-, -y-i-.x.

~ '- . Í ~ -- * <;; ~ï- , '',

"J

IHHI

Í 1 ^s 'J'

v-^-*-

I

1

If ID lb. N costs Í1.60 and eùrn

a bu., 1.20 bu. will buy 10 lb.

- 1.33). MPR is 165 lb. and yield

122 bu.

I I '

Il 1,1' -J I I I l_ I . I

1

80 160 240 320

10-lb units of nitrogen -Rates applied as indicated

Agriculture Handbook No. 93

UNITED STATES DEPARTMENT OF AGRICULTURE

Agricultural Research Service, Washington, D. C.

January 1956
PREFACE

Recent rate experiments have provided an

improved basis for exploring the nature of response to fertilizers. More rates of each of the

principal plant nutrients—^nitrogen, phosphoric

oxide, and potash—are now applied at different

levels of other yield-influencing factors than was

done in the past. As results from more of these

experiments become available, studies can be

made as to the relative merits of different yield

equations. Equations that are found to be satisfactory may be used in estimating yield response

and returns per acre at different combinations of

yield-influencingfactors. These studiesinvolve use

of statistical methods, so that statements may be

madewith confidence for eachimportant conclusion.

There is always need for continuing tests of

adequacy of different yield equations and for determining confidence limits of estimates developed

from the tests. Statistically trained researchers

will need to make these evaluations. However,

many research workers, teachers, and others will

find graphic methods useful in obtaining the

practical answers desired. Sums of squared

residuals can then be used to indicate the amount

of variability in the reported data. The method

presented here is intended for those who have

neither the time nor the facilities to obtain results

that are statistically precise^ but who are called

upon to make recommendations regarding profitable use of fertilizer.

CONTENTS

Page

Summary 1

The yield equation used 1

Graphic use of the exponential equation 2

Fitting the standard curve to reported yields^> . 2

Results from both graphic and precisely fitted

curves 3

Graphic estimate of yields and most profitable

rate with one nutrient varied 6

Graphic estimate of yields with two nutrients

varied 7

Yields at any combination 8

Economic combinations of nutrients 9

Minimum cost nutrient combinations for a

specified yield 11

Page

Exponential and quadratic square-root equations

compared 14

Measures of fit 14

Most profitable rates and confidence limits 16

Appendix 18

Calculation of most profitable rate of one nutrient. 18

Calculation of economic combinations of three

nutrients 18

Calculation of the most profitable combination

of nutrients 18

Forms 19

Values of l~ß* (table 9) 20

Washington, D. O. Issued January 1956

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

Washington 25. D. C. - Price 20 cents
A Graphic Method of

Interpreting Response to Fertilizer'

By D. B. ibach, agricultural economist, Production Economics Research Branch, Agricultural Research Service

SUMMARY

The exponential equation used here lends itself

well to graphic methods of estimating response to

fertilizers. The most profitable rate of a fertilizer

can be read directly from a graphically fitted ciirve.

The most profitable combination of two or more

nutrients may be calculated readily after the constants of the equation are found by graphic methods for each nutrient. When they are carefully

done, graphic methods yield results that closely approximate those obtained from mathematical determination of the constants. Graphic methods of

using the exponential equation are easily followed

by those unfamiliar with statistical techniques,

who often need to make reconamendations as to

use of fertilizer based on residts of rate

experiments.

The important use of a yield equation in connection with fertilizer rate experiments is to predict

the most profitable rates and combinations, and

the yields of the crops at those rates. Possibilities

of profitable substitution (economic) of one

nutrient for another are limited. A decline in the

price of the crop subsequent to application of

fertilizer at a given rate is of little significance with

respect to the optimum economic rate of applica-^

tion. The amount of reduction in return above

cost of fertilizer because of failure to apply the most

profitable rate depends on the shape of the curve

of response to additional applications. Existence

of risk factors will usually suggest applications

somewhat below the calculated most profitable

rates.

Use of different equations may produce results

that fit the reported yields about equally well for

that part of the surface that is of practical interest.

Ease of handling the work and goodness of fit over

the range within which applications would be most

profitable for farmers with either limited or

unlimited funds are important points to be considered in selecting an equation.

,-Î.This handbook supersedes U. S. Bur. Agr. Econ. F. M.

105, Determining Profitable Use of Fertilizers, 1953, by

^. B. Ibach and S. W. Mendum. Acknowledgment is due

0' W. Mendum and Burton L. French, Production Economics Research Branch, Agricultural Research Service,

tor helpful suggestions and assistance in the computations

s-na for consultation on statistical methods, respectively.

Experiments should include rates that encompass the total range in response, so that a reliable

estimate of the maximum yield can be obtained.

Also, an adequate number of rates in the area of

greatest curvature is needed to give reliability to

estimates of yield in that portion of the ciu-ve. If

these conditions are met, reconmaendations based

on fitted curves—graphic or otherwise—^need not

be questioned because of deviations on the steeper

part of the curve or on the portion to the right of

the most profitable rate. On the steep part of the

curve, returns usually are so high that this part of

the curve needs no special study even for the

farmer with limited funds. Then, as the calculated most profitable rate is approached, the

response to additional fertilizer is usually so small

that risk factors outweigh possible gains from

further increments in yield, even for farmers

whose funds are unlimited.

The first pages of this handbook include discussion and illustration of a graphic method of fitting

the curve^ and of the results obtained. This part

of the handbook, together with Appendix table 9,

should be useful in developing recoromendations

and other illustrative material based on suitable

response data. The rest of the handbook concerns

the question of minimum cost combinations for a

specâied yield. It contains also a comparison of

results from use of the exponential and the

quadratic square-root equations.

THE YIELD EQUATION USED

The equation illustrated—an exponential form—

was presented and discussed earlier.^ It rests on

the hypothesis that as fertilizer is added in units

of uniform size, with other factors unchanged,

increments in yield decrease at a constant per2 IBACH, D. B., and MENDUM, S. W. DETERMINING

PROFITABLE USE OP FERTILIZERS. U. S. Bur. AgT. Econ.

F. M. 105, 70 pp., iUus. 1953.

SPILLMAN, W. J. USE OP THE EXPONENTIAL YIELD CURVE

IN FERTILIZER EXPERIMENTS. U. S. Dept. Agr. Tech. Bui.

348, 66 pp., illiis. 1933.

SPILLMAN, W. J., and LANG, E. THE LAW OF DIMINISHING RETURNS. 178 pp., illus. Ncw York and Chicago.

1924.

1
centage rate. This ratio between successive increments in yield may have a different numerical

value, depending on growth conditions.^ Under

most field conditions, some factors usually operate

to limit the yield that could be obtained if all

conditions were ideal. For practical purposes,

therefore, it is necessary that the method used

allow for variation in the constants of the yield

equation, according to conditions of the experiment.

For purposes of illustration here, the equation

reflecting the hypothesis indicated above is written as y=M (1—fi^). In this equation, y is the

calculated yield at any specified level of the nutrient, M is the theoretical maximum yield attainable through use of fertilizer, for conditions

accompanying the reported yields, R is the ratio of

successive increments in yield—a constant having

a value more than 0 but less than 1.0— and x

is the quantity of the independent variable—

fertilizer—changes in which are found to be associated with changes in yield. The terrn "1—J?"^''

is expressed as a decimal fraction, which represents the ratio between a parti6ular yield and

the maximum yield. The quantity of a plant

nutrient (x) associated with yield refers to both

soil content and applied portion. For example,

when X refers to nitrogen (N), soil content is

designated as n, and a is used to indicate the

applied portion. Thus, x=n+a. Similarly, for

phosphoric oxide (P2O6), x=p+b; and for potash

(K2O), x=k+c. The term '^soil content" as used

here is not meant in a strictly literal sense. The

symbols n, p, or k designate constants that are

functions of the yield equation used.

Results that involve one or more than one

independent variable can be obtained. For combinations of independent variables, results are

obtained at present only by fitting each regression curve at a specified level of each of the other

nutrients. Thus far, no one has worked out for the

exponential model a convenient method of solving

the least-squares estimating equations when more

than one input is applied at different levels of

other inputs. Also, such a solution would not

lend itself to simple graphic treatment. For the

present, the multivariable form of this equation

is used for any particular experimental data

under the assumption that the rate (R) of response

to a nutrient is the same at different specified

levels of other nutrients. The results applied in

this way to three factorial experiments are compared with those obtained from simultaneous

estimates of values for another equation applied

to the same data.

3 E. A. Mitscherlich in Germany developed the hypothesis

that for a given size of unit of a specified nutrient the ratio

between successive increments in yield is constant for all

crops and soils, provided no other factors limit the yield.

This view is also advanced in the United States, primarilv

by 0. W. WiUcox. See » ^ j

MITSCHERLICH, E. A., DAS ERGEBNIS VON tJBER 27,000

FELD-DUNGUNGSvERsircHEN. Ztschr. f. Pflanzeuemähr

Düngung u. Bodenk. 38: 22-35. 1947.

WiLLCOX, O. W. ABC OF AGROBIOLOGY. 323 PP.. iUuS

New York. 1937. ^^ '

Graphic Use of the Exponential Equation

As the equation used is based on the assumption of a constant ratio (R) of decreasing increments in yield, a table of 1—Ä^ (yield ratio)

values based on some specified value of R can be

prepared. As Spillman had prepared such a

table in terms of R=0.8,^ this value is adopted

(see table 9, Appendix). This table of yield ratio

values may be put in the form of a graph, or

graphs, in which these values on the vertical scale

are plotted against number of units of the size

required to make R=O.S. Connecting the points

plotted in this way results in a ''standard yield

curve.'' Plotting this on different horizontal

scales (unit values) results in standard yield

curves of different shapes. For convenience, a

specified value of R (0.8) is used, but reflection

of different rates of response in different situations is permitted by the fact that the scale which

indicates the size of a unit is different for each

standard curve. The number of units associated

with a particular yield ratio value is the same

on all curves.

Several such curves should be prepared so that

use of some part of one of them will result in estimated constants that closely approximate those

mathematically determined, if rates used in the

experiment cover a range wide enough to characterize the response and, particularly, to locate

the area of the maximum yield. K rates do not

extend well beyond the steeper part of the curve,

only estimates of yield falKng within the range

of rates applied can be considered rehable. However, this limitation is often less serious with

respect to the exponential equation than with

others. If the pattern of reported yields gives

evidence that no further response would be obtained, reliability of the value of M read from the

curve for use in the prediction equation will be

greatly increased.

Standard yield curves that are clear and thin

should be drawn to dimensions that permit use

of overlays on which yields can be plotted on a

scale that is large enough to permit accurate

readings. Figure 1 illustrates part of a standard

yield curve. The horizontal line across the top

of figure 1 represents M, the maximimi yield

That is, at M the 1—S^, or yield ratio, value is 1.0.

For permanent use, standard yield curves may be

photographed on layers of opaque plastic material.

FiniNG THE STANDARD CURVE TO REPORTED YIELDS

Reported yields can be plotted against rates of

application on cross-section paper. By using the

overlay process, it is possible to find a portion of

one of the standard curves that appears to approximate a least-squares fit to the data. Figure 2

illustrates a set of data plotted on an overlay

sheet. Figure 3 shows figure 2 overlaid on the

standard curve of figure 1. Figure 3 is enlarged:

* See footnote 2, p. 1.
YIELD

RATIO

STANDARD YIELD CURVE

X-J I L J-.L

30 35 40

UNITS (X)

* R = 0.8

U.S. DEPARTMENT OF AGRICULTURE NEC. 55(5)-976 AGRICULTURAL RESEARCH SERVICE

Figure 1.—Section of standard yield curve when /?=0.8.

to permit more exact estimates of fractional units

and yield ratio values.

When a position on the standard curve is found

so that the curve appears to fit the data, further

refinement can be obtained by counting the deviations of reported yields both above and below

the standard yield curve. These may be recorded and totaled in 2 columns, 1 for ''+'' and

1 for "—'' deviations. When the sum of these 2

Columns is as nearly 0 as can be obtained, the

fit may be said to approximate the fit that could

be determined mathematically, provided that the

sum of squared deviations is smaller for the part

of the curve tested than it would be for some other

part of the curve or another standard yield curve.

When the estimate of best fit is obtained, the

laaximum yield (M) is read at the point at which

the yield scale on the overlay sheet intersects the

top line, indicating that 1—JS^=1.0 on the standard yield chart.

Thus, by repeated trials in which the position

01 the overlay sheet is shifted, it is possible by

^aphic means to obtain a fit to the curve that

closely approximates one that is mathematically

determined. Care must be taken to see that the

overlay sheet is held in a position that "squares"

with the grid marks at the edges of the sheet on

which the standard yield curve is shown.^

RESULTS FROM BOTH GRAPHIC AND PRECISELY FIHED

CURVES

Three experiments in 1952 and 1953, each carrying 12 rates of nitrogen on irrigated corn, were

used in comparing results from use of the graphic

method described with results obtained later when

the least-squares fit was precisely determined. In

determining the least-squares fit, the iterative

method demonstrated by Stevens was carried out

5 Even when an accurately computed curve is desired,

this graphic approach saves time, in that often only one

series of computations is required to find the true constants.

Experience thus far indicates that use of this combination

of the graphic approach plus one operation of the mathematical solution results in standard errors that are little

improved by further iterations. Thus, for most practical

purposes, the graphic device as a preliminary step naakes

use of the exponential function as easy and as rapid in

developing a least-squares fit as does use of other functions

that do not require iterative techniques.
REPORTED YIELDS OF CORN

YIELD

(BU./A.)

[

140

120

100

80

60

40

20 î

T—I—I—I—I—I—I—I—•—'—I—I ' ' ^

-1—I—I—I—I—I—I 1—i I > r0 i-uL I I I I I I I I r I I I I I

' ! '

0 40 80 120 160 200 240 280

LB. N APPLIED PER ACRE

320

U.S. DEPARTMENT OF AGRICULTURE NEC. 55(5)-977 AGRICULTURAL RESEARCH SERVICE

Figure 2.—Reported yields per acre of corn when nitrogen is the variable nutrient.

UQtil no improvement in the fit could be obtained.®

Results from the 2 methods are compared in table

1. Comparisons were made of all 12 rates of

nitrogen and also for a smaller number of rates

well distributed over the range. Results were

about the same; they indicated that, in these instances, 5 or 6 rates distributed over the range

were adequate for purposes of economic interpretation. In the 3 experiments, rates were carried

to levels sufficiently high to characterize the

response. Inclusion of such levels is important

in obtaining reliable answers through use of any

yield equation.

In 3 of the 6 comparisons between graphic and

precisely determined curves, differences in yields

at the most profitable rates were smaller than 1

standard error in the yield at the most profitable

rates on the mathematically fitted curves. (Compare differences between table 1, cols. 3 and 4,

with the standard errors shown in col. 5.) This

means that these differences in yields are no greater

than might be expected as a result of chance.

8 STEVENS, W. L. ASYMPTOTIC REGRESSION. Biometrics 7: 247-267. 1951.

The effect of differences in the most profitable

rates of nitrogen, as calculated from the two

methods, is brought out in columns 8 to 10, table 1.

In column 8 is shown the return above cost of

fertilizer when it is calculated from the graphically

fitted curve. Column 9 shows the same for the

mathematically fitted curve. Then, using constants of this curve, the return above cost of

fertilizer was calculated for the rate indicated as

most profitable on the graphically fitted curve.

These results are shown in colimin 10. Differences

within these three columns are negligible. Differences between columns 9 and 10 represent what a

farmer woidd lose per acre if he applied the

graphically estimated most profitable rates instead of those calculated with precision. This

assumes that conditions of the experiment were

duplicated exactly in the field. The probability

of this occurrence is remote.

Sums of squared residuals for the two curves are

also shown in table 1. These sums are approximately twice as great for the graphic as for the

mathematically fitted curve. Obviously, if the

problem were one in which precision, rather than
REPORTED YIELDS SUPERIMPOSED

YiEiD ON STANDARD CURVE

YIELD (BU./A)

RATIO

"1—I—I—^—r

.650 q ^,1

0 5

''''''' ' ' ' ' J_L.

120 160 200

LB. N APPLIED PER ACRE

240 280 320

10

—r- 15

I I I

20 25

UNITS (x)

* R-0.8

30

I

35 40

U.S. DEPARTMENT OF AGRICULTgRE NEC. 55(5)-978 AGRICULTURAL RESEARCH SERVICE

Figure 3.—Reported yields of corn at different rotes of nitrogen superimposed on standard yield curve.

a basis for approximate field recommendations

was needed, the mathematically fitted curve would

be necessary. But in interpreting results for

application in the field, approximations that can

be made by the method described produce results

so close to those obtained by the most accurate

methods that the additional time required by the

mathematically fitted curve is not justified. If

the exponential equation is suited to the data at

hand, the graphic method of fitting the curve is a

useful device for those who need reliable answers

quickly.

TABLE 1.—Comparison of most profitable rates of nitrogen for corn, yields, residual sums of squares, and return per acre, as

calculated from graphic approximation to least squares and from mathematically determined least-squares fits to the reported

data, 1962-53 i

Experiment and series of data used

Experiment 1, Hardy, Nebr.:

6 rates spaced over range.

_ All 12 rates

«•xperiment 2, Ontario, Oreg.:

6 rates spaced over range..

-, A1112rates __.!_..

iixperiment 3, Presser, Wash.:

6 rates spaced over rangeAll 12 rates _..__.

Most profitable

rate of nitrogen—

On graphic curve

(1)

Pounds

136

134

168

168

233

231

On fitted

curve

(2)

Pounds

153

164

151

164

Yield per acre at

most profitable

rateOn graphic curve

(3)

Bushels

llfl.ll

118.99

140.69

142.01

137.86

135.34

On fitted

curve

(4)

Bushels

119. 90

121.63

139.01

141.75

140. 92

140.25

Standard error

of yield

at most

profitable

rate on

fitted

curve

(5)

Bushels

±1.38

=bl.37

=b5.03

d=1.71

±2.71

±1.99

Residual sums of

squares—

On graphic curve

(6)

76.36

331.35

85.40

270.29

121.27

519.96

On fitted

curve

(7)

27.31

161.74

74.96

254. 78

52.80

307.20

Return per acre above cost

of fertilizer—

At most profitable

rateOn graphic curve

(8)

Dollars

147. 26

147.53

173.23

174.65

158. 52

155.24

On fitted

curve

(9)

Dollars

145.78

146.47

173.23

175.04

157.39

156.12

On fitted

curve at

most profitable

rate on

graphic

curve

(10)

Dollars

145.09

145.50

173.08

175.07

156.45

154.98

- , 1 Based on data contained in U. S. Dept. Agr. Tech. Bui. 1141, A Method of Economic Analysis Applied to Nitrogen Fertilizer Rate Experiments on

^igated Corn, by J. L. Paschal and B. L. French. [In oress.l
GRAPHIC ESTIMATE OF YIELDS AND MOST

PROFITABLE RATE WITH ONE NUTRIENT VARIED

An estimate of yield at any rate of application

is read from the standard curve traced on the

overlay sheet, which is scaled to show yields and

rates of application. When only 1 independent

variable is involved, it is not necessary to find

constants of the yield equation. The yields are

read directly from the curve, as indicated in

figure 3. In this illustration, the curve is based

on 9 rates of N, including no application, with no

other fertilizer variable.

The most profitable rate of application is

located by a simple calculation. Figure 4 represents the overlay portion of figure 3, with the

curve traced in. The abscissa of figure 4 is scaled

to indicate 10-pound units of application so that

increments in yield associated with each 10-pound

unit may be estimated. Suppose 10 pounds of N

costs $1.60, including cost of application, and that

a bushel of com standing in the field before harvest

is worth $1.40. Then 1.14 bushels of corn are

equal in value to 10 pounds of N ($1.60/$1.40).

This would mean that it would be profitable to

apply N until the last 10 pounds added would

result in an additional yield of 1.14 bushels of com.

So the most profitable rate is within the 13th

10-pound unit applied, because this unit is associated with the last increment in yield that is

equal to or exceeds 1.14 bushels. Thus, the most

profitable rate is between 120 and 130 pounds of

N, and the yield at that rate is between 120 and

122 bushels.

If the standard yield curve and the overlay

grid are plotted on a suflSciently large scale, these

estimates may be made easily with sufiicient

accuracy for any practical purpose. For example,

based on an accurately calculated least-squares

fit, the most profitable rate of N was found to be

129 pounds, and the calculated yield at that rate

is 121.6 bushels.

The particular shape of the curve in the area of

greatest curvature may affect substantially the

estimate of the most profitable rate. This serves

to emphasize the importance of an adequate

number of rates in that area, as well as in the area

of the maximum yield.

ESTIMATING MOST PROFITABLE RATE

From Corn Yield Response fo Nitrogen

Bu. PER ACRE ^zizz::::::::::::::::::::::::::^:::;;::^^

120

1001- y^ I Last yield increment

(10 LB. N WORTH 1.14 BU)

EXAMPLE:

COST OF 10 LB. N _ $1.60 _ , ,, o,.

PRICE PER BU. - JOO" ~ '•'* **"•

0 I" ' ''' ''' ''■■ I I '''' '''

0 40 80 120 160 200 240 280 320

LB. APPLIED PER ACRE

U. S. DEPARTMENT OF AGRICULTURE NEC. 55(5)-972 AGRICULTURAL RESEARCH SERVICE

Figure 4.—Estimating the most profitable rate from corn yield response to nitrogen.
GRAPHIC ESTIMATE OF YIELDS WITH TWO NUTRIENTS

VARIED

The illustration presented here is one in which

the reported yields at different rates of N, used in

developing the curve, were obtained when the

application of P2O5 was held constant at 160

pounds per acre. Similarly, fields used in finding

constants for the PgOg curve were obtained when

the application of N was held constant at 160

pounds. Graphic methods may be used in finding

the constants of the curve for each nutrient.

In figure 3 the abscissa of the overlay (fig. 2)

was purposely coordinated with the unit scale of

the standard curve of figure 1. This was done to

show the relationship between the scale in pounds

for the nutrient applied and the number and size

of units (x) when R=O.S for the curve as fitted.

Coordination of the overlay with the standard

curve in this way permitted direct reading of n

from the abscissa of the standard curve. It also

permitted use of the scales on the standard curve

for obtaining values from which is computed the

size of a unit of N needed to make 5=0.8, referred

to as Ua^ The reason for this careful coordination

was to illustrate more closely the relationship of

these constants to the curve.

However, in practice it is neither necessary nor

feasible to obtain the precise coordination of scales

on the overlay with those on the standard curve.

All that is required is that one of the standard

curves (many should be prepared) be found that

fits the data, and that it be used only to obtain

a reading of M on the ordinate of the overlay and

of the desired estimated yields on the cm^ve that

are coordinate with rates of application designated

on the abscissa of the overlay. The constants n

and Ua are derived from these readings.

The general method suggested for estimating

the values of the constants n and Ua (or p and Ui,

or k and Ue) involves use of two yields read from

the fitted curve, each subject to the following

restrictions applicable to some data. Near 0

application with low-level fertility, the curve rises

so steeply that it is diflSicult to locate the point

on that part of the curve that is coordinate with

a specified rate of application designated on the

abscissa of the overlay. Similarly, as the curve

approaches the maximum yield (ikO, there is a

wide range in rates of application indicated on

the abscissa within which it is difficult to deterDiine the rate that is coordinate with a specified

yield on the curve. The problem is to avoid these

extremes in order to obtain readings on the vertical

and horizontal scales that are coordinate with each

other, when the readings are to be used in estimating constants of the yield equation.

Accurate readings are more easily obtained when

these extremes are avoided. Then, accurate calculations made from these readings wiU result in

constants from which any yield calculated will

fall on the standard curve from which the two

yield readings are taken. A good general rule is

to take the two yield readings at points that encompass the area of greatest curvature. Accuracy

oí the constants can be verified by using them to

calculate a few yields over the range of observalí^^l* ^^^^ ^^ precision in reading yields from

the fitted curve, or in the computations for finding

constants, will have the greatest effect on accuracy

of calculated yields at points that are farthest

removed from the two readings taken.

Constants and Yields on the N Curve.—One of

the constants on the N curve, Ai, has been estimated graphically as about 126.4 bushels (fig. 3).

Next is needed the unit value, in terms of Ä=0.8,

of the two components of x—soil content and

applied portion. As indicated earlier, x for N is

n+a. When the unit value of n+a has been

found for any rate of application, its associated

yield ratio (1—Ä^) value multiplied by M results

in the calculated yield at that rate.

Referring to figure 3, n (in pounds) is found by

measuring the distance to the left from 0 application on the overlay sheet to the curve. In units,

n is represented by the distance on the unit scale,

x, between the point directly below zero apphcation and the point below the intersection of the

curve with the abscissa of the overlay sheet. The

unit value coordinate with 0 application is 5.75.

The curve crosses the abscissa of the standard

yield chart at a little more than 4.8 units, estimated at 4.85. Therefore, the graphic estimate

of n is 0.90 unit (5.75—4.85).

However, as described earlier, n is more easily

determined without requiring that the overlay be

tailored to fit the dimensions of the standard curve.

The yield at 0 application (yo) is read from the

curve as 23.3 bushels. At this yield, the yield

ratio is 0.184 (23.3^M, or 126.4). From a table

of 1—E^ values (table 9), the x value of 0.184 is

read as 0.91 unit. It differs only slightly from

that read directly from the standard yield chart.

This method does not require that fractional units

be indicated on the standard chart. Nor does it

require that the abscissa on the overlay sheet be

coordinate with 0 yield.

Next is found the number of pounds of a (applied N) required to make Ä=0.8, which is

referred to as Ua, Zero application is coordinate

with 5.75 units, and the 320-pound application

is coordinate with a little more than 39.5 units,

estimated at 39.55. The difference—33.80 units—

covers a range of 320 pounds. Thus, 320-^33.80=

9.47 pounds=t^a. (This value may be estimated

in this way from any 2 points on the curve.)

Then, pounds applied divided by Ua equals units

applied. For example, when a is 160 pounds, a

in units= 160-4-9.47= 16.89 units. Thus, 7i+a=

0.90+16.89=17.79 units, when the application

is 160 pounds per acre.

However, as mentioned earlier, Ua, as well as

71, is more easily determined in a way that does

not require that the overlay be taüored to fit the

standard curve. The following calculations illus363788—fifi o
trate the more practical use of the graphic method

for this purpose.

t/i6o^M=124.0-i-126.4=0.981 whose x value is 17.76

2/o-î-M=23.3-î-126.4 =0.184 whose x value is 0.91

Difference 16-85

Then Ua= 160-^16.85=9.50 pounds, close to the

value read from the standard yield chart. As in

the optional method of calculating n, x values

are read from a table of 1—Ä'^ values. The optional method may be faster if both a calculating

machine and atableof 1—5^ values are available.

The latter is needed in preparing the standard

yield curves. However, if the optional method is

used, the standard curves may be drawn to a

smaller scale. When estimates n and Ua are to be

read from the graph, the scale must be large

enough to permit reading of fractional units,

if results are to correspond closely with those

obtained with more precise methods.

The yield at 160 pounds of a may be read

directly from the curve as about 124 bushels.

It is necessary to use constants of the N curve,

together with those of other nutrient curves, in

calculating yields at different combinations of

nutrients. To illustrate use of the constants in

calculating the yield at 160 pounds of a, it is recalled that the unit value of n-\-a at 160 pounds of

a is 17.79, based on direct graphic estimates.

Keading upward from 17.79 on the unit scale, to

the curve and across to the left to the yield ratio

scale, the yield ratio is found to be about 0.981.

Then, 2/=M(l-Ä^) = 126.4 (0.981) = 124.0 bushels. The yield at any point on the curve is

calculated in the same way. Or, using the optional method, the value 0.981 may be read from

a table of 1—JB^ values.

Constants and Yields, ike P2O5 Curve,—These

are developed in the same way as illustrated for

the N curve. Constants are based on response to

9 rates of P2O6, with applied N held at 160 pounds

per acre. Read graphically, the constants are,

M, 121.5 bu., Í?, 0.50 unit, and u&, 7.63 pounds.

At the 160-pound rate, h contains 20.97 units

(160-^-7.63). Thus, at this rate, p+ô is 21.47

units (0.50+20.97). Reading from figure 3,

or from a table of 1—iZ* values, the yield ratio

value at 21.47 units is about 0.992. As y=

M(l-R^'), y=121.5 (0.992), or 120.5 bushels.

As the method is the same as that for the N

curve, a graphic illustration is not included.

Maximum Yield for Any Combination of Nutrients.-—The combination, 160 pounds of N and

160 pounds of P2O5, is found to occur on both

curves. But in estimating a least-squares fit of

each curve, the yields at this combinatioa are not

quite the same. Both curves are taken into

account in estimating the maximum for the combination, subsequently referred to as M2.

On the N curve, the yield at 160 pounds of N

and 160 pounds of PaOg is 124.0 bushels. Units

of n+a are 17.79 and the corresponding yield

ratio value is 0.981. At this combination on the

T2O5 curve the yield is 120.5 bushels. Units of

8

p+b are 21.47 and the corresponding yield ratio

value is 0.992. The product of these 2 yield

ratios is 0.973, or 0.981X0.992. Thus M2, as

calculated from the N curve, is 124-^0.973, or

127.4 bushels. On the P2O5 curve, M2 is calculated as 120.5-f-0.973, or 123.8 bushels. The accepted value of M2 for purposes of subsequent

calculations is 125.6, or the average of the values

calculated from the 2 curves. In calculatmg

yields at any combination, the process is reversed,

andy=M2(l-i;"+«)(l-5^+'). . ^

Inclusion of a third nutrient, K2O, involves the

same steps. When the graphic approximation to

least squares has been found, the maximum (M)

is read from the standard yield chart. As for

N and P2O5, x consists of 2 parts for K2O, referred to as k and c, respectively. Then, y on

the K2O curve is calculated as y=M{l—R^^^).

The yield on the K2O curve is estimated for the

combination that is common to the 3 curves.

Then, Mz is calculated from that point on each

yield curve as:

M3=2//(l-Ä"+'')(l-Ä^+')(l~ß*+0

and the average of the 3 values of Ms, thus obtained, is used in aU subsequent calculations that

involve combinations of N, P2O5, and K2O.

Yields at Any Combination

Using as M2 the value 125.6, the yield at any

combination of N and P2O5 is calculated as

2/=M(l—Ä"+^)(l—B^^). The value of n has

been established at 0.90 unit and the value of p

at 0.50 unit. On the N curve, Ua was found to

be 9.47 pounds and on the P2O5 curve Ui, was found

to be 7.63 pounds. Thus, a and b in units are

derived as pounds of applied N (a) and P2O5 (6)

divided by 9.47 and 7.63, respectively. Thus,

160 pounds of applied N is equivalent to 16.89

units so that n+a at 160 pounds of a is 17.79

units, whose 1—Ä^ value is 0.981. Similarly,

using values of p and u^, as found on the P2O5

curve, l—R^-^^ at 160 pounds b is 0.992. Thus,

the calculated yield at 160 pounds N and 160

pounds P2O5 is î/=M2(l-A"+")(l-B'^*) =

125.6(0.981)(0.992) = 125.6(0.973) =122.2 bushels.

Yields for any combination are calculated in the

same way.

The calculated yields in table 2, which are

based on mathematically determined constants,

are almost identical to those estimated when

constants are determined by a graphic approximation to least squares, as illustrated earlier.

Values of the constants are slightly different be^

cause of the smaU inaccuracies inherent in the

graphic readings. Mathematical determination

of the constants on which the calculated yields in

table 2 are based were developed by the method

suggested by Stevens.^ For purposes of table 2,

yields were calculated from accurately determined

7 See footnote 6, p. 4. In his article, Stevens illustrates determination of standard errors of constants of the

exponential equation.
constants so that standard errors could be computed.

The adequacy of the equation in describing

yield response in this experiment is indicated by

the multiple correlation coefficient of 0.9764. The

standard errors of M and A (increase in yield

attainable from use of fertilizer) on the N curve

are 3.80 and 8.55 bushels, respectively, under

the assumption that departures from the curve

are normally distributed; SR is 0.0746. On the

P2O5 curve, SM is 4.75 bushels, s^ is 11.69 bushels,

and SR is 0.1012. The standard error of estimate

of calculated yields for the 57 treatments included

in the experiment amounts to 10.43 bushels.

ECONOMIC COMBINATIONS OF NUTRIENTS

When 2 or more inputs are varied, each at different levels of the others, the estimated most

profitable combination cannot be read simply

from a standard yield curve, as may be done for

only 1 variable. But with a calculating machine,

economic combinations at specified nutrient costs,

price of the crop required to render each combinati()n most profitable, and other important determinations can be made easily. These calculations

are based on values that have been determined

by graphic methods. The work of computing

a series of economic combinations and associated

crop prices is reduced to a clerical job, if organized

as successive steps to be performed by a calculating

machine operator.

Table 3 presents a method of calculating most

profitable combinations associated with dSferent

prices of the crop, gross returns and returns above

cost of fertilizer, and reduction in returns from

adding more or less fertilizer than would be most

profitable. The method shown in table 3 also

permits easy calculation of returns per dollar

spent for fertilizer for each of a series of additional

combinations of fertilizer applied. These returns,

shown on the last line of table 3, are additional

retm^ns per dollar cost of the additional quantity

of fertilizer indicated. At precisely the most

profitable combination the added return per additional dollar is exactly $1.00.

Item 1 in table 3 consists of a series of arbitrarily

selected rates of one of the nutrients, in this case

P2O5. At the indicated cost per pound of each

nutrient, the rate of N that would be most profits

able to use with any rate of P2O5 is derived,

i^ollowmg this IS derived the price per unit of the

crop required to render a particular combination

most profitable.

Two additional values r' and r'' are introduced

at this point. These are the costs of the quantities (u) of N and P2O5, respectively, when Ä=0.8

For example, with N at $0.16 per pound, r'= $0 16

X9.47 pounds, or $1.52. Similarly, at $0.10 a

pound for P2O5, r''=$0.10X7.63 pounds, or

$0.763. In setting up the calculations, the cost

per pound of nutrient should include the estimated cost of application. The derived price of

the crop should be interpreted as the price of the

crop as it stands in the field before harvest. Each

combination will then represent the one at which

the farmer will reahze maximum profit from

fertilizer at the indicated crop price-fertilizer cost

relationship. Figure 5 shows the changes in the

most profitable combination over a wide range in

price of the crop. Data for figure 5 were prepared

as indicated in table 3, except that a wider range

in rates was used than is shown in that table.

In table 3, the combination 129 pounds N and

130 pounds P2O5 is associated with a price of

$1.42 a bushel. This combination may then be

regarded as the most profitable one at that price

for conditions of the experiment.^

Eßect qf Applying More or Less Than Most

Profitable Rates.—The effect of applying lower or

higher rates than those most profitable for a

specified price are indicated in figure 6, which is

based on data from table 3. Here the price

specified is $1.42 a bushel, and the combination

most profitable at that price is 129 pounds of N

and 130 pounds of P2O5. But if the appHcation

is only 81 pounds N and 90 pounds P2O5, with the

price for corn still $1.42, the return would be

$9.42 less per acre than if the combination most

profitable for that price had been applied. Different combinations are indicated in figure 6. One of

these, 129 pounds of N and 130 pounds of P2O6, is

8 See Appendix for method of calculating economic

combinations of 3 nutrients, most profitable rate of 1

nutrient, and most profitable combination of 3 nutrients.

lABLE 2.—Calculated yields per acre of corn for different combinations of N and P2O5 in a fertilizer rate experiment, Iowa, 1952 ^

Pounds of P2O6 applied per acre

Pounds of N applied per acre

0 40 80 120 160 200 240 280 320

0.... Bushels

2.46

16.89

21.27

22.60

23.00

23.12

23.16

23.17

23.18

Bushels

9.07

62.49

(78.69)

(83.60)

85.09

85.64

(85.67)

(86.71)

85.73

Bushels

11.64

(80.25)

101.05

(107.36)

109.27

(109.84)

110.02

(110.07)

110.09

Bushels

12.64

(87.12)

(109.70)

116. 55

118.62

(119.25)

(119.44)

119. 49

119. 52

Bushels

13.03

89.79

113.06

120.12

122.25

122.90

123.00

123.15

123.18

Bushels

13.18

90.83

(114.37)

(121. 51)

123.66

124. 32

(124. 61)

(124. 68)

124.60

Bushels

13.24

(91.23)

114.87

(122.04)

124.21

(124. 87)

125.06

(125.13)

125.15

Bushels

13.26

(91.38)

(116.07)

122.25

124.42

(125.08)

(125. 28)

126.34

125.36

Bushels

13 27 40... 91.44 80.... -

116 14 120.... 122 33 160 -

124.51 200... 126 16 240 126.36 280.... -

320... --

125.42

125.45

igy^JJ^^J on constants resulting from a mathematically fitted curve of response to N at the 160-pound level of P2O5, and of response to P2O6 at the 160-pound

HPI« W ^^^ shown in parentheses are at combinations not included in the experiment. See the followhig bulletm for reported yields of this experiment:

494 «t „A ^- ^'' PESEK, J. T., and BROWN, W. G. CROP RESPONSE SURFACES AND ECONOMIC OPTIMA IN FERTILIZER USE. Iowa Agr. Expt. Sta. Res. Bui.

*^. pp. 292-332, illus. 1955.
TABLE Z.—Steps in calculating economic combinations of N and P^O^for corn at specified costs per pound of the nutrients, price

of the crop required to render each combination most profitable, yields and returns per acre at each combination

Item and description or derivation

11.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

24.

25.

Ô, or applied P2O6 pounds.

6 in units [item 1 divided by Uh;Uh equals 7.631 units.

P+Ö [item 2 plus soil content (p); p equals 0.50] --—.do.—

l_-i?j>+b [read from standard yield curve or from table 9j.decimals_.

fip+b [1.0 minus Item 4]. - —do..-

(l_j?p+6) {Rp^b) [item 4 times item 5]-- do—.

T" (1—i?J^') [r" equals cost per pound PjO« (10 cents) times ui

(7.63 pounds), or $0.763].- numbers

T' (Rr>+b) [T' equals cost per pound N (16 cents) times Ua (9.47

pounds), or $1.52] numbers..

r" 0--Rp+f>)-\-T' (Rp+b) [item 7 plus item 8] do..-

(l_f?p+fc) iRp*-h)^(r" (l-i?»'+b)+r' (f?i^»)) [item 6 divided by

item 9] -— decimals-

/?«+« [item 8 divided by item 9] .do.-..

1—/?n+o [1.0 minus item 11].. _ -- do—

n4-a [read from standard yield curve or from table 9] units..

a [n+a minus n, n equals 0.91] do—

Most profitable rate of N to apply [item 14 times «a, or 9.47]

pounds..

Mi times item 10 [M2, or theoretical maximum yield, equals

126.64 bushels] - - numbers..

Item 16 times-Z710.8 [-ZnO.8 equals0.22314] _.do—.

», or value per unit of crop in field [1.0 divided by item 17]..dollars..

(l-jip+b) (i_/?«+a) [item 4 times Item 12] decimals..

y, or calculated yield [Ma (or 125.64 bushels) times item 19]

bushels..

Grossreturn at $1.42 per bushel [item 20 times $1.42] doDars.

Cost of fertilizer [item 2 times $0,763 plus item 14 times $1,620]

dollars..

Return above cost offertilizer [item 21 minus item 22] do...

Reduction in item 23 with com at $1.42, at different rates do—

Increase in gross return at additional combinations do—

Cost of additional rates of fertilizer do—

Changes in return per dollar cost of fertilizer, average each combination dollars.

Values

(1)

90

11.80

12.30

.936

.064

. 05990

. 71417

.09728

.81146

.07382

. 11988

.88012

9.51

81

9.2747

2.0696

.48

.824

103.53

147.01

22.07

124.94

9.42

(2)

100

13.11

13.61

.962

.048

. 04570

.72638

.07296

.79934

.05717

.09128

.90872

10.73

9.82

7.1828

1.6028

.62

.866

108. 68

154.33

24.93

129.40

4.96

7.32

2.86

2.66

(3)

110

14.42

14.92

.964

.036

.03470

. 73553

.05472

. 79026

. 04391

.06924

. 93076

11.97

11.06

106

6. 5169

1. 2310

.81

.897

112.70

160.03

27.81

132. 22

2.14

5.70

2.88

1.98

(4)

120

15.73

16.23

.973

.027

.02627

.74240

.04104

.78344

.03363

.05238

.94762

13.22

12.31

117

4.2127

.9400

1.06

.922

115.84

164.49

30.71

133.78

.68

4.46

2.90

1.64

(5)

130

17.04

17.54

.980

.020

.01960

. 74774

. 03040

. 77814

. 02619

.03907

.96093

14.63

13.62

129

3.1649

.7062

1.42

.942

118. 36

168.06

33.70

134.36

.00

3.67

2.99

1.19

(6)

140

18.35

18.85

.985

.015

.01477

. 75166

.02280

.77436

.01907

.02944

.97066

15.80

14.89

141

2.3960

.6346

1.87

.966

120.11

170. 66

36.63

133.93

.43

2.60

2.93

.85

(7)

150

19.66

20.16

.989

.011

.01088

.76461

.01672

.77133

. 01411

.02168

.97832

17.17

16.26

154

1.7728

.3956

2.53

121.62

172.70

39.72

132.98

1.38

2.14

1 Based on data from—HEADY, E. O., PESEK, J. T., and BROWN, W. G. CROP RESPONSE SURFACES AND ECONOMIC OPTIMA IN FERTILIZER USE. Iowa Agr.

Expt. Sta. Res. Bui. 424, pp. 292-332. illus. 1965.

most profitable if com is $1.42 a bushel. But, if 90

pounds of P2O5 are to be used, the best rate of N

will be 81 pounds. Table 3 shows that the price

at which this lower combination would be most

profitable is only $0.48 a bushel.

Some verification of estimates in figure 6 may

be had by comparing yields at the specified

combinations with those reported at combinations most nearly similar. For example, the

reported yields at 40 pounds of N and 40 pounds

of P2O5 was 71 bushels, compared with 66 bushels

calculated at 37 pounds of N and 50 pounds of

P2O5. Also, at 80 pounds each of N and P2O5, the

reported yield was 107 bushels compared with

calculated yields of 97 bushels at 70 pounds of N

and 80 pounds of P2O5, and 104 bushels at 81

pounds of N and 90 pounds of P2O5. Going back

to still lower rates, the calculated yield of 36

bushels at 17 pounds of N and 30 pounds of PaOs

appears to be a reasonable estimate in view of the

28-bushel yield reported at 0 pounds of N and 40

pounds of P2O5.

Effect of Change in Price of Crop After Application,—The effect of a change in price of the crop

from the price anticipated at the time of apphcation is indicated in figure 7. Here again, the price

expected is $1.42 a bushel, and the combination

apphed is the one most profitable at that price.

Suppose the price falls to only $1.00 a bushel.

How much are returns per acre reduced as compared with what they would have been if the combination most profitable for $1.00 corn had been

apphed? According to figure 7, the reduction in

10

this instance would be only about $0.50 per acre.

Figure 7 indicates that only an extreme drop in

price would make any substantial difference in

returns from application of what is expected to be

the most profitable rate and apphcation of the

rate that would have been most profitable for the

price actually received. The probability of reduction in returns from apphcation of too much

fertilizer because of a decline in price after apphcation apparently is a relatively minor risk.

But even if the price received is $1.42 a bushel,

there are other risks, and the farmer may not

choose to apply the rates calculated as most

profitable for that price. For example, the most

profitable combination at that price (129 pounds

N and 130 pounds P2O5) costs $5.89 more than the

one that would be most profitable for only $0.81

corn—105 pounds N and 110 pounds P2O5 (table

3). If the crop is worth $1.42 at harvesttime, the

return from apphcation of fertUizer at the lower

rates would be $2.14 per acre less than from

application at the higher rates indicated—the ones

most profitable at that price. But the natural

hazard may be so great that the better choice

would be to take the chance of losing the $2.14 if

everything goes right than to lose the additional

$5.89 if unfavorable weather or other damage were

to reduce the yield to that associated with the

lower rates, even though the higher rates are

applied.

These are the kinds of odds from which the

farmer must choose. The calculated most profitable rates provide starting points from which a
Most Profitable Combination

FERTILIZER FOR CORN

$ PER BU.

2.50

LB. APPLIED PER ACRE

* N AT 164 PER i-B.; P, O AT ?0< PER LB.

U.S. DEPARTMENT OF AGRICULTURE NEC. 55(5)-973 AGRICULTURAL RESEARCH SERVICE

Figure 5.—Most profitable combination of N and P2O4 for corn at different prices per bushel.

fanner may estimate the rates he should apply,

based on knowledge of the most probable environmental conditions. Some of the latter can be

controlled by his own actions. Use of good cultural practices greatly increases the chances of

realizing anticipated retmns from the calculated

most profitable rates.

MINIMUM COST NUTRIENT COMBINATIONS FOR A

SPECIFIED YIELD

The idea that different combinations of nutrients

may be appUed to obtain a specified yield imphes

possibilities of substitution. Physiologically, one

mitrient does not substitute for another in the

plant. At least, a minimum quantity of each

nutrient is required for healthy balanced growth.

The plant may draw more heavily on the soil

supply of a nutrient if the appUed portion of that

nutrient is inadequate to provide the physiological

balance needed for the yield obtained. Thus,

under certain nutrient-cost relationships, it may

pay to ''mine'' the soil in the short run in order to

obtain maximimi returns. This might mean that

later on the response to application of the depleted

nutrient would become so great that, even though

its cost per unit were still relatively high, more of

it should be added to obtain a specified yield at

minimum cost.

In deciding whether there are possibihties for

profitable substitution between appUed nutrients

in obtaining a specified yield, it is necessary to

know: (1) Different combinations of applied nutrients at which the specified yield would be obtained

(equal product combinations) and (2) the relationship between the costs per unit of the nutrients

that would be necessary in order to make each of

these equal product combinations most profitable.

The answer to the second problem will indicate

the marginal rate of substitution of one appHed

nutrient for another. Thus, the most profitable

combination was calculated with N at $0.16 and

P2O5 at $0.10 a pound. The ratio of the cost of

N to the cost of P2O5 is 1.6 to 1.0. Therefore, m

obtaining the yield at the most profitable combination for the conditions stated, 1 pound of apphed

P2O5 substitutes for 1.6 pounds of applied N; or 1

pound of applied N substitutes for 0.625 pound oí

apphed P2O5 (1.0/1.6). If the nutrient cost ratio

11
EFFECT ON RETURN

By Too iiffle or Too Much Ferfilizer

-10

S-20

<

ui -30

40

-50

/-

i\Aost profitable combination

corn at $1.42 per bu.

40 60 80 100 120 140 160

LB. OF P2P5

-H + +

20 40 60 80 100

LB. OF N.

120 140 160 180

U. S. DEPARTMENT OF AGRICULTURE NEC. 55(5)-974 AGRICULTURAL RESEARCH SERVICE

Figure 6.—Effect on return from corn by too little or too much fertilizer.

should change substantially and if it is possible

to obtain the specified yield at different combinations of nutrients, it would be worthwhile to know

the marginal rates of substitution over a considerable range of such combinations. The relevant

range would be just broad enough to include

nutrient cost ratios that are at all probable. For

example, it might be possible to obtain the specified yield at a combination that would be most

profitable only if N cost 10 or more times as much

per pound as P2O5. Obviously, such a cost relationship is not probable.

Determination of the marginal rate of substitution is illustrated here by referring to the yield at

the most profitable combination for the estimated

current price. The estimated current price is

$1.42 and the yield at the combination most

profitable at that price is 118.35 bushels (item 20,

col. 5, table 3). The theoretical maximum yield

(M2) was found to be 125.64 bushels. Then

t/-^M2= 118.35-^125.64=0.942 (item 19, col. 5,

table 3). Equal product combinations are determined as:

At the most profitable combination, l—R^'^^ is

0.96093 (item 12, col. 5, table 3). Then at the

rate of b required for the specified yield, l—R^'^^

is 0.942-^0.96093, or 0.980. Item 4, column 5,

table 3, shows that this is the l—R^ value associated with 130 pounds of P2O5, from which, at the

stated nutrient costs, is calculated the rate of N

(129 pounds) that would be most profitable to

use with that rate of P2O5. When 1—Ä^+^ is

0.980, p+b equals 17.54 units. As Uj, is 7.63

pounds, b (apphed P2O5) amounts to 130 pounds.

The discussion above illustrates calculation of

equal product combinations of two independent

variables, starting with a specified rate of one of

them. The marginal rate of substitution (MRS)

at each equal product combination is calculated as:

MRS in nmts=-^-^^'j—j^.

Substituting in the equation the values for the

most profitable combination when corn is $1.42

(col. 5, table 3):

(7/-^M2)-^(l-Ä«+-) = l-Äl'+ö; or,

(7/^Af2)-(l~Ä^+^) = l-7?'^+«

0.020 0.961 0.01922

0.039 0.980 0.03822=0.50288 unit of N, which

12
EFFECT ON RETURN

By Changes in Price of Crop offer Fertilizer is Applied

i I

$1.42

Price of corn *

J I L

1.00

$ PER BU.

1.50 2.00

* MOST PROFITABLE RATE OF APPLICATION IS MADE FOR CORN AT THIS PRICE.

U. S. DEPARTMENT OF AGRICULTURE NEC. 55(5)-975 AGRICULTURAL RESEARCH SERVICE

Figure 7.—Effect on return by changes in price of corn after fertilizer is applied.

substitutes for 1 unit of P2O5. Then, MRS in

pounds is (0.50288) (Ua-^u^,). As Ua is 9.47 pounds

and Uj, is 7.63 pounds, Ua-^u¡, is 1.24115. The

marginal rate of substitution in pounds is

(0.50288) (1.24115). When allowance is made for

rounding figures in the computations, this calculation reflects a substitution of 1 pound of N for

0.625 pound of P2O5. This was pointed out eariier

as the ratio of the cost per pound of P2O5 to the

cost per pound of N (0.10-^-0.16).

Using the rate of N shown in column 4, table 3,

as the starting point (117 pounds), the associated

rate of P2O6 required to produce the 3âeld of 118.35

bushels would be 172 pounds. At this combination, the marginal rate of substitution of N for

P2O5 is about 0.13 pound. Thus, the cost per

pound of P2O5 would need to be only 13 percent

that of N in order to justify this combination for

the specified yield. This is not a probable relationship.

Repeating the same calculations, but using 154

pounds of N as the starting point (col. 7, table 3),

109 pounds of P2O5 are required to produce the

specified yield. Here, the marginal rate of substitution is 2.16 pounds of N per pound of P2O5—

meaning that the cost per pound of P2O5 would

need to be 2.16 times that of N in order to justify

use of this combination in producing the yield

specified.

Based on this experiment, there is only a narrow

range within which substitution of 1 nutrient for

the other would be economically feasible. Both

combinations indicated in the 2 preceding paragraphs are outside that range, as the nutrient-cost

ratio associated with either of them is highly

improbable. However, many farmers may tend

to apply nutrients in relation to their cost per

unit without recognizing that the relative rates

of response are also important. For example, the

average cost of a mixture of 117 pounds of N and

172 pounds of P2O5 is about half a cent less per

pound than the average cost of 129 pounds of N

and 130 pounds of P2O5. But the latter is the

most profitable combination for the specified

price-cost relationship. Both these combinations

are calculated to result in the same yield. But at

the nutrient-cost ratio used in these calculations,

to obtain the same yield the farmer would pay

13
$2.28 more per acre by using the combination

that had the lower weighted average cost per

pound. Calculation of equal product combinations may be useful in emphasizing this point.

EXPONENTIAL AND QUADRATIC

SQUARE-ROOT EQUATIONS

COMPARED

The method described earlier and used in the

preceding illustration was applied to two other

experiments having the same design and the same

rates of application. These involved use of P2O5

and K2O as independent variables, one on alfalfa

and one on red clover.® Certain results from use

of the exponential and a quadratic square-root

equation are compared and presented in tables

4 and 5. For the exponential function, the sum

of squares explained by regression is based on

constants derived for each nutrient in the presence

of an application of 160 pounds of the other;

except that for the red clover experiment constants

for the K2O curve were developed when P2O6 was

held at the 80-pound level; and for the alfalfa

experiment, constants for the K2O curve were

developed when P2O0 was held at the 120-pound

level. A better fit was obtained for the K2O curve

in each of these two experiments when P2O6 was

held constant at the above indicated levels.

TABLE 4.—Comparison in sums of squared residuals

explained by exponential and quadratic square-root equations^ as applied to three 9X9 partial factorial experiments

Experiment

Sums of

squared

residuals

due to

treatments

Sums of squared

residuals explained by regression

Coefficients of correlation

Exponential 1

Quadratic

square

root»

Exponential Í

Quadratic

square

root 2

Com 233,811

26.75

13.66

222,927

20.78

9.69

222,899

22.98

11.52

0.9764

.8694

.8426

0.9764

.9229

.9184

Alfalfa

Red clover

1 Based on constants derived from 17 of the 57 treatment combinations;

that is, no convenient simultaneous solution is as yet available for this equation. The sums of squared residuals are computed for all 67 reported treatment mean yields.

3 Based on constants simultaneously derived, using reported yields for

all 57 treatment combinations.

Measures of Fit

The quadratic square-root equation produces ^°

a curve that turns down more slowly than the

parabola. The parabola is not discussed here.

» HEADT, E. 0., PESEK, J. T., and BROWN, W. G. CROP

RESPONSE SURFACES AND ECONOMIC OPTIMA IN FERTILIZER

USE. Iowa Agr. Expt. Sta. Res. Bui. 424, pp. 292-332

illus. 1955. '

10 HEADY, E. O., and PESEK, J. A FERTILIZER PRODUCTION SURFACE WITH SPECIFICATION OF ECONOMIC OPTIMA

FOR CORN GROWN ON CALCAREOUS IDA SILT LOAM. Jour

Farm Econ. 36: 466-482, illus. 1954.

The quadratic square-root equation where N and POOR

are the independent variables is written as:

y=o+biN -f- &2P H- 63VÑ -I- hi-y/T 4- 6ô-v

But, in contrast to the exponential function,

which approaches, though theoretically does not

reach, the calculated maximum, the quadratic

square-root equation provides for calculating

reduced yields as more fertilizer is added. If there

is merit in being able to calculate reduced yields

after the maximum attainable from fertilizer bias

been reached, the quadratic square-root equation

or the parabola would have an advantage over

the exponential function. As none of the equations mentioned provides for a phase of increasing

increments (sigmoid section), none would appear

to have an advantage when fitting to reported

yields that may sometimes follow that pattern

at low rates of application.

Comparison of the distribution in the deviations

of reported yields from those calculated by the

two equations is shown in tables 5 and 6. As

indicators of reliability of results for use in

recommending rates of application, deviations

assume importance primarily around the central

section of the surface. On soils that are materially deficient in either nutrient, it is impracticable to apply the other unless the level of the

deficient nutrient has been raised substantially.

If fertilizer is to be used at all, m nearly all cases

applications should be made until yields wül

somewhat exceed those on the steepest part of

the curve. This is true because of the substantial returns obtained at applications coincident with the steeper parts of the response

curve.

Some farmers have only limited funds. They

may, therefore, wish to apply less than calculated

most profitable rates. Also, there are possibilities of profitable substitution, in the short run,

of applied portions of one nutrient for applied

portions of another. However, as pointed out

earlier, the range within which it would be

profitable to substitute one nutrient for another

is limited. This is especially true when the soil

content of the nutrient for which a substitution

is considered is below the level at which the

curve of response to it would begin noticeably to

flatten. For this reason, departures from reported

yields on the steeper part of the curve do not

detract from the validity of recommendations

based on a good fit over that part of the curve

within which reconmiendations would fall.

Furthermore, for situationsin which the fertility

level is such that even the first few yield increments from added fertilizer are quite small, farmers with limited funds ordinarily will use the funds

for items other than fertilizer. Therefore, generally speaking, what happens at the "fringe" of

the production surface is of relatively little importance. The rate of response far to the left

on the curve is often too great to justify limiting

applications to that range, even for farmers with

limited funds. Far to the right, as the calculated

most profitable rate is approached, the response

to additional units is so small that risk factors

outweigh the possible gains from fm-ther incre14
TABLE 5.—Sums of squared deviations of reported yields in 3 factorial experiments from those calculated^ using different

equations

For regression of—

N on P2O6 P2O6 on K2O P2O5 on N K2O on PjOß

Level of fixed nutrient

(pounds) Com Alfalfa Red clover Com Alfalfa Red clover

Exponential

equation

Quadratic

square

root

Exponential

equation

Quadratic

square

root

Exponential

equation

Quadratic

square

root

Exponential

equation

Quadratic

square

root

Exponential

equation

Quadratic

square

root

Exponential

equation

Quadratic

square

root

0 549

514

531

670

409

850

460

317

1,142

1,141

239

522

254

619

806

317

592

995

1.9249

.3121

.0271

.2455

.0739

.1189

.1680

.1240

.2710

0.5138

.2313

.1096

.3011

.0958

.1710

.0958

.0877

.1882

0.5262

.1703

.1130

.1358

.0884

.1171

.1372

.2528

.4418

0.1711

.1263

.0594

,1370

.0915

.0512

.0669

.3852

.3141

950

1,230

245

464

738

560

153

215

887

970

948

208

687

641

618

356

236

821

0.9802

.2578

.1668

.0505

. 3277

.3983

.4553

.2327

.4061

0.6559

.2307

.0885

.1127

.1386

.2096

.1192

.2244

.0948

0.1310

.1690

.0762

.3014

.4558

.1042

.2786

.0523

.4132

0.1013

.1174

.0971

.3886

.1965

.0490

.1073

.0496

.2958

40

80

120

160 -

200

240

280

320

Total 5,442 5,485 3.2654 1. 7743 1.9816 1.4025 5,442 5,485 3.2654 1,7743 L9816 1.4025

TABLE 6.—Sums of squared deviations of reported yields in S factorial experiments from yields calculated at all levels of each

of the two nutrients except 0 and 320 pounds, using different equations

For regression of—

N on P2O5 P2O5 on K2O P2O5 on N K2O on P2OS

Level of fixed nutrient

(pounds) Com Alfalfa Red clover Com Alfalfa Red clover

Exponential

equation

Quadratic

square

root

. Exponential

equation

Quadratic

square

root

Exponential

equation

Quadratic

square

root

Exponential

equation

Quadratio

square

root

Exponential

equation

Quadratic

square

root

Exponential

equation

Quadratic

square

root

40 236

412

218

387

682

125

236

217

404

65

460

779

185

266

0.1618

.0253

.2180

.0731

.1086

.1236

.0357

0.2018

.0660

.2507

.0522

.1039

.0887

.0463

0.0731

.1119

.1137

.0619

.0326

.0271

.2224

0.0788

.0544

.1202

.0378

.0162

.0198

.2968

469

113

234

733

478

82

187

545

139

393

458

427

317

87

0.1592

.1201

.0333

.0919

.1262

.0408

.1745

0.1789

.0796

.0495

.1075

.1433

.0548

.1950

0.0059

.0559

.2643

.2083

.0031

.0767

.0284

0.0300

80... - .0295

120 .3197

160 .1110

200 .0431

240 .0626

280 _.„ .

.0381

Total _ 2,296 2,366 .7460 .8086 .6426 .6240 2,296 2,366 .7460 .8086 .6426 .6240

ments in yield, even though a farmer may have

unlimited funds. Thus, the principal function of a

yield equation is to estimate most profitable rates

and combinations of the different independent

variables. To do this efficiently requires an adequate number of rates in the portion of greatest

curvature and in the area of the maximum yield.

Other practical factors, such as knowledge of

probabilities of different types of risk and estimates of relative returns from différent inputs that

may compete for the farmer's dollar, are taken

mto account in deciding how much to apply.

Some further statistics that bear on this discussion are presented in table 6. This table shows

deviations from regression if deviations at the

0- and the 320-pound rates of both nutrients in

each case are omitted. Good experimental procedure requires the use of rates somewhat higher

than are expected to result in further increases

in yield if one is to be certain that practical maximum response has been attained for the conditions.

It is also necessary that rates be carried to such

levels to insure reliability of constants used in the

prediction equations, whether graphic or more

precise methods are used.

Results, as applied to the three experiments

noted here, are particularly useful in bringing out

the point that considerable segments of the curve

(the steep and the flat portions) are outside the

range of rate of application that would be recommended. The sou content of two of the nutrients

included in each case—N and P2O5—^was extremely

low, so that yields at low applications fell on steep

sections of the response curve. Also, the higher

rates applied residted in little or no further increase

in yields. Possibly other limiting factors were

present, in the absence of which response would

have been obtained at even higher rates than

were used.

Referring back to table 4, the coefficients of

correlation were the same for both equations in

the corn experiment, but they were higher for

the quadratic square-root equation as applied to

the alfalfa and red clover experiments. These

coefficients are based on deviation explained by

regression over the complete surface. But table

15
6 shows that if attetilioQ is centered only on the

parts of the surface that include rates which

would ordinarily be recommended, coefficients of

correlation would be slightly higher for the exponential equation for the corn and the alfalfa

experiments. For the red clover experiment the

quadratic square-root equation would have a

slight advantage.

Even when the quadratic square-root equation

is used, more than half (54 to 57 percent) of all

deviations from regression in the 3 experiments

are accounted for by deviations at the lowest and

the highest rates of both nutrients applied in the

experiment. These deviations are those around

the fringe of the production surface. Using the

exponential function, 58 percent of the deviation

from regression occurred in this fringe area for

the corn experiment, 68 percent for the red clover

experiment, and 77 percent for the alfalfa experiment. The quadratic square-root equation appears to fit the fringe areas better than the exponential function as applied to the alfalfa and the

red clover experiments. But the rest of the surface for the corn and alfalfa experiments was a

little better fitted by the exponential function.

Deviations from regression in the red clover experiment were slightly smaller for the rest of the

surface when the quadratic square-root equation

was used. Table 6 shows that when deviations

from regression for the portions of the surface

outside the range of recommendations are deducted, there is no appreciable difference in the

fit obtained through use of the 2 equations as

applied to these 3 experiments.

Most Profitable Rotes and Confidence Limits

Differences in most profitable rates as calculated, using the 2 equatioûs, are shown in table

7 for the 3 experiments. The differences are not

important for the purpose of using the results as

a basis for recommendations. Differences in

yields are small, even though there are substantial

differences in most profitable rates resulting from

use of the 2 equations. For the alfalfa and clover

experiments the differences in returns per acre

above cost of fertilizer are negligible. For the

corn experiment, the difference is about $6.00

per acre, which may be regarded as small when

viewed in the light of probabilities of variation

in yield because of weather and other risks. Also,

this difference is only half the value of 1 standard

error in yield at the most profitable rate, as calculated from the quadratic square-root equation.

The unimportance of the differences in results

obtained with the 2 equations is further brought

out by comparing yields and returns per acre in

table 7 with those* of table 8. For purposes of

table 8, the most profitable rates derived from

using each equation are substituted in the formula of the other. Comparison of the 2 tables

shows that the differences in results are too small

to affect any decisions the farmer might make.

The greatest difference in return above cost of

fertilizer occurs in the corn experiment—$128.30

in table 7 versus $140.59 in table 8. However,

this difference is approximately the same as the

value of 1 standard error in yield at the most

profitable rate.

Confidence limits in yield at the most profitable

combination have not been calculated for the exponential equation because of lack of a convenient

method for simultaneous solution of values of the

constants. Comparison of the residual sums of

squares for the whole surface and for the more relevant part of the surface was presented earlier.

For the quadratic square-root equation at the 67-

percent level of probability, confidence limits in

yields at the most profitable rates in the three experiments are ±8.60 bushels of corn, ibO.184 ton of

alfalfa, and ±0.077 ton of red clover. In terms of

dollar returns per acre above cost of fertilizer, these

limits are ±$12.21 for com, $2.76 for alfalfa, and

$1.16 for red clover. Only in the case of the com

TABLE 7.—Comparison of most profitable combinations of plant nutrients^ yields at these combinationSy and returns above cost

of fertilizer, as calculated by 2 equations applied to three O'XQ partial factorial experiments

Experiment

Item Corn Alfalfa Red clover

Exponential Quadratic

square root Exponential Quadratic

square root Exponential Quartratlc

square root

Most profitable rate of—

pounds..

do

129

130

160

P2O6 - 173 50

56

61

23

40

11

95

K2O2.- do—. 5

Total nutrients - do 259 333 106 74 51 100

- bu. or tons-.

dollars..

- do-.-.

— do-...

— - do..-.

Yield -

Price per unit of crop

Value of crop per acre

Cost of fertUizer

Returns per acre above cost of fertilizer.

n8.35

1.42

168.06

33.64

134.42

120.56

1.42

171.20

42.90

128.30

3.18

15.00

47.66

8.36

39.40

3.08

15.00

46.20

6.48

39.78

3.02

15.00

45.30

4.66

40.64

3.38

15.00

60.70

9.80

40.90

I N not a variable in alfalfa and red clover experiments.

* K2O not a variable in corn experiment.

16
experiment is the confidence limit possibly greater

than the margin of error in attempting to apply

the most profitable rate under farm conditions.

Were methods available to calculate comparable

confidence limits associated with use of the twovariable form of the exponential equation, they

might be expected to be of about the same order

of magnitude as those indicated above. This expectation is based on comparison of residual sums

of squares for the entire production surface and

for that remaining after eliminating from consideration a part of it that is of no practical importance.

TABLE 8.—Yields and returns per acre, if calculated hy substituting most profitable rates derived from each equation into the

formula of the other

Substituting most profitable rates derived—

Item From quadratic square-root equation

into exponential formula

From exponential equation into quadratic

square-root formula

Com Alfalfa Red clover Com Alfalfa Red clover

Yield

Value —-

Cost of fertilizer.-

Eeturn per acre above cost of fertilizerbu. or tons..

_ dollars..

- .do..-.

do....

122. 57

174.05

42.90

131.15

2.99

44.85

6.48

38.37

3.20

48.00

9.80

38.20

122.70

174.23

33.64

140. 59

3.16

47.40

8.36

39.04

3.18

47.70

4.66

43.04

17
APPENDIX

Calculation oí Most Profitable Rate of

One Nutrient

On page 9, reference was made to calculation of

the most profitable rate of one nutrient rather than

estimation directly from the curve, as shown in

figure 4. The graphic method requires that small

units of apphcation, such as 10 pounds, be indicated on the base scale of the overlay sheet so that

increments in yield can be estimated visually from

the curve. Small units of application and corresponding small increments in yield are necessary

if the graphic estimate of the most profitable rate

is to be close to the accurately calculated rate.

Those familiar with use of a machine may find it

easier to calculate the most profitable rate after

fitting the curve by graphic methods.

A simple way of doing this involves use of a

factor Q, which is calculated as 4.48142-î-!?M when

4.48142 is a constant for this computation,^^ v is

the price per unit of the crop before harvest, and

M is the theoretical maximum yield. Referring to

figure 3, M is 126.4 bushels and on figure 4, v is

$1.40. Thus: Ç=4.48142-î-1.40(126.4) =0.02532

then

Çr'=0.02532(1.52) =0.03849=

l-ß«+«=0.96151 at the most

profitable rate, thus,

72,-fa= 14.60 units (from table of

1—JS^ values), and as

71=0.90 unit (from fig. 3 or by

optional method described)

a= 13.70 units or

a= 129.7 pounds {a in units times

Uay or 9.47)

y=M(l -Ä'^+«) = 126.4(0.96151) =

121.5 bushels

This rate, 129.7 pounds, and this yield, 121.5

bushels, are substantially the same as the graphic

estimates shown in figure 3.

Calculation of Economic Combinations of Three

Nutrients

The most economic combinations of N, P2O5,

and K2O at specified costs per pound of the nutrients, prices of the crop required to render each

combination most profitable, and yields and

retmns at each combination, are calculated as

illustrated in table 3 (p. 10), in which only two

" 4.48142 =

1 1

0.22314 i-lnO.S)

nutrients were involved. The method of handling

the clerical steps after graphically fitting the

curves is outlined in form 1 (see p. 19). This

form is self-explanatory, as all terms used have

been defined and illustrated.

Calculation of the Most Profitable Combination

of Nutrients

Table 3 of the text and form 1 illustrate calculation of a series of combinations of 2 or of 3

nutrients, respectively, that would be most

economical when their costs have been specified.

With this method, the price of the crop at which

each of these combinations would be most profitable is derived.

If only the combination is used that would be

most profitable when both cost of fertilizer and

price of the crop are specified, steps in the calculation are different. Form 2 (p. 19) shows these

steps when 2 nutrients are involved; form 3

(p. 19) when all 3 nutrients are variables.

Form 3 represents a short-cut method of calculating the most profitable combination of 3

nutrients that produces results generally accurate

enough for practical pmT)oses. Exact determination of the most profitable combination involves

an iterative technique that is avoided in the

method presented here.

When 3 nutrients are involved, experience

indicates that if the value of p+b, item 25, form 3,

turns out to be lower than 10 to 14 units, the

answers obtained may contain enough error to

make a practical difference. However, a test of

this is readily made by calculating a factor T as

(Item 23) (Item 24) (Item 28)

(Item 31) (Item 32)

If T is found to differ from Q by no more than

about 0.0005, the calculated most profitable

combination wiU be substantially correct.

If a second trial is necessary, a value may be

substituted for p+b that is about the same

percentage above or below the first value found

as the percentage by which T exceeds or is smaller

than Q, When a new value is substituted for

p+b, item 25, form 3, a new associated value for

item 24 is read from a table of l-R'^ values.

Then 1.0 minus this new 1-Ä^+^ value, item 24,

will result in a new value for item 23, which is

JJP+&. ]SJQ adjustments need to be made back of

this point. The test may then be repeated if

necessary to render T—Q no greater than about

±0.0005.

18
Forms

T'onul.—Steps m calculating economic combinations of iV,

t^zUi, ancLKiO at specified costs per pound of the nutrients,

prices of the crop required to render each combination most

profitable^ and yields and returns at each combination

Item and description or derivation

6, or applied PjO« pounds

Ô in units [item 1 divided by «6;«»equals

pounds] _ ...unlti"

p+b [item 2 plus (p) ; p equals J ].„ do

l_ijp+6 [read from standard yield curve orfrom

table 9]_ _ decimals

R^^ [1.0 minus item 4]..,. do

r (ieiH-t) [r' equals cost per pound N (_cents)'

times Ua (_poimds), or $_J numbers

r fl-Äp+ft) [r" equals cost per pound PaOs

C_cents) times Ub Cpounds), or

., *-J--;—" numbers.. Item 6 plus item 7 do

Ä"+» [item 6 divided by item 81....decimals"

1-22"+« [1.0 minus item 9]..._ do .'.[

n+a [read from standard yield curve or from

table 9] units..

o [item 11 minus w;w equals _]. do

Most profitable rate of N to apply [item 12

„times u«; Ua equals _poundsl pounds..

14. r'" (ßp+6) [r'" times item 6; r^" equals $_j

Ti. « , ,x numbers.. Item 7 plus item 14 do

Ä*+« [item 14 divided by item 15]_.decimâls"'

l-Äk+c [1.0 minus item 16] do..!'

Ä+c [read from standard yield curve or from

table 9] _ units..

c [item 18 minus k; k equals _] do

Most profitable rate of Z2O to apply fitem" Í9

times UcjUo equals _pounds] pounds

atem 4) (item 5) (item 7) divided by (item 8)

(item 15) _ decimals

V [1.0 divided by item 21 (0.22314) (M3); M3

equ^s 1. dollars..

VlMz (item 4) (item 10) (item 17)1 yield..

Gross returns [item 23 times price] »..dollars

Cost of fertilizer [T' (item 12) plus T" (item 2)

plus T'" (item 19)] ....dollars

(1) (2) (3) (4)

1 It may be desirable to calculate the gross return for each combination at the

current price rather than at v, which is the price required to render a compmation the most profitable one. p, item 22, is different for each combinaK°«^-^®?l^^S steps as desired may be calculated in same way as items 3 to 27, table 3.

FORM 2.—Most profitable combination of 2 nutrients {N and

P2OS)

Crop^

Source of data

-Unit of yielcL

. Q=éASU2-^vM2, or-.

r'=cost per pound of N times i¿a, or $_

= cost per pound of PaO« times W6,or$_

Item

1. Qr'\...

2. -Qr' .""II.III.".: "."

3. Item 1 plus item 2 _

4. —1.0 plusitem 3 with changedsign..

5. One-half ofitem 41 _ '.'"/"

6. Item5squared3

7. Item 6 minusItem 1 I.

8. Square root ofitem 7 '

9. i2í»+6 [item 5 minusitem 8] "

10. 1—ÄP+fc [l.Omlnus item 9]

11. ©+6 [from standard curve or from table 9] '.

12. 6 in units [item 11 minus p; p equals j

13. Ô in pounds [item 12 timesUb;ubequals ^1.

14. Ä»+« [item 2 divided by item 10]»...-

15. 1—Ä»+« [1.0 minus item 14] ._

16. «+o [from standard curve or from table 9]

17. a in units [item lômtnus n, n equals ]

18. 0in pounds [item 17times««; ««equals ^j.

Value

1 Drop minus sign.

2 If item 6 is smaller than item 1, It means that a solution is not possible by

thisshortcut method and the method of table 3 issuggested.

FORM 3.—Most profitable combination of S nutrients (N, P2O5, and K2O)

Area_ Crop_ Unit of yield- Source of data_

!;=$_ M3: Q=4.48142-î-t;M3, or_

Item

-2.0 r" [r" equals 3

T'-r" [r' equals $__

T'*'-r'" [r'" equals $: ,

r" squared-...

Item 2 times r".

Item 3 times r"

Item 5 plus item 6

Item 2 times item 3

Item 4 times Q

Item 7 times Q

ItemStimesQ

r" minus item Í0

Item 1 minus item 11..

Item 13 divided bv r"

Item 12 divided by r"

Item 9 divided by r".

Item 15 divided by item 14

Item 16 divided by item 14

Item 17 divided by 2.0 [drop minus sign]..

Item 19 squared.

Item 18 plus item 201

Value Item

Square root of item 21

Item 19 minus item 22

1.0 minus item 23 _

p+b [from table 9]

0 [item 25 minus p; p equals ]

Item 26 times Uh [wb equals pounds]..

Item 24 times r"

Item 23 times r'

Item 23 times r'".

Item 28 plus item 29

Item 28 plus item 30

Item 29 divided by item 31

1.0 mtQus item 33.

7i-f-a [from table 9]

a [item 35 minus n; n equals ]

Item 36 times Ua{ua equals pounds]..

Item 30 divided by item 32

1.0 minus item 38..

k+c [from table 9]

c [item 40 minus k; k equals ]

Item 41 times w« [w« equals pounds]..

Value

> If item 21 is negative, it means that a solution is not possible by this method and the method of form 1 is suggested.

19
Values of l-/?'^

TABLE 9.—Values of l — R"" when R=0.8

(All values are decimal fractions)

X 0 1 2 3 4 6 6 7 8 9

0.00 000223

002452

000446

002674

000669

002897

000892

003119

001125

003342

001338

003564

001561

003786

001784

004009

002006

01 ""5Ó2229" 004231

02 004453 004676 004897 005119 005341 005563 005785 000007 006228 006450

03 006672 006894 007115 007337 007558 007780 008001 008222 008444 008665

04 008886 009107 009328 009549 009770 009991 010212 010433 010654 010874

05 011095 011315 011536 011757 011977 012198 012418 012639 012859 013079

06 013299 013519 013740 013960 014180 014400 014620 014839 015059 015279

07 015499 015718 015938 016168 016377 016597 016816 017035 017254 017474

08 017693 017912 018131 018350 018569 018788 019007 019226 019445 019664

09 019883 020101 020320 020538 020757 020975 021194 021412 021631 021849

10 022067 022285 022503 022721 022940 023158 023376 023593 023811 024029

11 024247 024465 024683 024900 025118 025335 025653 025770 025987 026205

12 026422 026639 026856 027073 027290 027508 027725 027942 028158 028375

13 028592 028809 029026 029242 029459 029675 029892 030108 030325 030541

14 030757 030974 031190 031406 031622 031838 032054 032270 032486 032702

15 032918 033133 033349 033565 033780 033996 034212 034427 034642 034858

16 035073 035288 035504 035719 035934 036149 036364 036579 036794 037009

17 037224 037439 037654 037868 038083 038298 038612 038726 038941 039156

18 039370 039584 039799 040013 040227 040441 040665 040869 041083 041297

19 041511 041725 041939 042153 042366 042580 042794 043007 043220 043434

20 043648 043861 044074 044288 044501 044714 044927 045140 045353 045566

21 045779 045992 046205 046418 046631 046843 047055 047268 047481 047794 22 047906 048119 048331 048543 048755 048968 049180 049392 049604 049816

23 050028 050240 050452 050664 050876 051088 051298 051511 051723 061934

24 052145 052357 052569 052780 052991 053203 053414 053625 053836 054047 25 054258 054469 054680 054891 055102 055313 056624 065735 055945 056156 26 056366 056577 056787 056998 057208 067419 057629 057839 058049 058260 27 058470 058680 058890 059100 059310 059520 059729 059939 060149 060358 28 06056S 060778 060987 061197 061406 061616 061825 062035 062244 062453 29 062662 062872 063081 063290 063499 063707 063916 064125 064334 064643

30 064752 064960 066169 065377 066586 065794 066003 066211 066420 066628 31 066836 067044 067253 067461 067669 067877 068085 068293 068500 068708 32 068916 069124 069332 069540 069747 069954 070162 070369 070577 070784 33 070992 071199 071406 071613 071820 072027 072234 072441 072648 072855 34 073062 073269 073476 073683 073889 074096 074302 074509 074715 074921 35 075128 075335 075541 076747 075953 076160 076366 076572 076778 076984 36 077190 077396 077602 077807 078013 078219 078424 078630 078836 079041 37 079247 079452 079658 079863 080068 080273 080478 080684 080889 081094 38 081299 081504 081709 081914 082119 082323 082528 082732 082937 083142 39 083347 083551 083756 083960 084165 084369 084573 084777 084982 085186

40

41

42

43

44

45

46

47

48

49

085390 085594 085798 086002 086206 086410 086613 086817 087021 087225 087429 087632 087836 088039 088243 088446 088649 088853 089056 089259 089463 089666 089869 090072 090275 090478 090681 090884 091086 091289 091492 091695 091898 092100 092303 092505 092708 092910 093113 093315 093517 093719 093922 094124 094326 094528 094730 094932 095134 096336 095538 095740 095941 096143 096345 096546 096748 096949 097151 097352 097554 097755 097956 098167 098359 098660 098761 098962 099163 099364 099565 099766 099967 100168 100369 100569 100770 100970 101171 101371 101572

103575

101773 101973 102173 102374 102574 102774 102974 103175 103375 103775 103975 104175 104374 104574 104774 104974 105173 105373

50

51

52

53

54

55

56

57

58

59

105573

107567

109556

111540

113521

115497

117468

119435

121398

123356

105772

107766

109754

111738

113719

115694

117665

119632

121594

123552

105972

107965

109953

111937

113916

115891

117862

119828

121790

123748

106171

108164

110152

112135

114114

116089

118059

120024

121986

123943

106371

108363

110350

112333

114312

116286

118256

120220

122182

124138

106570

108562

110549

112531

114509

116483

118452

120417

122378

124334

106770

108760

110747

112729

114707

116680

118649

120614

122574

124529

106969

108959

110945

112927

114904

116877

118846

120810

122769

124725

107168

109158

111143

113125

115102

117074

119042

121006

122965

124920

107367

109357

111342

113323

115299

117271

119239

121202

123160

125115

60

61

62

63

64

65

66

67

68

69

125310

127260

129206

131146

133083

135015

136943

13S867

140786

142701

125505

127455

129400

131340

133276

135208

137136

139059

140978

142893

125701

127649

129594

131534

133470

135401

137328

139251

141170

143084

125896

127844

129788

131728

133663

135594

137521

139443

141361

143275

126091

128039

129982

131921

133856

135787

137713

139635

141553

143466

126286

128233

130176

132115

134050

135980

137906

139827

141744

143657

126481

128428

130370

132308

134243

136172

138098

140019

141936

143848

126676

128622

130564

132502

134436

136366

138290

140211

142127

144039

126870

128817

130758

132696

134629

136558

138483

140403

142319

144230

127065

129011

130952

132889

134822

136750

138675

140595

142510

144421 70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

144612

146519

148421

160319

152213

154103

155988

157870

159747

161620

144803

146709

143611

150509

152403

154292

156177

158058

159934

161807

144994

146900

148801

150698

152592

154480

156365

158245

160122

161994

145185

147090

148991

150888

152781

154669

156553

158433

160309

162181

145376

147280

149181

151077

152970

154858

156741

158621

160497

162368

145566

147471

149371

151267

153159

156046

156930

158809

160684

162555

145757

147661

149561

151456

153348

155235

157118

158997

160871

162741

145947

147851

149750

151645

163536

156423

157306

159184

161058

162928

146138

148041

149940

151835

153725

165612

157494

159372

161245

163115

146328

148231

150130

152024

153914

155800

157682

159559

161433

163302 163488

165353

167213

169070

170922

172770

174614

176453

178289

180120

163675

165539

167399

169255

171107

172954

174798

176637

178472

180303

163S62

165725

167585

169440

171292

173139

174982

176821

178655

180486

164048

165912

167771

169625

171476

173323

175166

177004

178839

180669

164235

166098

167956

169811

171661

173508

175350

177188

179022

180852

164421

166284

168142

169996

171846

173692

175534

177371

179205

181035

164607

166470

168327

170181

172031

173876

175718

177655

179388

181217

164794

166656

168613

170366

172216

174061

175902

177738

179571

181400

164980

166841

168698

170552

172400

174245

176085

177922

179754

181583

165167

167027

168884

170737

172585

174429

176269

178105

179937

181765
TABLE Q.—

Values of1 -R'^ when fí

=0.5—Continued

[All values are decimal fractions)

X

0

1

2

3

4

6

6

7

8

9

0.90 181948 182130 182313 182495 182678 182860 183042 183225 183407 183589

91 183771 183953 184135 184317 184499 184681 184863 185045 185227 185409

92 185591 185772 185954 186136 186317 186499 186680 18Ö862 187043 187225

93 187406 187587 187768 187950 188131 188312 188493 188674 188855 189036

94 189217 189398 189579 189760 189940 190121 190302 190483 190663 190844

95 191024 191205 191385 191566 191746 191926 192107 192287 192467 192647

96 192827 193008 193188 193368 193548 193728 193907 194087 194267 194447

97 194627 194806 194986 195166 195345 195525 195704 195884 196063 196242

98 196422 196601 196780 196960 197139 197318 197497 197676 197855 198034

99 198213 198392 198671 198750 198928 199107 199286 199464 199643 199821

1.00 200000 200179 200357 200535 200714 200892 201070 201249 201427 201605

01 201783 201961 202139 202317 202495 202673 202851 203029 203207 203385

02 203563 203740 203918 204095 204273 204451 204628 204805 204983 205160

03 205338 205516 205692 205869 206047 206224 206401 206578 206755 206932

04 207109 207286 207463 207640 207816 207993 208170 208346 208523 208700

05 208876 209053 209229 209406 209582 209758 209935 210111 210287 210463

06 210640 210816 210992 211168 211344 211520 211696 211872 212047 212223

07 212399 212575 212750 212926 213102 213277 213453 213628 213804 213979

08 214154 214330 214505 214680 214856 215031 215206 215381 215556 215731

09 215906 216081 216256 216431 216606 216780 216955 217130 217304 217479

10 217654 217828 218003 218177 218352 218526 218700 218875 219049 219223

11 219398 219572 219746 219920 220094 220268 220442 220616 220790 220964

12 221138 221311 221485 221659 221832 222006 222180 222353 222526 222700

13 222874 223047 223220 223394 223567 223740 223913 224086 224260 224433

14 224606 224779 224952 225125 225298 225470 225643 225816 225989 226161

15 226334 226507 226679 226852 227024 227197 227369 227542 227714 227886

16 228059 228231 228403 228575 228747 228919 229091 229263 229435 229607

17 229779 229951 230123 230295 230466 230638 230810 230981 231153 231324

18 231496 231667 231839 232010 232182 232353 232524 232695 232867 233038

19 233209 233380 233551 233722 233893 234064 234235 234406 234577 234747

20 234918 236089 235260 235430 235601 235771 235942 236112 236282 236453

21 236623 236794 236964 237134 237304 237476 237645 237816 237985 238156

22 238325 238495 238666 238835 239004 239174 239344 239513 239683 239853

23 240023 240192 240362 240631 240701 240870 241039 241209 241378 241547

24 241717 241886 242055 242224 242393 242562 242731 242900 243069 243238

25 243407 243576 243744 243913 244082 244250 244419 244588 244756 244925

26 246093 246262 245430 246598 245767 245936 246103 246271 246440 246608

27 246776 246944 247112 247280 247448 247616 247784 247951 248119 248287

28 248455 248622 248790 248958 249125 249293 249460 249628 249795 249963

29 250130 250297 250464 250632 250799 250966 251133 251300 261467 251634

30 251801 251968 252135 252302 262469 252635 252802 252969 253135 263302

31 253469 263636 253802 253968 264135 254301 254468 254634 254800 254967

32 256133 266299 256465 265631 255797 265963 256130 256296 266461 256627

33 256793 266959 267125 257291 257456 267622 257788 257953 258119 258284

34 258450 258615 258781 268946 269111 259277 269442 269607 269772 259938

35 260103 260268 260433 260598 260763 260928 261093 261257 261422 261587

263233 36 261752 261917 262081 262246 262410 262575 262740 262904 263068

37 263397 263662 263726 263890 264066 264219 264383 264547 264711 264875

38 265039 266203 266367 265531 266696 265859 266022 266186 266360 266514

268149 39 266677 266841 267006 267168 267332 267496 267659 267822 267985

40 268312 268475 268638 268801 268964 269128 269291 269454 269617 269780

271407

273031

274652

276269

277882

279491

281097

282700

284299

41 269943 270106 270269 270431 270694 270767 270920 271082 271245

42 271570 271733 271895 272057 272220 272382 272545 272707 272869

43 273194 273356 273518 273680 273842 274004 274166 274328 274490

44 274814 274976 275137 275299 276461 275622 275784 275946 276107

45 276430 276592 276763 276914 277076 277237 277398 277669 277721

46

47

48

49

278043

279652

281268

282860

278204

279813

281418

283020

278365

279973

281578

283180

278526

280134

281739

283340

278687

280295

281899

283600

278848

280456

282069

283660

279009

280616

282219

283819

279170

280776

282380

283979

279331

280937

282540

284139

60

51

52

53

54

55

56

67

58

59

284468

286053

287644

289232

290816

292397

293976

295648

297118

298685

284618

286212

287803

289391

290976

292556

294132

296706

297276

298841

284777

286372

287962

289550

291133

292713

294290

295863

297432

298998

284937

286631

288121

289708

291291

292871

294447

296020

297589

299164

286097

286690

288280

289867

291449

293029

294604

296177

297746

299311

285256

286849

288439

290025

291607

293186

294762

296334

297902

299467

285416

287008

288598

290183

291765

293344

294919

296491

298059

299623

285575

287167

288756

290342

291923

293502

295077

296648

298216

299780

285734

287326

288915

290500

292081

293669

295234

296805

298372

299936

286894

287486

289074

290668

292239

293817

295391

296962

298528

300093

60

61

62

63

64

66

66

67

68

69

300249

301808

303364

304917

306466

308012

309656

311093

312629

314161

300406

301964

303520

306072

306621

308166

309709

311247

312782

314314

300561

302119

303675

305227

306776

308321

309863

311401

312936

314467

300717

302275

303830

305382

306931

308476

310017

311665

313089

314620

300873

302431

303986

306637

307085

308629

310171

311708

313242

314773

301029

302687

304141

305692

307240

308784

310325

311862

313396

314926

301186

302742

304296

305847

307394

308938

310478

312015

313549

316079

301341

302898

304452

306002

307549

309092

310632

312169

313702

316232

301496

303053

304607

306167

307703

309246

310786

312322

313855

315385

301652

303209

304762

308312

307858

309401

310940

312476

314008

315537

70

71

72

73

74

75

76

77

78

79

315690

317216

318737

320256

321771

323282

324791

326296

327798

329296

315843

317367

318889

320407

321923

323433

324941

326446

327947

329446

316996

317520

319041

320659

322074

323584

325092

326596

328097

329596

316148

317672

319193

320711

322226

323736

326243

326746

328247

329744

316300

317824

319345

320862

322376

323886

325393

326897

328397

329894

316453

317977

319497

321014

322527

324037

325644

327047

328547

330044

316606

318129

319648

321165

322678

324188

325694

327197

328697

330193

316758

318281

319800

321317

322829

324339

326845

327347

328847

330343

316910

318433

319952

321468

322980

324489

325996

327497

328997

330492

317063

318585

320104

321619

323131

324640

326145

327648

329146

330641

21
TABLE 9.—

Values of1— R'' when R=0.8—Continued

(All values are decimal fractions)

X

0

1

2

3

4

5

6

7

8

9

1.80 330791 330940 331089 331238 331388 331637 331686 331835 331984 332133

81 332282 332431 332580 332729 332878 333027 333176 333326 333474 333622

82 333771 333919 334068 334217 334366 334614 334762 334810 334959 336107

83 335256 335404 336562 336701 336849 336997 336146 336293 336441 336589

84 336737 336885 337033 337181 337329 337477 337626 337773 337920 338068

86 338216 338363 338611 338659 338806 338964 339101 339249 339396 339543

86 839691 339838 339985 340133 340280 340427 340674 340721 340868 341016

87 341163 341310 341457 341604 341750 341897 342044 342191 342338 342484

88 342631 342778 342924 343071 343218 343364 343611 343657 343803 343950

89 344096 344243 344389 344635 344681 344828 344974 346120 345266 345412

90 346558 345704 346860 345996 346142 346288 346434 346680 346726 346872

91 347017 347163 347308 347454 347600 347745 347891 348036 348182 348327

92 348472 348618 348763 348909 349054 349199 349344 349489 349634 349780

93 349925 360070 350216 350360 350506 360650 360794 350939 351084 351229

94 351374 351518 361663 361808 351962 352097 352241 352385 352530 352676

95 352820 352964 363108 353252 363397 353641 353686 363829 353974 354118

96 354262 354406 354550 364694 354838 364982 365126 356270 355414 356658

97 355701 355845 366989 356132 356276 356420 356663 356707 356851 356994

98 357137 357281 357424 357668 367711 367854 357998 358141 368284 358427

99 358570 358713 358856 359000 359143 359286 359428 359671 359714 359867

2.00 360000 360143 360286 360428 360671 360714 360866 360999 361141 361284

01 361426 361569 361711 361854 361996 362139 362281 362423 362566 362708

02 362850 362992 363134 363276 363418 363560 363702 363844 363986 364128

03 364270 364412 364654 364695 364837 364979 365121 366262 366404 366545

04 365687 365829 365970 366112 366253 366394 366636 366677 366818 366960

05 367101 367242 367383 367525 367666 367807 367948 368089 368230 368371

06 368612 368652 368793 368934 369075 369216 369357 369497 369638 369778

07 369919 370060 370200 370341 370481 370622 370762 370903 371043 371183

08 371323 371464 371604 371744 371884 372025 372166 372306 372446 372585

09 372725 372865 373006 373145 373285 373424 373664 373704 373844 373983

10 374123 374263 374402 374542 374681 374821 374960 375100 375240 375379

11 375518 375657 376797 375936 376075 376214 376364 376493 376632 376771

12 376910 377049 377188 377327 377466 377606 377744 377882 378021 378160

13 378299 378438 378576 378715 378854 378992 379131 379269 379408 379546

14 379685 379823 379961 380100 380238 380376 380515 380653 380791 380929

15 381067 381205 381343 381481 381619 381757 381895 382033 382171 382309

16 382447 382585 382722 382860 382998 383136 383273 383411 383648 383686

17 383823 383961 384098 384236 384373 384510 384648 384785 384922 385060

18 385197 385334 386471 385608 386745 385882 386019 386156 386293 386430

19 386567 386704 386841 386978 387114 387251 387388 387524 387661 387798

20 387934 388071 388207 388344 388480 388617 388763 388889 389026 389162

21 389299 389435 389671 389707 389843 389980 390116 390252 390388 390524

22 390660 390796 390932 391068 391204 391339 391475 391611 391747 391883

23 392018 392154 392289 392425 392561 392696 392831 392967 393102 393238

24 393373 393509 393644 393779 393914 394050 394185 394320 394465 394590

25 394725 394860 394995 395130 396266 396400 395535 396670 395805 396940 26 396075 396209 396344 396479 396614 396748 396883 397017 397151 397286 27 397421 397555 397689 397824 397968 398093 398227 398361 398495 398630 28 398764 398898 399032 399166 399300 399434 399568 399702 399836 399970 29 400104 400238 400372 400506 400639 400773 400907 401040 401174 401307

30 401441 401674 401708 401842 401975 402108 402242 402375 402509 402642 31 402775 402908 403042 403175 403308 403441 403574 403707 403840 403973 32 404106 404239 404372 404506 404638 404771 404904 405036 406169 405302 33 405435 405567 405700 406832 405964 406097 406230 406362 406495 406627 34 406760 406892 407025 407167 407289 407421 407553 407686 407818 407950 35 408082 408214 408346 408478 408610 408742 408874 409006 409138 409270 36 409401 409633 409665 409797 409928 410060 410192 410323 410455 410686 37 410718 410849 410981 411112 411244 411375 411506 411638 411769 411900 38 412031 412163 412294 412426 412556 412687 412818 412949 413080 413211 39 413342 413473 413604 413734 413865 413996 414127 414258 414388 414519

40 414650 414780 414911 415041 415172 415302 415433 416563 415694 415824 41 415954 416085 416216 416346 416475 416606 416736 416866 416996 417126 42 417256 417386 417616 417646 417776 417906 418036 418166 418295 418425 43 418555 418685 418814 418944 419074 419203 419333 419463 419592 419721 44 419851 419980 420110 420239 420369 420498 420627 420756 420886 421015 45

46

421144 421273 421402 421631 421660 421790 421919 422048 422177 422306 422434 422563 422692 422821 422950 423078 423207 423336 423465 423593 47

48

49

423722 423850 423979 424107 424236 424364 424493 424621 424760 424878 425006 425134 425263 425391 425619 425647 425775 426904 426032 426160 426288 426416 426544 426672 426800 426928 427055 427183 427311 427439

50

51

52

53

54

55

66

57

58

59

427567

428843

427694 427822 427950 428077 428205 428332 428460 428588 428715 428970 429097 429225 429362 429479 429607 429734 429861 429988 430116 430243 430370 430497 430624 430751 430878 431005 431132 431259 431386

432653

433918

435180

436439

437695

438948

431513 431640 431766 431893 432020 432147 432273 432400 432527 432780 432906 433033 433159 433286 433412 433639 433665 433792 434044 434170 434297 434423 434549 434675 434801 434928 435054 435306

436664

437820

439073

436432

436670

437946

439198

435568

436816

438071

439323

435684

436941

438197

439448

435809

437067

438322

439574

435935

437193

438447

439699

436061

437318

438672

439824

436187

437444

438698

439949

436313

437569

438823

440074

60

61

62

63

64

65

66

67

68

69

440199

441446

442691

443933

445173

446410

447644

448875

450103

451329

440323

441671

442816

444068

445297

446533

447767

448998

450226

451461

440448

441696

442940

444182

446420

446657

447890

449121

450349

451674

440573

441820

443064

444306

445544

446780

448013

449244

450471

451696

440698

441945

443189

444430

445668

446904

448136

449367

450594

461818

440823

442069

443313

444554

445792

447027

448260

449389

450716

451941

440948

442194

443437

444678

446916

447150

448383

449612

450839

452063

441072

442318

443561

444801

446039

447274

448506

449735

450962

462175

441197

442443

443685

444925

446163

447397

448629

449858

451084

452308

441322

442567

443809

445049

446286

447620

448762

449981

451207

452430

22
TABLE 9.—

Values ofl— R"" whenR=0.8—Continued

(All values are decimal fractions)

X

01

2

34

66

78

9

2.70 452552 452674 452796 452918 453040 453162 453284 453406 453628 453650

71 453772 453894 454016 454138 454259 454381 454503 454625 454746 454868

72 454990 455111 455233 455354 455476 455597 465719 455840 455962 456083

73 456204 456326 456447 456568 456690 456811 466932 457053 457174 457295

74 457416 457538 457659 457780 457901 458022 458142 458263 458384 458606

75 458626 458747 458867 458988 459109 459230 469350 459471 459692 459712

76 459833 459953 460074 460194 460315 460435 460555 460676 460796 460916

77 461037 461157 461277 461397 461517 461638 461758 461878 461998 462118

78 462238 462358 462478 462598 462718 462838 462958 463077 463197 463317

79 463437 463556 463676 463795 463915 464035 464155 464274 464394 464513

80 464633 464752 464871 464991 465110 465230 465349 465468 465587 465707

81 465826 465945 466064 466183 466302 466422 466541 466660 466779 466898

82 467017 467135 467254 467373 467492 467611 467730 467848 467967 468086

83 468205 468323 468442 468560 468679 468798 468916 469035 469153 469272

84 469390 469508 469627 469745 469863 469981 470100 470218 470336 470454

85 470573 470691 470809 470927 471045 471163 471281 471399 471517 471635

86 471753 471871 471988 472106 472224 472342 472460 472577 472695 472812

87 472930 473048 473165 473283 473400 473518 473635 473753 473870 473988

88 474105 474222 474339 474457 474574 474681 474808 474926 475043 475160

89 475277 475394 476511 475628 475745 475862 475979 476096 476213 476330

90 476447 476563 476680 476797 476914 477030 477147 477264 477380 477497

91 477614 477730 477847 477963 478080 478196 478312 478429 478546 478662

92 478778 478894 479011 479127 479243 479359 479475 479592 479708 479824

93 479940 480056 480172 480288 480404 480520 480636 480751 480867 480983

94 481099 481216 481331 481446 481562 481678 481793 481909 482025 482140

95 482256 482371 482487 482602 482717 482833 482948 483064 483179 483294

96 483410 483525 483640 483755 483871 483986 484101 484216 484331 484446

97 484561 484676 484791 484906 485021 485136 485251 485365 485480 485595

98 485710 485825 485939 486054 486169 486284 486398 486513 486627 486742

99 486856 486971 487085 487200 487314 487428 487543 487657 487772 487886

3.00 488000 488114 489228 488343 488457 488571 488685 488799 488913 489027

01 489141 489255 489369 489483 489597 489711 489825 489939 490052 490166

02 490280 490394 490507 490621 490735 490848 490961 491075 491189 491303

03 491416 491530 491643 491757 491870 491983 492096 492210 492323 492436

04 492550 492663 492776 492889 493002 493115 493229 493342 493455 493568

04 493681 493794 493907 494020 494132 494245 494358 494471 494584 494697

06 494809 494922 495035 495147 495260 495373 495485 495598 495710 495823

07 495935 496048 496160 496273 496385 496497 496610 496722 496834 496947

08 497059 497171 497283 497395 497508 497620 497732 497844 497956 498068

09 498180 498292 498404 498516 498628 498740 498851 498963 499075 499187

10 499298 499410 499522 499633 499745 499857 499968 500080 600191 500303

11 500414 600526 500637 500749 500860 500971 501083 501194 501306 501417

12 501528 601639 501750 501862 501973 502084 502195 502306 502417 502528

13 502639 502750 502861 602972 603083 503194 503305 503415 503526 503637

14 503748 503858 503969 604080 504190 504301 504412 504522 504633 504743

15 504854 504964 505076 605186 505296 505406 505516 506627 505737 505847

16 505958 506068 506178 506288 506398 506508 506618 506729 506839 506949

17 507059 507169 507279 507389 507498 507608 507718 507828 507938 508048

18 508157 508267 508377 608486 508596 508706 508815 508925 509035 509144

19 509254 509363 509473 509582 509691 509801 509910 510020 510129 510238

20 510348 510457 510566 510675 510784 510893 511003 611112 511221 511330

21 511439 511548 511657 611766 511875 511984 512093 512202 512310 512419

22 512528 512637 512745 612854 512963 513072 513180 613289 513397 513506

23 513615 513723 513831 613940 514048 514157 514265 514374 514482 514590

24 514699 514807 514915 515024 516132 615240 515348 515456 515564 515672

25 515780 515888 515996 516104 616212 616320 516428 516536 616644 516752

26 516860 516967 517075 617183 617291 617398 517506 517614 517721 517829

27 517936 518044 518152 518259 618367 618474 518582 518689 518796 518904

28 519011 519118 619226 619333 519440 519547 619654 519762 519869 519976

29 520083 620190 520297 520404 520511 520618 520725 520832 520939 521046

30 521153 521260 521366 521473 521580 521687 521794 521900 522007 522114

31 522220 522327 522433 622540 522646 522753 522859 522966 623072 523178

32 523285 523392 523498 523604 523710 523817 623923 524029 524135 524242

33 524348 524454 524560 524666 524772 624878 524984 525090 525196 525302

34 525408 525514 525620 625725 525831 625937 526043 526148 626254 626360

35 526466 526572 526677 526783 526888 526994 527099 527205 527310 527416

528469 36 527521 527627 527732 527837 627943 528048 528164 528269 528364

37 528574 528680 528785 528890 528995 529100 529205 529310 529415 529520

38 529625 529730 529835 529940 630045 530150 530255 530359 630464 530569

39 530674 630778 530883 630987 531092 631197 531302 531406 531511 531615

40

41

42

43

44

45

46

47

48

49

531720

532763

533805

534844

535881

536915

537947

538977

540005

641030

531824

532868

533909

534948

535984

537019

538050

539080

540108

641133

531929

532972

534013

536052

636088

537122

538154

539183

540210

541236

632033

533076

534117

536155

536191

537225

538257

539286

540313

541337

532137

533180

534221

535259

536295

537328

538360

539389

540415

541440

532242

533284

534325

535363

536398

537432

538463

539491

540518

541542

632346

533389

534429

535466

536502

537535

538566

539594

540620

541644

532451

533493

534532

535570

536605

537638

538669

539697

540723

541747

532655

533597

534636

535674

536709

537741

538772

539800

540826

641849

532659

533701

534740

535777

536812

537844

538874

539902

540928

541951

50

51

52

53

64

65

56

57

58

69

642053

543074

644092

645109

646123

647134

548144

649151

550156

651158

642155

643176

644194

545210

546224

547235

548245

549251

550256

551259

642258

543278

644296

546312

546325

547336

548345

549352

650357

551359

542360

543380

544398

545413

646426

547437

548446

549453

550457

551459

542462

543482

544499

545515

546528

547538

548547

549553

550557

551559

542564

543584

544601

545616

546629

547639

548648

549654

550657

551659

542666

543685

544702

545717

546730

547740

548748

649764

550758

551759

542768

543787

544804

545819

646831

547841

548849

649855

550858

551869

542870

543889

544906

645920

546932

547942

548950

549955

550958

551959

542972

543991

545007

646021

547033

548043

549050

550056

661058

552059

23
TABLE 9.—Values of 1 — R' when R=0.8—Continued

(All values are decimal fractions)

X 0 1 2 3 4 6 6 7 8 9

3.60 652169 552259 562359 652469 652568 652658 552758 662858 552958 653057

61 653167 663267 663356 553456 663566 553655 563755 663865 553954 654054

62 664153 654253 654352 654461 554551 554650 654750 664849 554948 566047

63 665147 655246 555345 656446 655544 555643 665742 555841 555940 556039

64 656138 556237 666336 656435 656534 556633 566732 656831 656930 557029

65 657128 567227 667325 667424 557523 567622 657720 567819 567917 568016

66 668115 668214 558313 658411 668509 558608 568706 658805 558903 659001

67 669100 669198 669297 659395 559493 659591 659690 569788 559886 559984

68 560082 560181 660279 660377 560476 560673 560671 560769 660867 560966

69 561063 661161 661259 661357 661455 561553 561650 561748 661846 561944

70 562041 562139 662237 662336 562432 562530 662627 562725 662823 562920

71 563018 563115 563213 663310 563408 563605 663602 563700 563797 563894

72 563992 664089 664186 664284 664381 564478 564675 564672 564769 664866

73 564964 565061 666158 566265 665352 565449 665546 565643 565739 665836

74 565933 566030 566127 666224 666320 566417 566514 566611 566707 566804

75 566901 566997 567094 567191 667287 567384 667480 567577 567673 667770 76 567866 667963 568059 568155 568252 568348 668444 568541 568637 668733

77 568829 568925 569022 669118 669214 569310 569406 669602 569598 669694 78 569790 669886 569982 670078 570174 570270 670366 570462 570558 670654 79 570749 670845 670941 671037 671132 571228 671324 571419 671516 671610

80 571706 671802 571897 671993 672088 672184 672279 572374 572470 672565 81 672661 572766 572851 672947 673042 673137 573232 573328 573423 673518 82 673613 673708 673803 573899 673994 574089 674184 574279 574374 574469 83 674564 674659 574753 574848 574943 575038 675133 575228 575322 576417 84 675512 576607 576701 675796 675891 575986 676080 576174 576269 576363 86 676468 676552 676647 676741 576836 576931 677025 577119 577213 577308 86 577402 577496 577591 577685 677779 677873 577967 578062 578156 578250 87 678344 678438 678532 678626 578720 678814 578908 579002 579096 579190 88 579284 679378 579472 679565 679659 579763 579847 679941 580034 580128 89 580222 580316 580409 580502 580696 680690 580783 580877 580970 581064

90 581157 581251 681344 581438 681531 681624 581718 581811 681904 581998 91 582091 582184 582277 682370 582464 682557 582660 582743 682836 582929 92 683022 583115 583208 683301 583394 583487 583580 583673 583766 583859 93 583952 684045 584137 584230 584323 584416 584608 584601 584787 94 684879 684972 585064 585167 585249 585342 685435 585627 585619 585712 95 585804 585897 585989 586082 586174 586266 586359 586451 586543 686635 96 586728 586820 586912 587004 587096 587189 587281 587373 687465 587567 97

98

99

587649 587741 587833 587925 588017 588109 588201 588292 588384 688476 588568 588660 588761 588843 588935 589027 589118 589210 589302 589393 589485 589577 589668 589760 589851 589943 590034 590126 590217 590309

4.0

1

2

3

4

5

6

7

8

9

59040

59944

60828

61692

62538

63364

64173

64963

65736

66493

59131

60033

60916

61778

62621

63446

64253

65041

65813

66567

59222

60122

61002

61863

62704

63527

64332

65119

65889

66642

59313

60211

61089

61948

62787

63609

64412

65197

65965

66716

59404

60300

61176

62033

62870

63690

64491

65275

66041

66790

59494

60388

61262

62117

62953

63771

64570

65352

66117

66864

59585

60477

61349

62202

63036

63851

64649

65429

66192

66938

69676

60565

61435

62286

63118

63932

64728

65606

66268

67012

69765

60653

61521

62370

63200

64012

64807

65583

66343

67085

59854

60740

61607

62454

63282

64093

64885

65660

66418

67159

6.0

1

2

3

4

6

6

7

8

9

67232

67955

68662

69354

70030

70691

71338

71971

72589

73194

67305

68027

68732

69422

70097

70757

71402

72033

72650

73254

67378

68098

68802

69490

70164

70822

71466

72095

72711

73313

67451

68169

68871

69558

70230

70887

71529

72158

V2772

73373

67523

68240

68941

69626

70296

70962

71592

72220

72833

73432

67596

68311

69010

69694

70363

71016

71656

72282

72893

73491

67668

68381

69079

69761

70429

71081

71719

72344

72964

73551

67740

68451

69148

69829

70495

71146

71782

72406

73014

73610

67812

68522

69217

69896

70560

71210

71845

72467

73074

73668

67883

68592

69286

69963

70626

71274

71908

72528

73134

73727

6.0

1

2

3

4

5

6

7

8

9

73786

74364

74930

75483

76024

76553

77071

77576

78071

78555

73844

74421

74986

75638

76077

76605

77122

77626

78120

78603

73902

74478

75041

75692

76131

76658

77173

77676

78169

78651

73960

74535

75097

75647

76184

76710

77224

77726

78218

78698

74018

74592

76163

75701

76237

76761

77274

77776

78266

78746

74076

74649

76208

75766

76290

76813

77325

77825

78315

78793

74134

74705

75263

75809

76343

76865

77375

77876

78363

78840

74192

74761

75318

75863

76396

76917

77426

77924

78411

78888

74249

74818

75373

75917

76448

76968

77476

77973

78459

78935

74307

74874

75428

75970

76501

77019

77526

78022

78507

78982

7.0

1

2

3

4

5

6

7

8

9

79028

79491

79944

80386

80819

81242

81656

82061

82457

82844

79075

79537

79989

80430

80862

81284

81697

82101

82496

82882

79122

79583

80033

80474

80905

81326

81738

82141

82535

82920

79168

79628

80078

80517

80947

81367

81779

82181

82574

82959

79215

79674

80122

80561

80990

81409

81819

82221

82613

82997

79261

79719

80166

80604

81032

81451

81860

82260

82652

83035

79307

79764

80211

80647

81074

81492

81900

82300

82690

83072

79354

79809

80255

80690

81117

81533

81941

82339

82729

83110

79400

79864

80299

80733

81159

81574

81981

82379

82768

83148

79445

79899

80343

80776

81201

81615

82021

82418

82806

83185 8.0

1

2

3

4

6

6

7

8

9

83223

83593

83955

84309

84655

84994

85325

85649

85966

86274

83260

83630

83991

84344

84690

85027

85358

85681

85997

86305

83297

83666

84027

84379

84724

85061

85391

85713

86028

86336

83335

83702

84062

84414

84758

85094

85423

85745

86059

86367

83372

83739

84098

84449

84792

85127

85456

85776

86090

86397

83409

83775

84133

84483

84826

85161

85488

85808

86121

86428

83446

83811

84168

84518

84859

85194

85520

85840

86162

86458

83483

83847

84204

84552

84893

85227

85552

85872

86183

86488

83520

83883

84239

84587

84927

86259

85685

85903

86214

86518

83557

83919

84274

84621

84960

85292

85617

85934

86245

86548
TABLE 9.—Values of

l— R"" whenR= 0.8-

(All values are decimal fractions)

-Continued

X

0

1

2

34

6

6

7

8

9

9.0 86578 86608 86638 86668 86697 86727 86767 86786 86816 86845 1 86875 86904 86933 86962 86991 87020 87049 87078 87107 87136 2 87164 87193 87221 87250 87278 87306 87335 87363 87391 87419 3 87447 87475 87503 87531 87669 87587 87614 87642 87669 87697 4 87724 87752 87779 87806 87833 87860 87887 87916 87942 87968 5 87995 88022 88049 88076 88102 88128 88155 88181 88208 88234 6 88260 88286 88312 88338 88364 88390 88416 88442 88468 88494 7 88519 88545 88570 88596 88621 88647 88672 88697 88722 88747 8 88773 88798 88823 88847 88872 88897 88922 88947 88971 88996 9 89020 89045 89069 89094 89118 89142 89166 89190 89215 89239

10.0 89263 89287 89310 89334 89368 89382 89405 89429 89453 89476 1 89500 89523 89546 89570 89593 89616 89639 89662 89685 89708 2 89731 89754 89777 89800 89822 89845 89868 89890 89913 89936 3 89958 89980 90003 90025 90047 90069 90091 90113 90135 90157 4 90179 90201 90223 90246 90267 90288 90310 90332 90353 90375 5 90396 90418 90439 90460 90482 90503 90624 90545 90566 90687 6 90608 90629 90660 90671 90692 90712 90733 90754 90774 90795 7 90815 90836 90856 90877 90897 90917 90937 90958 90978 90998 8 91018 91038 91058 91078 91098 91118 91138 91157 91177 91197 9 91216 91236 91255 91276 91294 91314 91333 91352 91372 91391

11.0 91410 91429 91448 91467 91486 91505 91524 91543 91662 91581 1 91600 91618 91637 91666 91674 91693 91711 91730 91748 91767 2 91785 91803 91822 91840 91858 91876 91894 91912 91930 91948 3 91966 91984 92002 92020 92038 92065 92073 92091 92108 92126 4 92144 92161 92179 92196 92213 92231 92248 92265 92283 92300 5 92317 92334 92351 92368 92385 92402 92419 92436 92453 92470 6 92487 92503 92520 92637 92553 92570 92586 92603 92619 92636 7 92662 92669 92685 92701 92718 92734 92750 92766 92782 92798 8 92814 92830 92846 92862 92878 92894 92910 92926 92942 92957 9 92973 92989 93004 93020 93035 93061 93066 93082 93097 93113

12.0 93128 93143 93159 93174 93189 93204 93219 93235 93250 93266 1 93280 93295 93310 93324 93339 93364 93369 93384 93399 93413 2 93428 93443 93457 93472 93486 93501 93515 93630 93544 93559 3 93573 93587 93602 93616 93630 93644 93668 93673 93687 93701 4 93716 93729 93743 93767 93771 93786 93798 93812 93826 93840 5 93853 93867 93881 93895 93908 93922 93935 93949 93962 93976 6 93989 94003 94016 94029 94043 94066 94069 94082 94096 94109 7 94122 94135 94148 94161 94174 94187 94200 94213 94226 94239 8 94252 94264 94277 94290 94303 94315 94328 94341 94353 94366 9 94378 94391 94403 94416 94428 94441 94453 94465 94478 94490

13.0 94502 94515 94627 94539 94651 94663 94575 94588 94600 94612 1 94624 94636 94648 94660 94671 94683 94695 94707 94719 94731 2 94742 94754 94766 94777 94789 94801 94812 94824 94835 94847 3 94858 94870 94881 94893 94904 94916 94927 94938 94949 94961 4 94972 94983 94994 95006 95017 95028 95039 95060 95061 95072 5 95083 95094 96105 95116 95126 95137 96148 96169 95170 95181 6 95191 95202 95213 95223 96234 95245 95255 96266 95276 95287 7 95297 95308 95318 95329 95339 95349 95360 95370 95381 95391 8 95401 95411 95422 96432 95442 95452 95462 95472 95483 95493 9 95503 95513 95623 95533 96643 95553 95663 95672 95682 96692

14.0 95602 95612 95621 95631 96641 95661 96660 96670 95680 95689

95784

95878

95969

96058

96144

96230

96313

96394

96474

1 95699 95709 96718 95728 95737 95747 95756 96766 95775 2 95794 95803 95813 96822 96831 95841 95860 95859 95868 3 95887 95896 95906 95914 96923 95932 96941 96961 96960 45

95978

96066

95987

96075

95996

96081

96004

96092

96013

96101

96022

96110

96031

96119

96040

96127

96049

96136

6789

96153

96238

96321

96402

96162

96246

96329

96410

96170

96255

96337

96418

96179

96263

96346

96426

96187

96271

96354

96434

96196

96280

96362

96442

96204

96288

96370

96450

96213

96296

96378

96468

96221

96304

96386

96466

15.0123456789

96482

96559

96635

96709

96782

96853

96923

96991

97057

97122

96489

96567

96643

96717

96789

96860

96929

96998

97063

97128

96497

96574

96650

96724

96796

96867

96936

97004

97070

97135

96605

96582

96658

96731

96803

96874

96943

97010

97076

97141

96513

96590

96665

96739

96811

96881

96950

97017

97083

97147

96621

96597

96672

96746

96818

96888

96957

97024

97089

97164

96528

96605

96680

96753

96825

96895

96963

97030

97096

97160

96536

96612

96687

96760

96832

96902

96970

97037

97102

97166

96544

96620

96695

96768

96839

96909

96977

97044

97109

97173

96561

96628

96702

96775

96846

96916

96984

97050

97115

97179

16.0123456789

97185

97247

97308

97368

97426

97482

97538

97592

97645

97697

97192

97253

97314

97373

97431

97488

97544

97598

97651

97702

97198

97260

97320

97379

97437

97494

97549

97603

97656

97708

97204

97266

97326

97385

97443

97499

97654

97608

97661'

97713

97210

97272

97332

97391

97448

97505

97660

97614

97666

97718

97216

97278

97338

97397

97464

97510

97566

97619

97671

97723

97223

97284

97344

97403

97460

97616

97571

97624

97677

97728

97229

97290

97360

97408

97465

97521

97576

97630

97682

97733

97235

97296

97356

97414

97471

97527

97581

97635

97687

97738

97241

97302

97362

97420

97477

97532

97587

97640

97692

97743

17.0123456789

97748

97798

97846

97894

97940

97986

98030

98074

98116

98168

97753

97803

97851

97899

97945

97990

98035

98078

98120

98162

97758

97808

97856

97903

97950

97995

98039

98082

98125

98166

97763

97813

97861

97908

97954

97999

98043

98087

98129

98170

97768

97817

97866

97913

97969

98004

98048

98091

98133

98174

97773

97822

97870

97917

97963

98008

98052

98095

98137

98178

97778

97827

97876

97922

97968

98013

98056

98099

98141

98182

97783

97832

97880

97927

97972

98017

98061

98104

98146

98186

97788

97837

97885

97931

97977

98022

98065

98108

98150

98190

97793

97842

97889 , 97936

97981

98026

98069

98112

98154

98194

25
TABLE 9.—

Values of1— R'' whenR=0.8—Continued

(All values are decimal fractions)

26

X0

1

23

4

66

7

.

9

18.0 98199 98203 98207 98211 98215 98219 98223 98226 98230 98234 1 98238 98242 98246 98260 98264 98258 98262 98266 98269 98273 2 98277 98281 98285 98289 98292 98296 98300 98304 98308 98311 3 98315 98319 98323 98326 98330 98334 98338 98341 98345 98349 4 98352 98356 98360 98363 98367 98371 98374 98378 98382 98386 6 98389 98392 98396 98399 98403 98406 98410 98414 98417 98421 6 98424 98428 98431 98435 98438 98442 98445 98449 98452 98456 7 98459 98462 98466 98469 98472 98476 98480 98483 98486 98490 8 98494 98497 98500 98603 98506 98510 98513 98516 98520 98623 9 98526 98530 98633 98636 98639 98543 98646 98549 98562 98566

19.0 98559 98662 98565 98568 98572 98576 98578 98581 98584 98587 1 98591 98594 98597 98600 98603 98606 98609 98612 98616 98619 2 98622 98625 98628 98631 98634 98637 98640 98643 98646 98649 3 98652 98655 98658 98661 98664 98667 98670 98673 98676 98679 4 98682 98685 98688 98691 98694 98697 98699 98702 98705 98708 6 98711 98714 98717 98720 98722 98726 98728 98731 98734 98737 6 98739 98742 98745 98748 98751 98763 98756 98759 98762 98765 7 98767 98770 98773 98775 98778 98781 98784 98786 98789 98792 8 98794 98797 98800 98802 98805 98808 98810 98813 98816 98819 9 98821 98824 98826 98829 98832 98834 98837 98839 98842 98845

20.0 98847 98850 98852 98855 98867 98860 98862 98865 98867 98870 1 98873 98876 98878 98880 98883 98885 98888 98890 98892 98895 2 98897 98900 98902 98905 98907 98910 98912 98914 98917 98919 3 98922 98924 98927 98929 98931 98934 98936 98938 98941 98943 4 98945 98948 98950 98953 98955 98967 98960 98962 98964 98966 5 98969 98971 98973 98976 98978 98980 98982 98985 98988 98990 6 98992 98994 98996 98998 99001 99003 99005 99007 99009 99012 7 99014 99016 99018 99020 99023 99025 99027 99029 99031 99034 8 99036 99038 99040 99042 99044 99046 99048 99051 99063 99065 9 99057 99059 99061 99063 99065 99067 99069 99071 99073 99075

21.0 99077 99080 99082 99084 99086 99088 99090 99092 99094 99096 1 99098 99100 99102 99104 99106 99108 99110 99112 99114 99116 2 99118 99120 99122 99124 99126 99128 99130 99132 99134 99136 3 99137 99139 99141 99143 99146 99147 99149 99161 99153 99155 4 99157 99158 99160 99162 99164 99166 99168 99170 99171 99173 66789

99175 99177 99179 99181 99182 99184 99186 99188 99190 99192 99193 99195 99197 99199 99201 99202 99204 99206 99208 99209 99211 99213 99215 99216 99218 99220 99222 99223 99225 99227 99229 99230 99232 99234 99235 99237 99239 99241 99242 99244 99246 99247 99249 99251 99252 99264 99256 99257 99259 99261

22.0 99262 99264 99266 99267 99269 99270 99272 99274 99275 99277 123456789

99279

99294

99310

99325

99340

99355

99369

99383

99396

99280

99296

99312

99327

99341

99356

99370

99384

99398

99282

99298

99313

99328

99343

99358

99372

99386

99399

99283

99299

99315

99330

99344

99369

99373

99387

99401

99285

99300

99316

99331

99346

99360

99374

99388

99402

99286

99302

99318

99333

99347

99362

99376

99390

99403

99288

99304

99319

99334

99349

99363

99377

99391

99405

99290

99306

99321

99336

99360

99365

99379

99392

99406

99291

99307

99322

99338

99362

99366

99380

99394

99407

99293

99308

99324

99339

99353

99368

99381

99395

99408

23.0123456789

99410

99423

99436

99448

99460

99472

99484

99495

99506

99517

99411

99424

99437

99449

99461

99473

99485

99496

99507

99518

99412

99425

99438

99450

99463

99474

99486

99497

99508

99619

99414

99427

99439

99462

99464

99476

99487

99498

99509

99520

99415

99428

99440

99453

99465

99477

99488

99500

99511

99521

99416

99429

99442

99454

99466

99478

99490

99501

99612

99622

99418

99431

99443

99455

99467

99479

99491

99502

99513

99524

99419

99432

99445

99456

99468

99480

99492

99603

99514

99525

99420

99433

99446

99458

99470

99481

99493

99504

99615

99526

99421

99434

99447

99459

99471

99483

99494

99505

99516

99527

24.0123456789

99528

99538

99548

99558

99568

99578

99587

99596

99605

99614

99529

99639

99549

99559

99569

99579

99588

99597

99606

99615

99630

99540

99550

99660

99570

99580

99589

99698

99607

99616

99531

99541

99551

99561

99571

99581

99690

99699

99608

99616

99532

99542

99552

99562

99572

99581

99691

99600

99609

99617

99633

99543

99553

99563

99573

99582

99592

99601

99609

99618

99634

99644

99654

99564

99574

99583

99693

99602

99610

99619

99635

99645

99656

99565

99575

99584

99593

99602

99611

99620

99536

99546

99566

99666

99576

99585

99594

99603

99612

99621

99537

99647

99557

99567

99577

99586

99595

99604

99613

99621

25.0123456789

99622

99630

99639

99647

99655

99662

99670

99677

99684

99691

99623

99631

99640

99648

99656

99663

99670

99678

99685

99692

99624

99632

99640

99648

99666

99664

99671

99678

99686

99692

99626

99633

99641

99649

99667

99664

99672

99679

99686

99693

99626

99634

99642

99650

99658

99665

99672

99680

99687

99694

99626

99635

99643

99651

99659

99666

99673

99681

99687

99694

99627

99636

99644

99651

99659

99667

99674

99681

99688

99695

99628

99636

99644

99652

99660

99668

99675

99682

99689

99696

99628

99637

99645

99653

99661

99668

99675

99683

99690

99696

99629

99638

99646

99664

99661

99669

99676

99683

99690

99697

26.0123456

99698

99704

99711

99717

99724

99730

99736

99699

99705

99712

99718

99724

99730

99736

99699

99706

99712

99719

99726

99731

99737

99700

99706

99713

99719

99726

99731

99738

99743

99749

99755

99700

99707

99714

99720

99726

99732

99738

99744

99750

99765

99701

99707

99714

99720

99727

99733

99702

99708

99715

99721

99727

99733

99703

99708

99716

99722

99728

99734

99703

99709

99716

99722

99728

99734

99704

99710

99717

99723

99729

99735

789

99741

99747

99753

99742

99748

99753

99743

99749

99764

99739

99744

99750

99756

99739

99745

99751

99756

99740

99746

99751

99767

99740

99746

99752

99757

99741

99747

99752

99758
TABLE 9.—Values of l — R"" when R =0.8—Continued

(All values are decimal fractions)

X 0 1 2 3 4 5 6 7 8 9

27.0 99759 99759 99759 99760 99760 99761 99761 99762 99762 99763 1 99764 99764 99765 99765 99766 99766 99767 99767 99768 99768 2 99769 99769 99770 99770 99771 99771 99772 99772 99773 99773 3 99774 99774 99775 99775 99776 99776 99777 99777 99778 99778 4 99779 99779 99780 99780 99781 99781 99782 99782 99783 99783 5 99784 99784 99785 99785 99786 99786 99787 99787 99788 99788 6 99789 99789 99789 99790 99790 99791 99791 99792 99792 99793 7 99793 99794 99794 99795 99795 99796 99796 99796 99797 99797 8 99798 99798 99799 99799 99800 99800 99800 99801 99801 99802 9 99802 99803 99803 99804 99804 99805 99805 99805 99806 99806

28.0 99807 99807 99807 99808 99808 99809 99809 99810 99810 99810 1 99811 99811 99812 99812 99813 99813 99813 99814 99814 99815 2 99815 99816 99816 99816 99817 99817 99817 99818 99818 99819 3 99819 99820 99820 99820 99821 99821 99821 99822 99822 99823 4 99823 99823 99824 99824 99825 99825 99825 99826 99826 99827 5 99827 99827 99828 9982S 99829 99829 99829 99830 99830 99830 6 99831 99831 99832 99832 99832 99833 99833 99833 99834 99834 7 99835 99835 99835 99836 99836 99836 99836 99837 99837 99838 8 99838 99839 99839 99839 99840 99840 99840 99841 99841 99841 9 99842 99842 99842 99843 99843 99844 99844 99844 99845 99845

29.0 99845 99846 99846 99846 99847 99847 99847 99848 99848 99848

1 99849 99849 99849 99850 99850 99850 99851 99851 99851 99852

2 99852 99852 99853 99853 99853 99854 99854 99854 99855 99855

3 99855 99856 99856 99856 99857 99857 99857 99858 99858 99858

4 99858 99859 99859 99859 99860 99860 99860 99861 99861 99861

6 99862 99862 99862 99862 99863 99863 99863 99864 99864 99864

6 99865 99865 99865 99866 99866 99866 99866 99867 99867 99867

7 99868 99868 99868 99869 99869 99869 99869 99870 99870 99870

8 99871 99871 99871 99871 99872 99872 99872 99873 99873 99873

9 99873 99874 99874 99874 99875 99875 99875 99875 99876 99876

30 998762

999010

999208

999366

999493

999594

999675

999740

999792

999834

999866

31

32

33

34

35

36

37

38

39

40

27

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