

1. 70

1. 72

1.74

1.76

1.78

1. 80

1.82

1.84

1.86

1.88

1.90

1. 92

1.94

1.96

1.98

2.00

2.02

2.04

2.06

2.08

0. 3202 6140

. 3306 3932

. 3410 0347

. 3513 4911

. 3616 7186

. 3719 6769

. 3822 3287

. 3924 6400

. 4026 5795

. 4128 1188

. 4229 2319

. 4329 8952

. 4430 0873

. 4529 7891

Acno QQOO . IvWo yooo

. 4727 6544

. 4825 7886

. 4923 3737

. 5020 3991

. 5116 8554

. 5212 7346

. 5308 0297

. 5402 7349

. 5496 8455

. 5590 3574

. 5683 2678

. 5775 5743

. 5867 2755

. 5958 3705

. 6048 8590

. 6138 7414

. 6228 0186

. 6316 6917

. 6404 7626

. 6492 2334

. 6579 1065

. 6665 3848

.6751 0712

. 6836 1693

. 6920 6825

. 7004 6146

. 7087 9696

. 7170 7518

. 7252 9652

. 7334 6146

. 7415 7042

. 7496 2389

. 7576 2233

. 7655 6623

. 7734 5608

. 7812 9236

. 7890 7557

. 7968 0622

. 8044 8480

.8121 1183

. 8196 8780

. 8272 1322

. 8346 8860

. 8421 1445

. 8494 9126

.8568 1954

-865

-1377

-1851

-2289

-2693

-3064

-3405

-3717

-4002

-4262

-4498

-4711

-4904

-5076

ROOT

-5369

-5491

-5598

-5691

-5772

-5841

-5899

-5947

-5986

-6016

-6038

-6054

-6062

-6064

-6061

-6053

-6040

-6022

-6001

-5977

-5949

-5918

-5884

-5848

-5811

-5771

-5729

-5686

-5642

-5597

-5550

-5502

-5454

-5406

-5356

-5307

-5257

-5206

-5156

-5105

-5055

-5004

-4954

-4903

-4853

-4803

-512

-474

-438

-404

-372

-341

-312

-285

-260

-236

-213

-192

-173

-155

1 QQ loo

-122

-107

-93

-81

-69

-58

-48

-39

-30

-22

-15

-8

-2

+ 3

8

13

18

21

25

28

31

34

36

38

40

42

43

44

45

47

48

48

49

49

50

50

50

50

50

51

51

50

50

50

50

50

40

38

36

34

32

31

29

27

25

24

22

21

20

18

17 

16

15

14

12

12

11

\

10

9

9

8

7

7

6

5

5

5

4

4

4

3

3

3

2

2

2

2

1

2

1

2

1

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0
98 APPARENT RESISTIVITY OP A SINGLE UNIFORM OVERBURDEN

TABLE 38. Modified potential Continued

a -w Ai, A!. Al,

Q=+1.0; tabular spacing=0.02 Continued

2. 10

2.12

2.14

2.16

2. 18

2.20

0. 8640 9980

. 8713 3251

. 8785 1819

. 8856 5733

. 8927 5040

. 8997 9791

-4753

-4704

-4655

-4606

-4557

50

49

49

49

48

0

0

0

0

1

Q=+1.0; tabular sj>acing=0.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3.0

3. 1

3.2

3.3

3.4

3.5

3.6

3.7

3.8

3.9

4.0

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

5.0

5. 1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

5.9

6.0

6. 1

6.2

6.3

6.4

6.5

6.6

6.7

6.8

6.9

7.0

7. 1

7.2

7.3

7.4

0. 8997 9791

. 9343 6857

. 9678 7038

1. 0003 5912

1. 0318 8757

1. 0625 0541

1. 0922 5932

1. 1211 9301

1. 1493 4744

1. 1767 6092

1. 2034 6929

1. 2295 0610

1. 2549 0274

1. 2796 8860

1. 3038 9122

1. 3275 3644

1. 3506 4850

1. 3732 5017

1. 3953 6285

1. 4170 0668

1. 4382 0063

1. 4589 6258

1. 4793 0939

1. 4992 5699

1. 5188 2044

1.5380 1397

1. 5568 5107

1. 5753 4451

1. 5935 0640

1. 6113 4824

1. 6288 8096

1. 6461 1493

1. 6630 6003

1. 6797 2567

1. 6961 2080

1. 7122 5396

1. 7281 3330

1. 7437 6660

1. 7591 6129

1. 7743 2447

1. 7892 6294

1. 8039 8320

1. 8184 9147

1. 8327 9374

1. 8468 9571

1. 8608 0288

1. 8745 2051

1. 8880 5366

1. 9014 0720

1. 9145 8580

1. 9275 9396

1. 9404 3598

1. 9531 1605

-11 2745

-10 6885

-10 1306

-9 6030

-9 1060

-8 6394

-8 2021

-7 7926

-7 4095

-7 0510

-6 7156

-6 4018

-6 1078

-5 8323

-5 5740

-5 3316

-5 1039

-4 8899

-4 6885

-4 4988

-4 3200

-4 1514

-3 9921

-3 8416

-3 6992

-3 5644

-3 4366

-3 3155

-3 2005

-3 0913

-2 9874

-2 8887

-2 7947

-2 7051

-2 6197

-2 5382

-2 4604

-2 3861

-2 3151

-2 2471

 2 1821

-2 1198

-2 0601

-2 0029

-1 9480

-1 8954

-1 8448

-1 7961

-1 7494

-1 7045

-1 6612

-1 6196

-1 5795

5860

5579

5277

4969

4666

4373

4095

3831

3584

3354

3139

2940

2755

2583

2424

2277

2140

2014

1897

1788

1687

1593

1505

1424

1348

1278

1211

1150

1092

1038

988

940

896

854

815

778

743

710

679

650

623

597

572

549

527

506

486

467

449

433

416

401

386

-241

-281

-302

-308

-303

-294

-278

-264

-247

-230

-215

-199

-185

-172

-159

-147

-137

-126

-117

-109

-101

-94

-87

-82

-76

-71

-66

-62

-58

-54

-51

-48

-44

-42

-40

-37

-35

-33

-31

-29

-28

-26

-25

-23

-22

-21

-20

-19

-18

-17

-16

-15

-15

TABLE 38. Modified potential Continued

a

-W Al, Al, Al,

Q=+1.0; tabular spacing=0.1 Continued

7.5

7.6

7.7

7.8

7.9

8.0

8. 1

8.2

8.3

8.4

8.5

8. 6

8. 7

8. 8

8. 9

9.0

9. 1

9.2

9 3

9.4

9.5

9.6

9.7

9. 8

9.9

10.0

1. 9656 3817

1. 9780 0620

1. 9902 2388

2. 0022 9478

2. 0142 2238

2.0260 1003

2. 0376 6094

2.0491 7825

2. 0605. 6496

2. 0718 2400

2. 0829 5818

2. 0939 7022

2. 1048 6278

2. 1156 3839

2. 1262 9954

2. 1368 4862

2. 1472 8795

2. 1576 1980

2. 1678 4633

2. 1779 6966

2. 1879 9186

2. 1979 1492

2. 2077 4076

2. 2174 7128

2. 2271 0829

2. 2366 5358

-1 5409

-1 5036

-1 4677

-1 4330

-1 3996

-1 3673

-1 3361

-1 3059

-1 2768

-1 2486

-1 2213

-1 1949

-1 1694

-1 1446

-1 1207

-1 0974

-1 0749

-1 0531

-1 0319

-1 0114

-9914

-9721

-9533

-9350

-9173

373

359

347

334

323

312

301

292

282

273

264

256

247

240

232

225

218

212

206

199

194

188

183

177

172

-14

-14

-12

-12

-12

-11

-11

-10

-10

_ g

-9

-8

-8

__ Q

_ _ O

-7

-7

_ 17

-6

-6

-6

-6

-5

-5

-5

Q=+1.0; tabular spacing=0.2

10.0

10. 2

10.4

10.6

10. 8

11.0

11. 2

11.4

11.6

11.8

12.0

12.2

12.4

12. 6

12.8

13.0

13.2

13.4

13.6

13. 8

14.0

14. 2

14.4

14. 6

14. 8

15.0

15. 2

15. 4

15. 6

15.8

16.0

16.2

2. 2366 5358

2. 2554 7581

2. 2739 5121

2. 2920 9230

2. 3099 1092

2. 3274 1830

2. 3446 2512

2. 3615 4149

2. 3781 7703

2. 3945 4090

2. 4106 4180

2. 4264 8804

2. 4420 8754

2. 4574 4784

2. 4725 7616

2. 4874 7938

2. 5021 6410

2. 5166 3662

2. 5309 0298

2. 5449 6896

2. 5588 4010

2. 5725 2172

2. 5860 1891

2. 5993 3659

2. 6124 7945

2. 6254 5202

2. 6382 5865

2. 6509 0353

2. 6633 9068

2. 6757 2399

2. 6879 0721

2. 6999 4392

-3 6006

-3 4683

-3 3432

-3 2246

-3 1123

-3 0057

-2 9045

-2 8083

-2 7168

-2 6296

-2 5466

-2 4675

-2 3920

-2 3198

-2 2509

-2 1850

-2 1220

-2 0616

-2 0038

-1 9484

-1 8952

 1 8442

-1 7952

-1 7481

-1 7029

-1.6594

-1 6175

-1 5772

-1 5384

-1 5010

-1 4650

-1 4302

1323

1251

1185

1124

1066

1012

962

915

871

830

792

755

721

689

659

630

604

578

554

532

510

490

471

452

435

419

403

388

374

360

348

336

-77

-72

-66

-62

-58

-54

-50

-47

-44

-41

-38

-36

-34

-32

-30

-29

-27

-26

-24

-22

-22

-20

-19

-18

-17

-16

-16

-15

-14

-14

-12

-12
REFERENCES CITED 99

TABLE 38. Modified potential Continued

a -w Ai, Ai, Ai,

Q=+1.0; tabular spacing =0.2 Continued

16.4 

16. 6

16.8

17.0

2. 7118 3762 -1 3966 

2. 7235 9166 -1 3642 

2. 7352 0928 -1 3330

2. 7466 9359

324 

313 

302

-12 

-11 

-11

Q=+1.0; tabular spacinj;=0.5

17.0

17.5

18.0 

18. 5 

19.0 

19.5

20.0 

20. 5 

21.0 

21. 5 

22.0

2. 7466 9359

2. 7748 4080 

2. 8022 1802 

2. 8288 6624 

2. 8548 2325 

2. 8801 2398

2. 9048 0076 

2. 9288 8362 

2. 9524 0045 

2. 9753 7723 

2. 9978 3821

-8 1450

-7 6998 

-7 2900 

-6 9121 

-6 5629 

-6 2394

-5 9392 

-5 6602 

-5 4004 

-5 1581 

-4 9317

4452

4097 

3779 

3493 

3234 

3002

2790 

2598 

2424 

2264 

2118

-397

-355 

-318 

-286 

-258 -233

-212 

-192 

-174 

-160 -146

TABLE 38. Modified potential Continued

a

-W Ai, Ai, Ai,

Q=+1.0; tabular spacing=0.5 Continued

22. 5

23.0

23.5

24.0

24 5

25.0

25. 5

26. 0

26.5

27.0

27.5

28.0

28.5

29. 0

29.5

30.0

3. 0198 0602

3. 0413 0183

3. 0623 4550

3. 0829 5565

3. 1031 4976

3. 1229 4431

3. 1423 5478

3. 1613 9579

3. 1800 8114

3. 1984 2387

3. 2164 3631

3. 2341 3014

3. 2515 1643

3. 2686 0569

3. 2854 0790

3. 3019 3253

-4 7199

-4 5214

-4 3352

-4 1603

-3 9957

-3 8407

-3 6946

-3 5566

-3 4262

-3 3029

-3 1861

-3 0754

-2 9703

-2 8706

-2 7757

1985

1862

1749

1646

1550

1461

1380

1304

1234

1168

1107

1050

998

948

902

-133

-123

-112

-104

-96

-89

-82

-76

-70

-65

-61

-57

-53

-49

-46

REFERENCES CITED

Barus, Carl, 1882, On the electrical activity of ore bodies, in

George F. Becker, 1882, Geology of the Comstock Lode and

the Washoe district: U.S. Geol. Survey Monograph 3, p.

309-367. 

Charlier, C. V. L., 1907, Mechanik des Himmels: v. 2, Von Veit,

Berlin.

Fox, R. W., 1830, On the electromagnetic properties of metalliferous veins in the mines of Cornwall: Phil. Trans. Royal

Soc., v. 2, p. 399. 

Gish, O. H., and Rooney, W. J., 1925, Measurement of resistivity

of large masses of undisturbed earth: Terrestrial Magnetism

and Atmospheric Electricity, v. 30, p. 161-188. 

Heiland, C. A., 1940: Geophysical exploration; New York,

Prentice-Hall, 1013 p. 

Hummel, J. N., 1929a, Der scheinbare spezifische Widerstand:

Zeitschr. Geophysik, v. 5, p. 89-105. 

Hummel, J. N., 1929b, Der scheinbare spezifische Widerstand

bei vier planparallelen Schichten: Zeitschr. Geophysik,

v. 5, p. 228-238. 

Jakosky, J. J., 1950, Exploration geophysics: Trija Pub. Co.,

Los Angeles, 2d ed., 786 p. 

Jeans, J. H., 1925, Electricity and magnetism: Univ. Press,

Cambridge, 5th ed., 652 p. 

Jolley, L. B. W., 1925, Summation of series: Chapman and

Hall, London., 232 p. 

Keller, Joseph B., 1953, The scope of the image method: Commun. pure and appl. math. (Inst. Math. Sci., New York

Univ.), v. 6, p. 505-512.

Mathematical Tables Project, 1944: Tables of Lagrangian 

interpolation coefficients, Columbia Univ., New York, 392 p.

Maxwell, J. C., 1892, Electricity and magnetism: Univ. Press, 

Oxford, 3d ed., v. 1.

Roman, Irwin, 1931, How to compute tables for determining 

electrical resistivity of underlying beds and their application 

to geophysical problems: U.S. Bur. Mines Tech. Paper 502.

 = 1936, An Euler summation formula: Am. Math. Monthly, 

v. 43, p. 9-21.

 '1951, Resistivity reconnaissance in Symposium on 

surface and subsurface reconnaissance: Am. Soc. Testing 

Materials, Spec. Tech. Pub. 122 p. 171-220.

This paper contains a bibliography of 127 entries covering 

various aspects of resistivity prospecting.

Scarborough, J. B., 1950, Numerical mathematical analysis: 

Johns Hopkins Press, Baltimore, 2d ed., 511 p.

Steffensen, J. F., 1927: Interpolation, Williams and Wilkins,

Baltimore, 248 p. 

Thomson, J. J., 1897, Elements of the mathematical theory of

electricity and magnetism: Univ. Press, Cambridge, 2d ed.

Thomson, Sir William (First Baron-Lord Kelvin) 1884: Reprints 

on electrostatics and magnetism, Macmillan, London.

Wenner, Frank, 1915, A method of measuring earth-resistivity: 

U.S. Bur. Standards, Sci. Paper 258, p. 469-478.

Willers, F. A., 1948: Practical analysis: Dover, New York, 422 p.

O