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NBS MONOGRAPH 2

Temperature-Induced Stresses in

Solids of Elementary Shape

U.S. DEPARTMENT OF COMMERCE

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UNITED STATES DEPARTMENT OF CO\RIERCE • Frederick H. Mueller, Secret, NATIOX.\L BUREAU OF ST.\XDARDS • A. X. Astin. Director

Temperature-Induced Stresses in

Solids of Elementary Shape

Leason H. Adams and Roy M. Waxier

National Bureau of Standards Monograph 2

Issued June 21, 1960

For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington 25, D.C. - Price 25 cents

Contents

Page

1. Introduction 1 2. Sudden change of surface temperature 1 Slab 1 Cylinder 3

Sphere ^ 4

3. Linear heating 5

4. References 27

m

Temperature-Induced Stresses in Solids

of Elementary Shape

Leason H. Adams and Roy M. Waxier

In general, a solid subjected to non-uniform temperature change develops internal

stresses. These are determined by, (1) the temperature distribution within the solid, and (2) certain physical constants of the material. Although the necessary fundamental

relations are known, the computation of stresses in even simple solids has heretofore been

a tedious operation. For two varieties of heating, the equations determining stress have now been put in convenient form for practical use, and tables of certain temperature functions

are offered as a means of quickly determining stresses in a slab, in a cylinder, or in a sphere

subjected to either of two modes of heating. The temperature-distribution tables inde- pendently will provide a useful means for the ready estimation of temperature gradients.

1. Introduction

Stresses in an elastic solid caused by variations

in temperature may be evaluated from the temperature change, the coefficient of thermal ex- pansion, and the elastic constants. In principle,

this calculation of stress, at least for elementary

solids, is quite simple, but in actual practice it may involve laborious and time-consuming computations. There are given here some equations

in convenient form for calculating the stresses

in solids of certain shapes when subjected (a) to a sudden change of surface temperatm'e, or (b) to a

siurface temperature that increases or decreases at

'*a constant rate. (It is assumed that there is no

creep or 3-ielding in the material.)

2. Sudden Change of Surface

Temperature

Slab.—For the case of a slab that is assmned

to be of large enough extent so that end effects can be neglected, the stress is in a direction parallel

to the face of slab, and it varies, in both magnitude and sign, from sm-face to the center of the

slab. In the following treatment, average values

of the elastic constants and the coefficient of thermal expansion over the temperatm-e range are assumed to be adequate. A basis for calculation

is foimd in the equations given by Hopkinson

[1],^ Williamson [2], Williamson and Adams [3], and Adams and Wdliamson [4]. In order to visualize the stress relations in a

slab, let us consider first the case of a slab consisting of a number of detached layers. Upon

' being heated uniformly on both faces, the outer

layers will expand more than the inner layers, as

in figure la, but if the layers are not separated,

the tendency of the outer layers to expand rela- tively to the inner layers will be partiaUy offset

' Figures in brackets indicate the literatvire references at the end of this Mon(^aph.

by the constraint imposed by adjacent laj'ers and

the block will retain its original shape, as in

figtu-e lb. It is as if in a solid block the final effect were the sum of two effects, namely, (1) the lengthening of each layer by an amount determined by the temperature change, and (2) a

shortening or lengthening of the layer owmg to the

influence of other lavers. The stress distribution is foimd to be simply

related to the thermal expansion and the elastic constants. Let F be the stress (positive for a

tension and negative for a compression), d the

temperature at a distance x from the center of

the slab, and a the thermal expansibility. Furthermore, let e and / be, respectively, the relative

longitudinal lengthening and transverse shortenX

]

Figure la. The relative changes in dimension in a slab which is being heated when ike slab consists of detached

layers.

I

"

I

^ I

I

FiGTTHE lb. There are no relative changes in dimension in

a slab which is being heated if the slab is one solid -piece.

1
ing produced by unit stress. From symmetry

considerations it is evident that in an infinite slab

the stresses are the same in all directions parallel to the faces of the slab. It is to be noted that

these stresses are purely tensile or compressive

and there is no shearing force in planes parallel to the faces of the slab. Considering a small rectangular element in any

plane parallel to the face of the slab, a stress in a given direction will produce a lengthening equal

to Fe, while the equal stress in a transverse direction will produce a shortening (in the initial direction) equal to Ff, the aggregate effect being F

(e—/). The change in length of any element is also dependent on the temperature; and the

thermal and the elastic effects are found to be

related in such a way that the smn of the two

effects is a constant independent of x. This is the

fundamental basis of the present treatment, and

may be illustrated as follows. Suppose that, similar to figure 1, we have a slab

consisting of two layers on each side of central

plane, and suppose that the layers, originally

identical in size, are caused to expand symmetrically by applying a temperature gradient such

that O—gx, the distance from the central plane to the middle of the slab being x and the temperatiu-e

gradient being g. Then, if we take x as equal to

1 and 2, respectively, for the inside and the outside layers, and if the thermal expansibility is a, the relative lengthening in each transverse du'ec- tion will be ag and 2ag. Now suppose that we

bring the inner and the outer layers to the same

length by stretching the former by ^ag and compressing the letter by ^ag. The corresponding

stresses F will be iag/{e—f) and —^ag/{e—f).

Finally, we fasten the adjacent sxu-faces together

and remove the external restraints, noting that by

reasons of symmetry there will be no overall

warping or bending. Simple addition (algebraically) of the effects (neglecting those of the second

order) shows that for each of the layers F{e—f) ad

is equal to S/2ag.

Applying the same procedm-e to a slab with four

layers on each side, first dealing with each outer

and each inner pau" as in the above example, and

then treating the groups of two pairs m the same

fashion, we find that in this case F{e~J)-{-aB is equal to a constant, 5/2ag, for all layers; and it is clear that by further subdivision of the slab into

thinner and thinner layers the constancy of the

sum of the thermal and mechanical effects will continue to exist, and furthermore that in the

limit F{e—f)-\-aO is accm'ately a constant, which

in the simple cases considered here will always

be a times the average temperatirre from center to surface. We have, then.

F=C-ad

(e-f)

or F^a ad

(1)

Ci being a constant. It may be noted that our fundamental relation is equivalent to the state- ment that by the aggregate effect of temperature

and stress the plane surfaces in the slab remain

plane.

It may be noted that lie is Young's modulus,

Jle is Poisson's ratio, and e—/=l/6(?+2/9ii:, G

and K being, respectively, the rigidity modulus , and the bulk modulus, and (7i is a constant to be evaluated from the condition of mechanical

equilibrium to the effect that the algebraic summation of the forces m any direction parallel to the faces of the slab is zero. That is,

0

so that

edx. (2)

For convenience we employ instead of 6, a temperature function which may be called the frac- tional temperature excess for a slab, and is defined by

4>i (3)

in which di is the initial uniform temperature of

the material, and df is the final constant temperature. Inserting in eq (1) the value of Ci from

eq (2) and transforming, we have

F=H{ri-<i>i),

in which

H- <e,-df)

e-f

and is the definite integi'al

(4)

(5)

(6)

Qm2

-2y'e~^^^ V Qm cos I Qm -h

The function \}/[ is merely the average fractional

temperature excess at any given time (averaged

from center to surface). It may be noted that

in applying eq (1) the stress, F, is in the same

units as the elastic moduli, G and K.

To calculate the stress, then, we need to determine (1) the temperature at a specified time and

location, and (2) the average temperature from

center to surface at the specified time.

There are two types of equations for calculating

the temperature distribution in a slab the temperature of which is changing, a cosine-exponential

form and a probability-integral form. The for- mer, for sudden change of surface temperature

from an originallv uniform temperature, is as follows [3].

(7)

2
r jin which m means the succession of integers

- Ibeginning with unity; and Q?n = (2m— l)(7r/2).

! jOf the two dimensionless parameters, xja and g, the latter is defined by:

I IK being the thermal difi^usivity which is the

I thermal conductivity divided by the product of

jthe specific heat and density. It may be noted

' that at the niddle of the slab eq (7) simplifies to

:

m / \\m+l

=2S. • (9)

The cosine-exponential series converges rapidly

I for large values of the time, or small values of a

(or both). Indeed, if an accuracy of 0.0001 in

is sufficient, only one term of the series is necessary

whenever g is less than 0.8 (e.g., when t is greater

than 100 sec, k being 0.004 cm^sec and a being

ij 1 cm). On the other hand, for short times or

large dimensions many terms of the series may be

required. _

An alternate equation, [5, 6], for temperatm-es

in a slab is the infinite series of probability

I j integrals:

l|0i=erf g(l-//a)+X] [erf g(2m+3-y/a) .

' 0 -erf g(2m+l+a:/a)](-l)'"+S (10)

2 _zi

in which erf {A) is written for -p I e dZ, Z

Vtt Jo

I j being merely an integration variable.

In contrast with the .cos—exp equation, the

I' series (10) converges rapidly for short times, or

large dimensions (or both). For an accmracy of 0.0001 in 4>i only one term of the series is required

whenever g is greater than 2.7 (e.g., when t is less than 8 sec, if k=0.004 and a = l). In general the

' convergence is more rapid near the surface than

l' near the center. The same is true for the cos—exp

' formula.

As an example, for t=\2Q sec and a=l cm,

I (g=0.72), one term of the cos—exp formula is

li sufficient for obtaining an accuracy of 0.0001 in

I the fractional temperature excess, while three

terms of the prob—int series are required for the same accuracy with a given q. The latter series

,1 at longer times requires still more terms—as many

I as six for i=700, a= 1(2= 0.3). By contrast, for

;| the relatively short time, t=%, a=l(g=2.8) the

i| cos—exp formula requires five terms and the

prob—int only one. There is an intermediate range of times (in

relation to slab dimensions) for which neither

I formula can be depended on to converge suffi- ciently well with one term.

The average temperature, one of the basic

factors for determining the temperature-induced

stresses, can be computed by applying (6) to either

(7) or (10). Thus from the cos—exp formula we

have

m ^p[=2^ e-«'" /*«7^w2. (11)

Here, we have been able to integrate analytically, but when it is advantageous to use the

proh—int formula (as, for example, when t is relatively smaU) it is necessary to have recourse

to numerical integration. In this case, a table of

values of ^^'i can be prepared by calculating

for each desired value of g at several values of

xja, and then applying eq (6) by numerical integration. Alternatively, as a more convenient

method for a certain range in t (or more properly

in q), the procedure may be as follows: First,

for the various values of g, we compute s/'n, which is here defined as the average value of the

first term of the summation in eq (11) as xia

varies from 0 to 1. Then from the -^[I's it turns out that we may readily compute i^J for the given

2 by the expression

^[ (2) ='/'u (2) - [ (3 V'n (32) -2^n (22) ) -

(2^n(22)-'^n(2))]

+[(5./'n (52) -4i^n (42) ) - {^^P'u (42) -

-[{li'n (72) - e./'n (62)) - (6i/'n (62) -

5^n(52))l+ . . . (12)

This procedure will permit an appreciable saving in time for calculating ip'i when high-speed

computing devices are not available. As will appear below, the principal shortcut, for this and

other shapes, is achieved through the tabulation

of certain temperature-distribution functions. The

series given in eq (12) converges rapidly except for

relatively small values of 2- Thus for 2=0.6 only

two of the terms in square brackets of eq (12) are required for an accuracy of 0.0001 in yp'i] for

2=1.0 only one term (in square brackets); and

for 2=2-5 ^[ equals the simply calculated ^'n within the above specified precision.

Cylinder.—The stresses which develop in a cylinder having a radial temperature gradient may

be visuahzed if the cylinder is first thought of as

consisting of a number of separate, concentric,

cylindrical shells. These shells would separate if

a' temperature gradient were established, and there

would be radial, tangential and axial strains in

each shell. However, since a solid cyhnder does

not consist of separate shells, internal stresses are

created, and the combination of the thermal and

elastic effects gives the total strain. The stresses

in the cyhnder may be evaluated from these equations for strain and additional equations which

express the mechanical equilibrium which must

exist [2]. 3
For the radial stress, Fr, in a cylinder, we have

an extension and modification of the relations

given by Williamson [2].

Fr=Coie (13)

in which d is a constant to be evaluated from the

condition that Fr=0 when x—a. Thus we have ^ - r xddx. (14)

Making the transformations similar to those employed for the slab, we find that the convenient

working equation for the cylinder is

in which

F, =H(;A2-\^2) (15)

(16)

(t>2 being the fractional temperature excess for a

cylinder; and 1^2 is the value of 1^2 at x/a=l. It may be noted that \p2 is one-half the average value

of </)2 when averaged with respect to {xjaY. End

eft'ects being neglected, we have for a cylindrical

rod a radius a, the surface temperature of which

has been suddenly changed, the following equation for <^2, at the time t and the fractional dis- tance xja from the center:

1

1 J^mJl{Rm)

Jo(Rmxla), (17)

in which Jo and Ji, respectively, stand for Bessel

functions of the zero'th and first order, and the

Rm's are the successive roots of Jo=0.

The tangential stress Ft is related to the radial

stress as follows,

dF

and is readily found to be

F,=H{^l^'2+^l^o-<t>2).

(18)

(19)

The function i/'2 can be calculated by numerical

integration from previously determined ^2, or by substituting (17) into (16) and integrating.

Thus

In determining i/'2,the value of \f/2 at x=a, it is to be noted that when x=a, Ji{RmX/a)=Ji(Rm)-

Sphere. —^In a manner analogous to that used to analyze the thermal stresses which develop in a

slab and in a cylinder, a sphere may be imagined

which consists of a number of detached, concentric,

spherical shells. These shells will separate when

a temperature gradient is established and show

radial and tangential strains. These thermal

effects may also be seen in a sphere which is not

segmented, and, in addition, there are elastic

effects in this case, because the solid sphere acts

as an entity, and it may be imagined that the

shells are bound together. The equations for total strain, together with those for the mechanical equilibrium which must exist in the sphere,

suffice to calculate the internal stresses. The fol- lowing equations are extensions of those given byj

Wilhamson [2]. In a sphere the radial stress is determined by

Fr^Cs2a J_ p , '(e-f) x'jo^'

edx. (21)

The constant C3, as in the case of the cylinder, is evaluated by the condition that Fr=0 where x=a. Thus

e J d Jo ! and we therefore have

F,=2H(xl.',-rP,).

The function yj/^ is defined by

(23)

*'=(^r<'/"''*'*(i)

in terms of <f>3 the fractional temperature excess

for a sphere; and i^-s is the value of ^ps for xja=l.

It may be noted that 1^3 is one-third of the average

value of (f>3 when averaged with respect to {xja)?

For the tangential stress

and the working equation is, thus

i<', =77(2,A3 +^3-^3).

(25) i

(26)

The fractional temperature excess 4>3 in the

sphere, as in the case of the slab, can be calculated

from either of two equations. The former is

(-1)m+l , S„

'- sin (S„,x/a) (27)

in which Sm is written for mx. Alternately, we may use the probability-integral equation [5,6]:

03 —a . a r

X x\ erf q(l-x/a)+J2[eri q{3m+S-x/a) — erf q{2m-{-l-\-x/a)] }

(28)

4
Similarly to the cases for the slab and the cylinder

the function ^3 can be computed by numerical

ji. integration; or for the sphere it can be determined

icij bv substituting (27) into (24) and integrating.

V Thus

si 9 m I

cos(5„jya))(-l)"'+i. (29a)

mI It is readily found that

si, m ^P'z=2^e "o/Sl- (29b)

1

3. Linear Heating

For the important case in which the surface

temperatm-e is changed at a constant linear rate,

^ the internal temperatm-e, as shown by Williamson

)| and Adams [3], can be calculated by equations

; consisting of three terms, a linear one, a parabolic

f one, and a third one with an exponential involving

; the time t. When t becomes large with respect to the size a, the exponential term becomes vanish-

)

I ingly small, the temperature distribution is es- sentially parabolic and the equations for 6 take

the simple forms for the "steady state":

Slab e-e,=ht-ha'-x') (30)

Cylinder e-e.^ht-^y-x"-) (31)

Sphere e-et=ht~(a'-x')

OK

(32)

in"which h is the rate at which the surface is being

heated. Having these temperatiu-es we may applv

eqsl (1), (2), (1.3), (14), (21), and (22) to (30),

(31), and (32), and we find that the stresses are

Slab F=^ ia'—3x') (33)

Cylinder F,=^ (a'-x') (34)

^'=v%

-•'^^'^ (35)

Sphere (36)

F,=Y^ {a'-2x'). (37)

In these equations

J ah

538523—60 2

Especially for the case of sudden change of

surface temperature, the complexity of the equations for temperature and stress have made the

calculation of numerical values a time-consuming

process. Extended tables of various temperature

functions have now been prepared by the Computation Section of the Applied Mathematics

Division of the National Bureau of Standards.

By the use of these tables the calculation of stres- ses in many important instances becomes a matter

of simple addition and multiplication. At present

there are five tables, namely for (together with

^i), <t>2, 4^2, 4>z, and and these are designated by

numbers 5 through 9, respectively.

Quite apart from the utility of the tables in

connection with stress calculation, the tables for the <^'s may prove to have wide application as a

labor-saving device in problems involving the

estimation of temperature distribution. In this

regard it is important to note that for the case of

sudden change of surface temperature the tables

may be used for calculating temperatures in cer- tain other shapes. Thus, from the complete

equation for temperatures in a parallelepiped, it turns out that such temperatures may be obtained

by multiplying together 3 </)i's each taken from

the table with its appropriate parameter. Furthermore, temperatures in a finite cylinder, for

this species of temperature change, may be obtained by multiplying the appropriate and ^2- A numerical example of the use of the tables

for determining stresses in simple solids subjected

to a sudden change of surface temperature is as

follows:

The example chosen is for the case of a slab of

Pyrex glass 7740, 2 cm in thickness (a=l), the

surface temperature having been changed suddenly from 100° to 0° C. The appropriate parameters are found in tables 1 and 2. Since

q=a/2^jKt, after a time lapse of 60 sec q will

be 0.786.

Table 1. Parameters for the calculation of stress

Material

Coefficient

of thermal

expansion, a

(range 0°-100° C)

Diffusivitv, K (at 100° C)

Parameter for elastic effects (e-f)

(at 100° C)

Plate glass 7.80X10-6 5. 49X103 1. 18X10-* Pyrex 7740_ 3. 46 6.74 1.27 Vvcor. . - - 0.83 8. 82 1.27 Fused silica .48 9.80 1. 16 Pure aluminum 23.5 964 0.951 Pure iron,- 12.1 198 .359

For 2=0.786 it is necessary to interpolate

between the values of fractional temperature excess given in table 5. To find the distribution of

stress we apply the working equation for the slab,

F=H{rP[-<l>r) (4)

5

I
Table 2. Calculated thermal stresses in a slab of Pyrex 7740

5=0.7800'* ?=0.8000 =0.786 F, kgjcm? Temp.

° C

0.0 0. 4619 0. 4856 0. 4690 -0. 1704 -46.3 46.9

. 1 .4662 .4796 .4632 -. 1646 -44.8 46.3

. 2 .4393 .4618 .4461 -. 1475 -40.1 44.6

.3 .4116 .4327 .4179 -. 1193 -32.4 41.8

.4 .3737 .3929 . 3795 -. 0809 -22.0 38.0

. 5 . 3267 . 3435 . 3317 —. 0331 —9. 0 33. 2

.6 .2716 . 2855 .2758 .0228 6.2 27.6

.7 .2098 .2206 .2130 .0856 23.3 21.3

.8 .4128 . 1501 .1450 .1536 41.8 14,5 .9 .0723 .0760 .0734 .2252 61.3 7.3 1.0 .0000 .0000 .0000 .2986 81.2 0

.2941 .3092 .2986

•Distance from center plane/slab semi-thickness. a '*q= — where t is the elapsed time and k is the thermal difiusitivitv. 2^1 Kt

is the fractional temperatm-e excess and ({J is the average fractional temperature excess (see p. 2). where

H- (5)

and, substituting numerical values, we have

H=272. With appropriate interpolations, values

of fractional temperature excess and of the stress are found as in table 2. A graphical representation of this variation of

stress with position in the slab is shown in figure 2,

together with curves for other values of elapsed

time.

To find the temperature at various points in the

slab for the 100 deg temperature change it is

necessary merely to multiply by a factor of 100

the values of fractional temperature excess given

in the fourth column of table 2. A plot of this

variation of temperature with position in the slab

is given in figure 3, together with curves for other

values of elapsed time.

In order to gain a better conception of these

eftects in different materials and with difi^erent slab thicknesses and elapsed times, tables 3 and 4

have been prepared, temperature distribution

being shown in table 3 and stress distribution in

table 4. For illustrative purposes it was felt to be of interest to consider the case of pure aluminum and of pure iron, although the parameters for

these materials are not as well known as the

corresponding ones for glasses.

Figure 4 shows the distribution of stress in a 2-cm thick slab of plate glass at selected time

intervals after the surface temperature has been

changed suddenly from 100° to 0° C. Figure 5

affords a comparison of the distribution of stress in four commercial glasses which have undergone

a sudden change in surface temperature under

the same boundary conditions.

The parameters which were needed for these

calculations were obtained from the literature

[7, 8, 9], and are given in table 1. The values for

plate glass, iron and aluminum are only

approximate.

For further illustration an example of stress in a

cylinder of plate glass with linear surface heating

is given in figure 6.

300

1 second

2 minutes

1.0 0.8 0.6 0.4 0.2 Distance from

0

the

0.2 0.4 0.6 0.8 1.0 center plane, cm Figure 2. Curves showing the calculated distribution of stress in a slab of Pyrex 7740 2-cm. thick, when the surface temperature has been changed suddenly from 100° to 0° C. The curves show the distribution of stress which exists after a time lapse of 1 sec, 10 sec, 1 mln, and 2 mln. 6
1-0 .9 .6 .7 .6 .5 .4 .3 .2 ,1 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 Distonce from the center plone, cm Figure 3. Curves showing the calculated distribution of temperature in a slab of Pyrex 7740 2-cm

thick, when the surface temperature has been changed suddenly from 100° to 0° C. The curves show the distribution of temperature which exists after a time lapse of 1 sec, 10 sec, 1 min, and 2 min.

1.0 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1.0 Distonce from the center plane, cm Figure 4. Curves showing the calculated distribution of stress in a slab of plate glass 2-cm

thick, when the surface temperature has been changed suddenly from 100 to 0° C. The curves show the distribution of stress which exists after a time lapse of 1 sec, 10 sec, 1 min, and 2 min.
Figure 6. Cvrves showing the calculated distribution of radial and tangential stresses in a, cylinder of plate glass 20 cm in diameter which is being heated linearly at a rate of 2° C/min.
Table 3. Temperature distribution}, ° C in various

materials suddenly cooled*

Pvrex Aluminuin Aluminum Aluminum Iron Iron a=l 0 = 1 0 = 1 0 = 1 0=10 0 = 1 0 = 1 «=60"" t=2 (=2 «=1 t=l «=2 t=l

0.0 47 100 1 12 100 48 77

. 1 46 100 1 12 100 47 76

2 45 100 1 11 100 46 74 .3 42 100 1 11 100 43 69

.4 38 100 10 100 39 63

. 5 33 100 } 8 100 34 55 28 99 100 28

21 97 5 97 22 36

.8 15 84 0 4 85 15 24

. 9 52 0 2 53 8 12

1.0 0 0 0 0 0 0 0

•See p. 6. "Distance from center plane/slab semithictness. •"Time in seconds.

Table 4. Thermal stress, kgjcm? in various materials suddenly cooled*

Pvrex Pyrex £\.nilll ixit Jill ^\ niiii|imiii A InTninnTTi 1 1 1 1 1 ' ' 1 1 1 1 iJ Iron Iron

xia** 0 = 1 0 = 1 0=10 0 = 1 0 = 1 /=60*** t=2 i = 2 1 t=l t=2 t=l

0. 0 —46 —31 ^0 — 106 —274 537 933

_ I 45 3j JQ —103 —274 5f;8 902

_ 2 —40 —31 9 —92 274 5Q8 gjj

. 3 —32 —31 _7 —74 —274 —411 —612

'.i -22 -31 -5 -50 -274 -279 -454

.5 -9 -31 -2 -21 -273 -114 -193

.6 6 -30 1 14 -264 79 -116

. 7 23 -22 5 53 -198 295 466

.8 42 13 9 95 97 629 850

.9 61 100 13 140 892 776 1,256

1.0 81 241 17 185 2,197 1,029 1,674

•See p. 6. "Distance from center plane/slab semithiokness. •••Time in seconds. Table 5. Fractional temperature excess, 4>i, and generalized temperature averages, i/-'], for a slab

0. 3000 0.3200 0.3400 0. 3600 0. 3800 0. 4000 0.4200 0. 4400 0.4600 0. 4800

0.0.10

0.20

0.30 0.40

0.50 0.60

0.70

0.80

0.90

1.00

0. 0013

.0013

.0013

. 0012

.0011 .0010

.0008

.0006 .0004

.0002

.0000

0. 0031 .0030 .0029

.0027 .0025

.0022 .0018

.0014

.0010 .0005

.0000

0. C061

.0061 .0058

. 0055

.0050 .0043

.0036

.0028

.0019

.0010

.0000

0. 0109

.0108

.0104 .0097

.0088

.0077

.0064

.0050 .0034

.0017 0000

0. 0178

.0176

.0169

. 0158 .0144

.0126

.0104

.0081 .0055

.0028 .0000

0. 0270

.0266

.0256

.0240 .0218

.0191 .0158

.0122 .0083

.0042

.0000

0. 0386

.0381 .0367

.0344 .0312 .0273

.0227

.0175

.0119

.0060 .0000

0. 0526

.0520 .0500

. 0469 .0426

.0372

.0309 .0239

.0163 .C082

.0000

0. 0690

.0682

.0656

.0615 .0558

.0488

.0406

.0313

.0213 .0108

.0000

0. 0875

. 0865

.0832

. 0780

.0708

.0619

.0515

.0397

.0270

.0137

. 0000 Average ^i' 0.C009 0. 0020 0. 0039 0. 0069 0. 0113 0.0172 0. 0246 0. 0335 0. 0439 0. 0557

g xIa 0.5000 0. 5200 0. 5400 0. 5600 0. 5800 0.6000 0.6200 0. 6400 0. 6600 0. 6800

0.0.10

0.20

0.30 o!40

0.50

0.60

0.70

0.80

0.90

1.00

0. 1080 .1066

.1027

. 0962

.0874 0764

.0635

.0490

.0334

.0169

.0000

0. 1301 .1285 .1237

. 1159

1052 .0920

.0765

. 0591" .0402

.0203

.0000

0. 1535 .1516

: 1460

. 1368

1242 .1086

.0902

.0697

.0474

.0240 .0000

0. 1781

. 1759 1694

. 1587

1441

. 12.59

. 1047 .0809

.0550

.0279

.0000

0. 2035

.2010 1935

. 1813 .1646

. 14.39 .1196

.0924

.0629

.0318

.0000

0. 2295

.2267 2183

. 2045

1857

. 1623 .1349

.1042

.0709 .0359

.0000

0. 2559

.2527 2433

.2280 2070

.1809

.1504

.1162

.0791 .0400

.0000

0. 2824 .2789

2686

. 2516

2285

.1997

.1660

. 1282 .0873

.0442

.0000

0. 3090

.3052 2938

. 2753

2500

.2185 .1816

.1403 .0955

.0483

.0000

0 3354

.3313 3190

. 2988

2713

. 2372

. 1971 .1523

. 1036 .0525

.0000 Average <l/\' 0. 0687 0. 0828 0. 0977 0. 1134 0. 1295 0. 1461 0. 1629 0. 1798 0. 1967 0. 2135

Q

xja \

0.7000 0. 7200 0. 7400 0. 7600 0.7800 0.8000 0. 8200 0.8400 0. 8600 0.8800

0.0.10

0.20

0.30

0.40

0.50

0.60

0.70 0.80

0.90

1.00

0. 3616

.3571 .3439

.3222

.2925 .2557

.2125

.1642

. 1117 .0566

.0000

0. 3874

.3826

.3684 .3452

.3134

.2739 .2277

.1759

.1197

.0606 .0000

0. 4127 .4077

.3925

.3678

.3339

.2919

.2426

.1874 .1276

.0646

.0000

0. 4376

.4322

.4162 .3899

.3541 .3095 .2573

.1987 .1353

.0685 .0000

0. 4619

.4562 .4393

.4116

.3737

.3267

.2716 .2098

.1428

.0723

.0000

0. 4856 .4796

.4618

.4327

.3929

.3435

.2855

.2206

.1501 .0760

.0000

0. 5086

.5024 .4838

.4533

.4116

.3598

.2991 .2311 .1573

.0796

.0000

0. 5310

.5245

.5051 .4733 .4298

.3757

.3124 .2413

. 1643 .0832

.0000

0. 5527 .5459

.5258

.4927 .4474

.3912

.3252 .2513

. 1711 .0866

.0000

0. 5738

.5667 .5458

.5115 .4645

.4062

.3377 .2609

. 1777 .0900

.0000

Average 0.2302 0. 2466 0. 2628 0. 2786 0. 2941 0. 3092 0. 3239 0. 3382 0. 3521 0. 3655

I/O \^ 0.9000 0.9200 0. 9400 0. 9600 0.9800 1.0000 1. 0200 1. 0400 1.0600 1. 080O

0.0.10

0.20 0.30

0.40

0.50

0.60

0.70

0.80 0.90

1.00

0.5941 .5868

.5652

.5297

.4811 .4207

.3499

.2704

.1841 .0932 0000

0. 6137

.6062 .5839

.5473

.4972 .4348

.3617 .2795

. 1903 .0964

.0000

0. 6327 .6250 .6020

.5643

.5127

.4485

.3731 .2884

. 1964 .0995

.0000

0. 6509

.6430

.6195

.5807 .5278

.4617

.3842

.2970 .2023

. 1025 .0000

0. 6685

.6604

,6363 .5966

.5423

.4746

.3950 .3054

.2081

. 1054 .0000

0.6854

.6772

.6525

.6119

.5564

.4870 .4054

.3136

.2137

. 1082 .0000

0. 7017 .6933

.6681 .6267

.5700

.4991 .4156

.3215

.2191

. 1110 .0000

0. 7173 .7087

.6831 .6410

.5831 .5108

.4255

.3293

.2245

. 1137 .0000

0. 7323

.7236

.6976

.6547 .5958

.5221

.4351 .3368

.2297 .1164

.0000

0. 7467

. 7378 .7114

. 6680

.6081 .5331

. 4445

.3442

. 2348

. 1190 .0000 k verage ^i' 0. 3786 0. 3912 0. 4034 0. 4153 0. 4267 0. 4378 0. 4485 0. 4588 0. 4688 0. 4784

9
Table 5. Fractional temperature excess, 4>i, and generalized temperature averages, ^'i, for a slab—Continued

1. 1000 1. 1200 1. 1400 1. 1600 1. 1800 1. 2000 1. 2200 1. 2400 1. 2600 1. 2800

0. 0. 7604 0. 7736 0. 7862 0. 7982 0. 8097 0. 8206 0. 8311 0. 8410 0. 8505

0. 10 . 7515 . 7645 . 7771 . 7890 . 8005 . 8114 . 8218 . 8318 . 8412 . 8503

0.20 . 7248 . 7376 . 7498 . 7616 . 7729 . 7837 . 7941 .8040 . 8135 . 8226

0. 30 . 6807 . 6930 . 7048 . 7162 . 7272 . 7378 . 7480 . 7578 . 7672 . 7763

0.40 . 6200 . 6315 . 6426 . 6534 .6638 . 6739 . 6837 '(5932 . 7024 . 7113

o!50 .5438 .5542 .5643 . 5741 .5837 .5930 .6020 !6109 !6195 !6280

0.60 . 4536 . 4625 .4712 . 4797 . 4880 .4962 . 5042 . 5120 .5197 .5272

0.70 .3514 .3585 . 3655 .3723 .3789 .3855 .3920 .3984 .4046 .4109

0.80 .2398 .2447 .2496 .2543 .2590 . 2636 .2682 .2727 .2772 .2816

0.90 .1216 .1241 . 1266 .1291 .1315 .1339 .1362 .1386 .1409 .1432

1.00 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 Average 0. 4878 0. 4968 0. 5055 0. 5140 0, 5221 0. 5301 0. 5377 0. 5451 0. 5523 0. 5593

1. 3000 1. 3200 1. 3400 1. 3600 1. 3800 1. 4000 1. 4200 1. 4400 1. 4600 1. 4800 -—

0. 0. 8680 0. 8761 0. 8838 0. 8911 0. 8980 0. 9046 0. 9108 0. 9166 0. 9221 0. 9273

0.10 . 8589 . 8670 . 8748 .8822 . 8892 . 8958 .9021 .9081 . 9137 . 9191

0.20 . 8313 . 8396 . 8475 . 8551 . 8624 . 8693 . 8759 .8822 '. 8882 . 8939

0.30 '. 7850 . 7935 .8016 . 8094 . 8169 .8242 . 8312 . 8379 '. 8444 . 8506

0.40 . 7200 . 7283 .7365 . 7444 .7521 . 7596 . 7668 . 7739 .7808 . 7874

oiso .6362 .6443 .6522 .6599 .6674 .6748 .6821 .6892 !6962 !7030

0.60 . 5346 . 5419 . 5491 . 5562 . 5632 . 6701 . 5769 . 5836 .5902 .5967

0.70 .4170 .4231 .4291 .4350 .4409 .4467 . 4525 .4582 .4639 .4696

0.80 .2860 .2904 .2947 .2990 .3033 .3075 .3118 .3160 .3202 .3243

0.90 .1455 . 1478 .1501 . 1523 .1546 . 1568 .1590 .1613 .1635 .1657

1.00 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 Average ^[ 0. 5661 0. 5726 0. 5790 0. 5852 0. 5912 0. 5970 0. 6027 0. 6082 0. 6136 0. 6188

—

1. 5000 1. 5200 1. 5400 1. 5600 1. 5800 1. 6000 1. 6200 1. 6400 1. 6600 1. 6800

0. 0. 9322 0. 9368 0. 9412 0. 9453 0. 9491 0. 9527 0. 9561 0. 9592 0. 9622 0. 9650

0.10 . 9241 . 9289 .9334 .9377 . 9417 .9455 . 9491 .9524 . 9556 . 9585

0.20 . 8994 . 9046 . 9096 .9143 . 9188 . 9231 . 9272 .9311 . 9348 . 9383

0^30 '. 8566 '. 8624 !8680 . 8734 '. 8785 . 8835 .8883 !8930 . 8974 '9017

0.40 . 7939 .8002 . 8064 . 8124 .8182 .8239 . 8294 . 8348 .8400 . 8451

0.50 !7097 .7163 .7227 .7291 .7353 .7414 .7474 .7533 .7591 !7648

0.60 .6032 .6095 .6158 .6221 .6282 .6343 .6403 .6462 . 6.521 . 6579

0.70 .4752 .4807 .4863 .4918 .4972 .5026 . 5080 .5134 .5187 .5240

0.80 .3285 .3326 .3368 .3409 .3450 .3491 .3532 .3572 .3613 .3653

0.90 .1679 .1702 . 1724 .1746 .1768 .1790 .1812 .1834 .1856 .1878

1.00 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 Average 0. 6239 0. 6288 0. 6336 0. 6383 0.6429 0. 6474 0.6517 0. 6560 0. 6601 0. 6642

1. 7000 1. 7200 1. 7400 1. 7600 1. 7800 1. 8000 1. 8200 1. 8400 1. 8600 1. 8800 —

0. 0. 9676 0. 9700 0. 9723 0. 9744 0. 9763 0. 9782 0. 9799 0. 9815 0. 9829 0.9843

0.10 . 9613 . 9640 . 9664 !9687 ^9709 . 9729 !9748 ] 9766 .9783 !9798

0.20 . 9416 '. 9448 . 9479 .9507 . 9535 . 9560 . 9585 . 9608 .9631 . 9652

0.30 !9058 '. 9098 '. 9136 19173 .9209 . 9243 . 9276 .9308 .9338 [9367

0.40 '. 8501 '. 8549 '. 8596 . 8642 '. 8686 . 8730 . 8772 . 8813 ". 8853

' 8891

o!50 !7704 !7758 !7812 !7865 !7917 .7968 .8018 .8067 !8115 !8162

0.60 .6637 .6693 .6749 .6805 .6860 .6914 . 6967 .7020 .7073 .7124

0.70 .5292 .5344 .5396 .5447 .5498 .5549 .5600 .5650 .5700 .5749

0.80 .3693 .3734 .3774 .3814 .3854 .3893 .3933 .3972 .4012 .4051

0.90 .1900 .1922 . 1944 .1966 .1987 .2009 .2031 .2053 .2075 .2097

1.00 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 Average 0. 6681 0. 6720 0. 6758 0. 6794 0. 6830 0. 6866 0. 6900 0. 6934 0. 6967 0. 6999 \— —

1. 9000 1. 9200 1. 9400 1. 9600 1. 9800 2. 0000 2. 0200 2. 0400 2. 0600 2. 0800

0. 0. 9856 0. 9868 0. 9878 0. 9889 0. 9898 0. 9906 0. 9914 0. 9922 0. 9928 0. 9935

0.10 . 9813 . 9826 . 9839 . 9851 . 9862 . 9872 .9882 .9891 .9899 .9907

0.20 . 9672 . 9690 . 9708 . 9725 . 9741 . 9757 . 9771 . 9785 . 9798 .9810

0.30 . 9395 . 9422 . 9448 . 9473 . 9497 . 9520 . 9543 . 9564 . 9584 .9604

0.40 . 8929 . 8966 . 9001 . 9036 . 9070 . 9102 . 9134 . 9165 .9195 . 9224

0^50 !8208 .' 8254 !8298 !8342 !8385 .8427 .8468 .8508 .8548 .8586

0.60 .7175 .7226 .7275 .7324 .7373 . 7421 .7468 .7515 .7561 .7607

0.70 .5798 .5847 .5895 .5943 .5991 .6039 .6086 .6132 .6179 .6225

0.80 .4090 .4129 .4168 .4207 .4245 .4284 .4322 .4361 .4399 .4437

0.90 .2118 .2140 .2162 .2184 .2205 .2227 .2249 .2270 .2292 .2314

1.00 .0000 .0000 .lOOO .0000 .0000 .0000 .0000 .0000 .0000 .0000 Average lAi 0. 7031 0. 7062 0. 7092 0. 7121 0. 7151 0. 7179 0. 7207 0. 7234 0. 7261 0.7288

10
Table 5. Fractional temperature excess, 4>i, and generalized temperature averages, for a slab—Continued

I/a q

2. 1000 2. 1200 2. 1400 2. 1600 2. 1800 2. 2000 2. 2200 2. 2400 2. 2600 2.2800

0. 0. 9940 0. 9946 0. 99ol

_ 0. yyoo 0. 9959 0. 9963 0. 9966 0. 9969 0. 9972 0. 9975

0.10 . 9914 . 9921 . 9927 , 9932 . 9938 . 9943 . 9947 .9951 . 9955 . 9959

0.20 . 9821 . 9832 . 9842 . 9852 , 9861 . 9870 . 9878 . 9886 .8893 .9900

0.30 . 9623 . 9641 . 9658 . 9674 . 9690 . 9705 .9720 .9734 .9747 .9760

0.40 . 9252 . 9279 . 9306 . 9332 . 9356 . 9381 . 9404 . 9426 .9448 . 9470

0.50 . 8624 . 8661 . 8698 . 8733 . 8768 . 8802 . 8835 .8868 .8900 .8931

0.60 .7651 .7696 .7739 .7782 .7825 .7867 .7908 .7949 .7989 .8029 0.70 .6270 .6316 .6361 .6405 .6450 .6494 .6537 .6581 .6624 .6666

0.80 .4475 .4512 .4550 .4588 .4625 .4662 .4699 .4736 .4773 .4810

0 90 .2335 .2357 .2378 .2400 .2421 . 2443 . 2464 . 2486 2507 2529

lioo .0000 .0000 .0000 .0000 .0000 .0000 ]oooo !oooo !oooo !oooo Average 0. 7313 0. 7339 0.7364 0. 7388 0. 7412 0. 7436 0. 7459 0. 7481 0.7504 0. 7525

— —

2. 3000 2.3200 2.3400 2. 3600 2. 3800 2. 4000 2. 4200 2. 4400 2. 4600 2. 4800

9. 0. 99* / 0. 99/9 0. 9981 u. yybo 0. 9985 0. 9986 0. 9988 0. 9989 0. 9990 0. 9991

9. 10 . 9962 . 9965 . 9968 . 9971 . 9973 . 9976 . 9978 . 9980 . 9981 . 9983

9. 20 . 9906 . 9912 . 9918 . 9924 .9929 . 9933 . 9938 . 9942 . 9946 . 9950

0. 30 . 9772 . 9783 . 9795 . 9805 .9815 . 9825 . 9834 . 9843 . 9851 .9859

9. 40 . 9490 . 9510 . 9529 . 9548 . 9566 . 9583 . 9600 . 9616 . 9631 . 9647

0. 50 . 8961 . 8991 . 9020 . 9048 . 9076 . 9103 . 9130 . 9155 . 9180 . 9205

0.60 .8068 .8106 .8144 .8181 .8218 .8254 .8290 .8325 .8360 .8394 0.70 .6708 .6750 .6792 .6833 .6874 .6914 .6954 .6994 .7034 .7073 0.80 .4847 .4883 .4919 .4956 .4992 .5027 .5063 .5099 .5134 .5170 0.90 . 2550 . 2572 .2593 .2614 .2636 . 2657 . 2678 . 2700 . 2721 . 2742 LOO .0000 .0000 .0000 .OOCO .0000 !oooo !oooo !oooo !oooo [oooo Average i^i —

0. 7547 0. 7568 0.7589 0. 7609 0.7629 0. 7649 0. 7669 0. 7688 0. 7707 0. 7725

X/fl ? 2. 5000 2. 5200 2. 5400 2. 5600 2. 5800 2. 6000 2. 6200 2.6400 2. 6600 2.6800

.

S. U. 9992 u. yyyo u. yyyo U. 9994 0. 9995 0. 9995 0. 9996 0. 999b 0. 9997 0. 9997

0. 10 . 9984 . 9986 . 9987 . 9988 . 9989 . 9990 . 9991 . 9992 . 9993 . 9993

0. 20 . 9953 . 9956 . 9959 . 9962 . 9965 . 9967 . 9970 . 9972 . 9974 . 9976

0. 30 . 9867 . 9874 . 9881 . 9887 . 9894 . 9899 . 9905 . 9910 . 9915 . 9920

0.40 . 9661 . 9675 . 9689 . 9702 . 9714 . 9726 . 9738 . 9749 . 9760 . 9770

O.SO . 9229 . 9252 . 9275 . 9297 . 9319 . 9340 . 9361 . 9381 . 9400 . 9419 0.60 .8427 .8460 .8492 .8524 .8556 .8586 .8617 .8647 .8676 .8705 0.70 .7112 . 7150 .7188 .7226 .7263 .7300 .7337 .7373 .7409 .7445 0.80 .5205 .5240 .5275 . .5310 .5344 .5379 .5413 .5448 .5482 .5516 0.90 .2763 . 2784 . 2806 .2827 . 2848 .2869 .2890 .2911 .2932 .2953 LOO .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 Average 0. 7743 0. 7761 0.7779 0. 7796 0. 7813 0.7830 0.7847 0. 7863 0. 7879 0. 7895 ^-

l/fl Q—2. 7000 2. 7200 2. 7400 2.7600 2. 7800 2. 8000 2. 8200 2. 8400 2. 8600 2.8800

0. 0. 9997 0. 9998 0. 9998 0. 9998 0. 9998 n Anno

0. 9998 n nnnn0. 9999 u. yyyy u. yyyy 0. yyyy

0. 10 . 9994 . 9994 . 9995 . 9995 , 9996 . 9996 . 9997 . 9997 , 9997 . 9997

0. 20 . 9977 . 9979 . 9981 . 9982 . 9983 . 9985 . 9986 . 9987 . 9988 . 9989 0.30 . 9925 . 9929 . 9933 . 9937 , 9941 . 9944 . 9948 . 9951 . 9954 . 9956

0. 40 . 9780 . 9790 . 9799 . 9808 . 9817 . 9825 . 9833 . 9840 . 9848 . 9855

0. 50 . 9438 . 9456 . 9473 . 9490 . 9507 . 9523 . 9539 . 9554 . 9569 . 9583

0.60 .8733 .8761 .8789 .8815 .8842 .8868 .8893 .8918 .8943 .8967

0.70 .7480 .7515 . 7550 .7584 .7618 .7651 .7685 .7718 .7750 .7782 0.80 .5549 .5583 .5617 .5650 .5683 .5716 .5749 .5782 .5814 .5847

0 90 2974 . 2995 . 3016 . 3037 3058 . 3079 . 3100 .3120 . 3141 . 3162 LOO '.0000 !oooo '.0000 '.0000 !oooo loooo .0000 .0000 .0000 .0000 Average 0. 7910 0. 7926 0. 7941 0. 7956 0. 7971 0. 7985 0. 7999 0. 8013 0. 8027 0.8041

2. 9000 2. 9200 2.9400 2. 9600 2. 9800 3. 0000 3. 0200 3. 0400 3. 0600 3.0800

0. 0. 9999 0. 9999 0. 9999 0. 9999 0. 9999 1. 0000 1. 0000 1. 0000 1. 0000 nnnn

1. 0000

0. 10 . 9998 . 9998 . 9998 . 9998 . 9998 0. 9999 0. yyyy u. yyyy u. yyyy u. yyyy

0. 20 . 9990 . 9990 . 9991 . 9992 . 9993 . 9993 . 9994 . 9994 . 9995 . 9995 0.30 .9959 . 9962 . 9964 . 9966 . 9968 . 9970 . 9972 . 9974 , 9975 . 9977

0. 40 . 9861 . 9868 . 9874 . 9880 . 9885 . 9891 . 9896 . 9901 . 9906 . 9910

0. 50 .9597 . 9611 .9624 . 9637 . 9649 . 9661 . 9673 . 9684 . 9695 . 9706

0.60 .8991 .9014 .9037 .9060 .9082 .9103 .9124 .9145 .9165 .9185 0.70 .7814 .7846 . 7877 .7908 .7939 .7969 .7999 .8029 .8058 .8087 0.80 .5879 .5911 .5943 .5975 .6007 .6039 .6070 .6101 .6132 .6163

0. 90 . 3183 . 3204 . 3224 . 3245 . 3266 . 3286 . 3307 3327 3348 3369

1.00 .0000 .0000 .0000 .0000 .0000 .0000 .0000 !oooo !oooo !oooo

Average 0. 8055 0.8068 0. 8081 0.8094 0. 8107 0. 8119 0. 8132 0.8144 0. 8156 0.8168

11
Table 5. Fractional temperature excess, 4>i, and generalized temperature averages, \p'i, for a slab—Continued

x}a ^-^^^^ 3. 1000 3. 2000 3.3000 3. 4000 3. 5000 3. 6000 3. 7000 3. 8000 3. 9000 4.0000

0. 1. 0000 1. 0000 1. 0000 1 . 0000 1. 0000 1. 0000 1. 0000 1 0000 1 0000 1 Aflflfl 0. 10 0. 9999 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1 0000 1 0000 1 0000 i. UuUU 1 0. 20 . 9995 0. 9997 0 QQQ8 n QQQQ 0 QQQQ 1 0000 1 0000 1 0000 1 0000 1. uuuu

0. 30 . 9979 . 9985 . 9989 . 9992 . 9995 n QQQ^ flu. QQOQ yyyy

0. 40 . 9915 9934 . 9949 . 9961 . 9970 . 9977 9983 9987 9991 9993

0 50 . 9716 . 9763 . 9804 . 9838 . 9867 . 9891 9911 9928 9942 . yyoo ' o! 60 [9205 !9297 !9381 !9456 9523 !9583 !9637 19684 !9726 .9763 0.70 .8116 .8254 .8385 .8508 .8624 .8733 .8835 .8931 .9020 .9103 0.80 .6194 .6346 .6494 .6638 .6778 .6914 .7047 .7175 .7300 .7421

0. 90 .3389 .3491 .3593 .3694 .3794 .3893 .3992 .4090 .4187 .4284 1.00 .0000 .0000 .0000 .0000 ,0000 .0000 .0000 .0000 .0000 .0000 Average i^i 0. 8180 0. 8237 0. 8290 0. 8341 0.8388 0. 8433 0. 8475 0. 8515 0. 8553 0. 8590

4. 1000 4. 2000 4. 3000 4. 4000 4. 5000 4. 6000 4. 7000 4. 8000 4. 9000 5.0000 :

nu. 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1 0000 1 0000 1 0000 0 10 1. 0000 ] 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1 0000 1. 0000 1 0000 0 20 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1 0000 1 0000

u. ou 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1 0000 1 0000

0. 40 0. 9995 0. 9996 0. 9997 0. 9998 0. 9999 0. 9999 0. 9999 1. 0000 1. 0000 1 0000

0 50 . 9963 . 9970 . 9976 . 9981 . 9985 . 9989 . 9991 n QQQ^ n QQQ^i u. yyyo oieo !9796 !9825 ;9850 !9872 ^9891 !9907 !9922 .9934 .9944 .9953 . 0.70 .9181 .9252 .9319 .9381 .9438 .9490 . 9539 .9583 .9624 .9661 0.80 .7538 .7651 .7761 .7867 ,7969 .8068 .8163 .8254 .8342 .8427 0.90 .4380 .4475 .4569 ,4662 .4755 .4847 . 4937 .5028 .5117 .5205

1. 00 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 L Average 4/i 0. 8624 0. 8657 0. 8688 0. 8718 0. 8746 0. 8774 0. 8800 0. 8825 0. 8849 0. 8872

5. 1000 5. 2000 5. 3000 5. 4000 5. 5000 5. 6000 5. 7000 5. 8000 5. 9000 6.0000 j

0. 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1 . 0000 1. 0000 1 0000

w.iu 1. 0000 1. 0000 1. 0000 1. 0000 1, 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000

0. 20 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1 0000 I. UUUU ' 0.30 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1i. nnnn UUUVI i

1 0.40 1. 0000 1. 0000 1. 0000 1,0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1 . UUUU

0.50 0. 9997 0 QQQ8 0 QQQQ 0 QQQQ 0 Qqqq 0 QQQQ 1. 0000 1 0000 11. nnnn UUUU 1 0.60 !9961 .9967 .9973 .9977 .9981 .9985 .9987 o! 9990 o! 9992 0.9993 1 0.70 .9695 .9726 .9755 .9780 .9804 .9825 .9844 .9861 .9877 .9891 ! 0.80 .8508 .8587 .8661 .8733 .8802 .8868 .8931 .8991 .9048 .9103

0.90 .5292 .5379 .5465 .5549 .5633 .5716 .5978 .6879 .5959 .6039

1.00 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 Average 0. 8894 0. 8915 0. 8935 0. 8955 0. 8974 0. 8993 0. 9010 0. 9027 0. 9044 0.9060

6. 1000 6. 2000 6. 3000 6. 4000 6. 5000 6. 6000 6. 7000 6.8000 6. 9000 7. 0000

0. 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1 0000

0.10 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000

0.20 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 0 30 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 0 40 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1 0000

u.ou 1 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1 0000

0.60 0. 9994 o'. 9995 O'. 9996 o! 9997 O'. 9998 O'. 9998 o'. 9998 o' 9999 o'. 9999 0. 9999

0.70 .9903 .9915 .9925 .9934 .9942 .9949 .9955 .9961 .9966 .9970

0.80 .9155 .9205 .9252 .9297 .9340 .9381 .9419 .9456 .9490 .9523

0.90 .6117 .6194 .6270 .6346 .6420 .6494 .6566 .6638 .6708 .6778

1.00 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 Average 4>\ 0. 9075 0. 9090 0. 9104 0. 9118 0. 9132 0. 9145 0. 9158 0. 9170 0. 9182 —0. 9194

xja 7. 1000 7. 2000 7. 3000 7. 4000 7. 5000 7. 6000 7. 7000 7. 8000 7. 9000 8. 0000

Q 1 0000 1 0000 1 0000 1 0000 1 0000 1. 0000 1. 0000 1 0000 1 0000 1 0000

0 10 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1 0000

0 20 1 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1 0000

0 30 1 0000 1 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1 0000

0 40 1 0000 1 0000 1 0000 1 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1 0000

0.50 1 0000 1 0000 1 0000 1 0000 1. 0000 1. 0000 1. 0000 1 0000 1 0000 1 0000

0.60 o! 9999 L 0000 I'oooo l' 0000 l' 0000 1. 0000 l' 0000 i! 0000 i! 0000 i! 0000

0.70 .9974 0. 9977 0. 9980 0. 9983 0. 9985 0. 9987 0. 9989 0. 9991 0. 9992 0. 9993

0.80 .9554 .9583 .9611 .9637 .9661 .9684 . 9706 .9726 .9745 .9763

0.90 .6847 .6914 .6981 .7047 .7112 .7175 .7238 .7300 .7361 .7421

1.00 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 Average i^l

j

0. 9205 0, 9216 0. 9227 0. 9238 0. 9248 0. 9258 0. 9267 0. 9277 0. 9286 0. 9295

12
Table 5. Fractional temperature excess, <^i, and generalized temperature averages, xj/'^, for a slab—Continued

8. 1000 8. 2000 8. 3000 8. 4000 o. uuuu o. ouuu o. /UUU o. 8000 8. 9000 9. OOOO

0. 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000

0.10 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1 0000 1. OOOO 0^0 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. OOOO

0.30 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. OOOO

0.40 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1.0000 1. 0000 1. 0000 1. 0000 1. OOOO

0.50 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. OOOO

0.60 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1 OOOO

o!70 0.9994 0. 9995 0. 9996 0. 9996 0. 9997 0. 9997 0. 9998 0. 9998 0. 9998

. 9882

O' 9999

0.80 .9891 .9780 .9796 .9811 .9825 .9838 .9850 .9861 .9872

0.90 . 7480 . 7538 . 7595 7651 7707 77A1 . 7814 . 7867 . 7918 . 7969

1.00 !oooo !oooo ." 0000 .' 0000 .' 0000 .0000 .0000 .0000 .0000 .OOOO Average i^-'i 0. 9303 0. 9312 0. 9320 0. 9328 0. 9336 0. 9344 0. 9351 0. 9359 0. 9366 0. 9373

j/a ^-^^ 9 1000 9. 2000 V. ouuu 9. 4000 y. ouuu y. Duuu y. /OUU a. oOUO 9. 9000 10. OOOO

0. 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. OOOO

0.10 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. OOOO

0.20 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. OOOO

0.30 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. OOOO

0.40 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. OOOO

0.50 1.0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. OOOO

0.60 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. OOOO

0.70 0. 9999 0.9999 0. 9999 0. 9999 0. 9999 1. 0000 1. 0000 1. 0000 1. 0000 L OOOO

0.80 .9899 .9907 .9915 .9922 .9928 0. 9934 0. 9939 0. 9944 0. 9949 0. 9953

0.90 8068 8116 fil fi^l

. 8209 . 8254 . 8299 . 8342 . 8385 . 8427

1.00 .0000 ^0000 .'oooo .0000 .0000 .0000 .0000 .0000 .0000 .0000 Average i^I —

0.9380 0. 9387 0. 9393 0. 9400 0. 9406 0. 9412 0. 9418 0. 9424 0. 9430 0. 9436-

Z/fl in inon in 9nnn in "^nnn inxu. 4nnn ^uuu 1 n t^nnn 1 n f\nnn iU. DUUU lU. /uUU in onnn 10. yUOO 11. OOOO'

0. 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 LOOOO

0.10 1.0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. OOOO

0.20 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. OOOO. 0.30 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1.0000 1. 0000 1. 0000 1. 0000 1. OOOO

0.40 1.0000 1. 0000 1.0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 ; 1. 0000^ 0.50 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. OOOO

0.60 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1.0000 1. 0000 1. 0000 1. OOOO' 0.70 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 I'oooo 1. OOOO. 0.80 0. 9957 0. 9961 0.9964 '

0. 9967 0. 9970 0. 9973 0. 9975 0. 9977 0. 9979 0. 9981

0.90 . 8468 . 8508 . 8548 . 8586 . 8624 . 8661 . 8698 . 8733 . 8768 . 8802

1 00 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 .0000 . OOOO. Average t^'i 0. 9441 0. 9447 0. 9452 0. 9457 0. 9463 0. 9468 0. 9473 0. 9478 0. 9482 0. 9487

Table 6. Fractional temperature excess, 4>-i, for a cylinde r

0. 4000 0. 4200 0.4400 0. 4600 0. 4800 0. 5000 0. 5200 0. 5400 0. 5600 0. 5800'

0. 0. 0002 0. 0004 0. 0009 0. 0017 0. 0030 0. 0049 0. 0076 0.0113 0. 0159 0. 0218. 0.10 .0002 .0004 .0009 .0017 .0030 .0049 .0075 .0111 .0157 . 0215

0.20 .0002 .0004 .0009 .0016 .0028 .0047 .0072 .0106 .0150 . 0205

0.30 .0002 .0004 .0008 .0015 .0026 .0043 .0067 .0098 .0139 .0190

0.40 .0001 .0003 .0007 .0013 .0024 .0039 .0060 .0088 .0125 .0170

0.50 .0001 .0003 .0006 .0012 .0020 .0033 .0051 .0075 .0107 .0146

0.60 .0001 .0002 .0005 .0009 .0016 .0027 .0041 .0061 .0087 . oiia

3 70 nnni . UUUA . uuuz .0004 .0007 .0012 .0020 .0031 . 0046 . 0065 . 0089

o!80 .0001 .0001 .0002 .0005 .0008 .0013 .0020 .0030 .0043 . 0058

0.90 .0000 .0001 .0001 .0002 .0004 .0006 .0010 .0015 .0021 . 0028.

1.00 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

0. 6000 0. 6200 0. 6400 0. 6600 0. 6800 0. 7000 0. 7200 0. 7400 0. 7600 0. 7800'

0. 0. 0289 0. 0373 0. 0470 0. 0580 0. 0703 0. 0838 0. 0985 0. 1143 0. 1311 0. 1488

0.10 .0285 .0367 .0463 .0571 .0693 .0826 .0971 . 1126 . 1292 . 1467

0.20 .0272 .0351 .0443 .0547 .0663 .0790 .0929 . 1078 . 1236 . 1403; 0.30 .0252 .0326 .0410 .0507 .0614 .0732 .0861 .0999 1146 .1301

0.40 .0226 .0291 .0367 .0453 .0549 .0655 .0770 .0893 . 1025 . 1163; 0.50 .0193 .0250 .0315 .0388 .0471 .0561 .0660 .0766 .0878 .0997

0.60 .0157 .0202 . 0255 0315 .0382 .0456 .0535 .0621 .0712 . 0809'

ft Tft

. 0118 . 0152 .0191 .0236 .0286 .0342 .0401 . 0466 . 0534 . 0606

0.80 .0077 .0100 .0126 .0155 .0188 .0225 .0264 .0306 .0351 . 0399- 0.90 .0038 .0049 .0061 . 0076 .0092 .0109 .0128 .0149 .0171 . 0194, 1.00 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

13.
Table 6. Fractional temperature excess, <j>2, for a cylinder—Continued

0. 8000 0. 8200 0. 8400 0. 8600 0. 8800 0. 9000 0. 9200 0. 9400 0. 9600 0. 9800

0. 0. 1673 0. 1866 0. 2064 0. 2268 0. 2476 0. 2687 0. 2901 0.3117 0. 3334 0. 3551

0.10 . 1649 . 1839 .2304 .2235 .2440 .2649 .2860 .3072 .3286 .3500

O.20 . 1578 .1759 . 1947 .2139 .2335 .2534 .2736 .2940 .3145 . 3350

0.30 .1463 .1631 .1804 .1983 .2164 .2349 .2537 .2725 .2915 .3106

O.40 . 1308 .1458 .1614 .1773 .1936 .2101 .2269 .2438 .2608 .2779

0.50 . 1121 . 1250 .1383 .1520 . 1659 . 1801 . 1945 .2090 .2236 .2382 0.60 .0909 .1014 .1122 .1233 .1346 .1461 .1578 .1696 . 1814 .1933

0.70 .0682 .0760 .0841 .0924 . 1010 . 1096 . 1184 .1272 . 1361 .1451

0.80 .0448 .0500 .0553 .0608 .0664 .0721 .0778 .0836 .0895 .0954

0.90 .0218 .0243 .0269 .0295 .0323 .0350 .0378 .0407 .0435 .0464

1 nn Q 0. 0. 0. 0. 0. 0. 0. Q u.

1.0000 1. 0200 1. 0400 1. 0600 1. 0800 1. 1000 1. 1200 1. 1400 1. 1600 1. 1800

0. 0. 3768 0. 3984 0. 4199 0. 4412 0. 4622 0. 4830 0. 5035 0. 5236 0. 5433 0. 5627

0.10 .3714 .3927 .4139 .4349 . 4557 .4762 .4964 .5162 . 5357 . 5548

0.20 .3555 .3759 .3962 .4163 .4362 .4559 .4753 .4944 .5132 .5316 0.30 .3296 .3486 .3674 .3862 .4047 .4231 .4412 .4590 .4766 .4938

0.40 .2949 .3119 .3289 .3457 .3624 .3789 .3952 .4114 .4272 . 4429

O.60 .2529 .2675 .2821 .2966 .3110 .3252 . 3394 .3533 .3671 .3807 O.60 .2053 .2172 .2290 .2408 .2526 .2642 . 2758 .2872 .2986 .3098

0.70 . 1540 .1630 . 1719 . 1808 . 1897 . 1985 . 2072 .2159 .2245 .2330

O.80 . 1013 . 1072 .1131 .1190 . 1248 . 1306 .1364 .1421 .1478 . 1535 0.90 .0493 .0521 .0550 .0579 .0607 .0636 .0664 .0692 .0720 .0747

1 nn Q 0. 0. 0. 0. 0. 0. Q Q u.

1. 2000 1. 2200 1. 2400 1. 2600 1. 2800 1. 3000 1. 3200 1. 3400 1. 3600 1. 3800

0. 0. 5816 0. 6001 0. 6181 0. 6356 0. 6527 0. 6692 0. 6853 0. 7008 0. 7158 0. 7303

0.10 .5735 .5918 .6096 .6270 .6439 .6603 .6762 .6916 .7066 .7210 O.20 .5497 .5673 .5845 .6014 .6178 .6337 .6492 .6643 .6789 .6931 0.30 .5108 . 5274 .5436 .5595 .5751 .5902 .6050 .6194 .6335 .6471 O.40 .4583 .4734 .4882 .5028 .5171 .5311 .5448 .5582 .5713 .5841 O.50 .3941 .4073 .4204 .4332 .4458 . 4582 .4703 .4823 .4941 .5056 0.60 .3208 .3318 .3426 .3532 . 3638 .3741 .3844 .3945 .4044 .4142 O.70 .2414 .2497 .2580 .2662 .2743 .2823 .2902 .2980 .3058 .3135 O.80 . 1591 . 1647 . 1702 .17.56 .1811 . 1864 1918 1971 2023 .2075 O.90 .0775 .0802 .0829 .0856 .0883 .0909 .0935 .0962 .0988 .1013

1 fin l.UU Q 0. 0. 0. 0. 0. 0. 0. Q

1. 4000 1.4200 1. 4400 1. 4600 1. 4800 1. 5000 1. 5200 1. 5400 1. 5600 1. 5800

0. 0. 7443 0. 7578 0. 7708 0.7833 0. 7952 0. 8067 0. 8177 0.8282 0. 8382 0. 847

0.10 .7349 .7484 .7613 .7738 . 7857 .7972 .8082 .8188 .8289 .83868

0.20 .7068 .7201 .7329 .7453 .7572 .7687 .7798 . 7905 .8007 .8106

0.30 .6604 .6733 .6858 .6980 .7097 .7212 .7322 .7429 .7533 . 7633

O.40 .5966 .6088 .6207 .6323 .6437 . 6547 . 6655 .6760 . 6863 . Oflirj 0.50 .5169 .5280 .5389 .5496 .5601 .5704 .5805 .5905 .6002 . 60',)7 O.60 .4239 .4335 .4429 .4522 .4613 .4704 .4793 .4881 .4967 . 5053

0.70 .3210 .3286 .3360 .3434 .3507 .3579 . 3651 .3722 .3793 . 38112 0.80 .2127 .2178 .2229 .2280 .2330 .2380 .2430 .2480 .2529 . 2fu>i O.90 .1039 .1065 . 1090 . 1116 . 1141 .1166 . 1191 . 1216 . 1240 . 1265

1.00 Q Q Q Q Q 0. Q Q Q 0.

1. 6000 1. 6200 1.6400 1. 6600 1. 6800 1. 7000 1. 7200 1. 7400 1. 7600 1. 7800

0. 0. 8570 0. 8657 0. 8740 0. 8818 0. 8893 0. 8964 0. 9031 0. 9095 0. 9155 0. 9212

O.IO .8478 .8566 .8650 .8730 .8806 .8879 .8947 .9013 .9075 .9133

0.20 .8200 .8291 .8378 .8461 .8541 . 8617 .8690 .8760 .8827 . 8890

O.30 .7730 .7823 .7914 .8001 .8085 .8166 .8244 .8320 . 8393 .8463 0.40 .7060 .7154 .7246 .7336 . 7423 . 7.508 .7590 .7670 .7748 . 7824

0.50 .6191 .6282 .6372 .6460 .6547 .6631 .6714 .6796 .6876 .6954

0.60 .5137 .5220 .5302 .5383 .5463 . 5542 .5620 .5697 . 5772 .5847 O.70 .3931 .4000 .4068 .4135 .4202 .4268 .4334 .4399 .4464 . 4528

0.80 .2626 .2674 .2723 .2770 .2818 .2865 .2913 .2960 .3006 .3053 O.90 . 1290 . 1314 . 1339 .1363 . 1387 .1412 .1436 .1460 . 1484 .1508

1.00 0. Q Q Q Q Q Q u. u.

1. 8000 1.8200 1. 8400 1. 8600 1. 8800 1.9000 1. 9200 1.9400 1. 9600 1. 9800

0. 0. 9265 0. 9316 0. 9363 0. 9408 0. 9450 0. 9489 0. 9527 0. 9561 0. 9594 0. 9624

0.10 .9189 . 9241 .9291 .9337 .9381 .9423 .9462 .9499 . 9534 .9566 0.20 . 8951 .9008 .9063 .9116 .9166 .9213 .9258 .9301 .9341 .9380 0.30 .8530 .8595 .8657 .8717 .8775 .8830 .8884 .8935 .8984 .9031

0.40 .7898 .7969 .8039 .8106 .8172 .8236 .8297 .8357 .8416 .8472 O.50 .7030 .7105 . 7179 .7251 .7322 .7391 .7458 .7525 .7590 . 7653

0.60 . 5921 . 5994 . 6066 . 6136 . 6206 . 6275 . 6343 . 6410 . 6476 . 6542

O.70 .4592 .4655 .4717 .4779 .4841 .4902 .4963 .5023 .5082 5141 O.80 .3099 .3145 .3191 .3237 .3282 .3328 .3373 .3418 .3463 .3507

•0.90 .1532 . 1556 . 1580 . 1604 . 1628 . 1651 1675 1699 . 1722 . 1746

1.00 0. 0. 0. 0. 0 0. 0. 0. 0. 0.

I

14
Table 6. Fractional temperature excess, <j>2, for a cylinder—Continued

If

2. 0000 2. 0200 2. 0400 2. 0600 2. 0800 2. 1000 2. 1200 2. 1400 2. 1600 2. 1800

Q 0, 9679 0. 9704 0. 9727 0. 9748 0 9768 0 9787 n QQfi/i U. you^ U. yoJU 0. 9So5

0 10 9597 . 9625 . 9652 . 9677 . 9701 9723 . y/'±o . 9762 . 9780 . 9797 0 20 . 9416 . 9451 . 9484 . 9515 9544 9572 Ql^QQ

. yoyo . yO/o . 9646 . 9669

v.OU . 9077 . 9120 . 9162 . 9202 9240 9276 Q^l 1

. 9345 . 9377 . 9408

0 40 . 8527 . 8580 8632 . 8682 . 8730 8777 8822 . 8866 . 8909 . 8950

o!50 !7716 . 7776 17836 !7894 !7951 ^8007 !8062 .8115 .8167 .8218

0.60 .6606 .6669 .6732 .6794 .6854 .6914 .6973 .7032 .7089 . 7146

0.70 .5200 .5258 .5316 .5373 .5430 -5486 .5542 .5597 .5652 .5707

0.80 .3552 .3596 .3640 .3684 .3728 .3771 .3815 .3858 .3901 .3944

0.90 .1769 .1793 .1816 .1840 .1863 .1887 . 1910 .1933 .1956 .1980

1.00 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

\' 2. 2000 2. 2200 2. 2400 2. 2600 2. 2800 2. 3000 2. 3200 2. 3400 2. 3600 2. 3800

n

U. 0 9849 0. 9861 0. 9873 0. 9884 0 9894 u. yyuo 0 9912 fl QQ9A n QQ97

0 10 981^ . 9827 . 9840 9853 9864 9875 9885 9894 . 9903 .9911

0 20 9690 9709 . 9728 9745 9762 9792 . youo . 9818 . 9830

u.ou 9437 9466 9492 9518 9543 9566 9588 0A1

A

9630 . 9650

yj.w 8990 9066 9102 9137 9171 . yzoo . 9266 . 9295

0.50 '. 8268 !8317 !8365 !s4il .' 8457 !8501 .8544 .8587 .8628 .8669

0.60 . 7202 .7256 .7311 .7364 .7416 .7468 .7519 .7569 .7619 .7667

0.70 .5760 .5814 .5867 .5920 .5972 . 6023 .6075 .6125 .6176 .6226

0.80 .3986 .4029 .4071 .4113 .4155 .4197 .4239 .4280 . 4321 .4362

0.90 .2003 .2026 .2049 .2072 .2095 .2118 .2141 .216-4 .2187 .2210

1.00 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

2 4200 2 4400 2 4600 2 4800 2 5200

_ - J. odUu

-

o conA

2. ooOO

u. n QQ^Q 0 9945 u. yyou u. yyoo 0 99.59 u. yyoij n QQfifi u. yyoD 0 9970 0 9972 u. yy/o

n iA u.iu 9918 9925 9931 9937 9942 9947 9952 9956 9960 9963

n on 9842 9853 9863 9872 9881 9889 9897 9904 9911 QQ17 . yyi/ 9668 9686 9703 9718 9734 9748 9762 9775 9787 Q7QQ . y/yy

0 40 9324 9351 . 9377 . 9403 . 9428 9451 . 9474 . 9496 9517 9538

oiso !8708 !8746 !8784 !8820 !8856 !8891 !8925 !8958 !8990 [9021

0.60 .7715 .7762 .7809 .7854 .7899 .7943 .7987 .8029 .8072 .8113

0.70 .6275 .6324 .6373 .6421 .6468 .6515 .6562 .6608 .6654 .6700

0.80 .4403 . llli .4485 .4525 .4565 .4605 .4645 .4685 .4724 .4764

0,90 .2233 !2256 .2279 .2301 .2324 .2347 .2370 .2392 .2415 .2437

1.00 0. 0. 0. 0. 0. 0. 0. 0 0. 0.

xfa^ 2. 6000 2. 6200 2.6400 2. 6600 2.6800 2. 7000 2. 7200 2. 7400 2. 7600 2. 7800

fi\f. 0 9978 n QQSiA u. yyou 0 9982 u. yyor« u. yyou 0 9987 u. yyoo u. yyoy 0 9991 nu. QQQ9 yyy^

n in 9967 9970 9972 9975 9977 . 9979 . 9981 9983 9984 9986

9923 9928 9934 9938 9943 . 9947 9951 9955 9958 9961

0.30 . 9810 . 9821 • . yooi 9841 9850 . 9859 . 9867 . 9875 9882 9889

0.40 . yoDo . 9577 . 9595 9613 9629 9646 . 9661 . 9676 9691 9705

0.50 .9052 .9082 .9111 !9139 !9166 !9193 !9219 !9244 !9269 '. 9293

0.60 .8154 .8193 .8233 .8271 .8309 .8346 .8383 .8419 .8454 .8489

0.70 .6745 .6789 .6833 .6877 .6920 .6963 .7006 .7048 .7089 .7131

0.80 .4803 .4842 .4881 .4919 .4958 .4996 .5034 .5072 .5110 .5148

0.90 .2460 .2483 .2505 .2528 .2550 .2572 .2595 .2617 .2639 .2662

1.00 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

g 2.8000 2.8200 2.8400 2.8600 2.8800 2. 9000 2. 9200 2.9400 2. 9600 2.9800

J. u. yyyo n QQQQ u. yyy^ n QQQ^ u. yyyo u. yyyo u. yyyu u. yyyu 0. 9997 0. 9997 0 9997 A in QOfifi . yyoo 9990 9991 9991 . 9992 . 9993 . 9994 . 9994 . 9995 X20 9964 OOP7 . yyo* 9969 9972 9974 . 9976 . 9978 . 9979 . 9981 . 9983

}.30 . 9895 . 9902 QOflU . yyuo 9913 9918 . 9923 . 9928 . 9932 . 9937 . 9941

}.40 . 9718 . y/ox . y/ 9755 9766 9777 . 9787 . 9797 . 9807 . 9816

0.50 .9316 .9338 .9360 !9382 !9402 !9423 !9442 !9461 !9480 !9497

0.60 .8523 .8557 .8589 .8622 .8653 .8684 .8715 .8745 .8774 .8803

0.70 .7171 .7212 .7251 .7291 .7330 .7369 .7407 .7445 .7482 .7519

0.80 .5185 .5223 .5260 .5297 .5333 .5370 .6406 .5442 .5478 .5514

0.90 .2684 .2706 .2728 .2751 .2773 .2795 .2817 .2839 .2861 .2883

1.00 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

g 3.0000 3. 1000 3.2000 3. 3000 3.4000 3. 5000 3. 6000 3. 7000 3. 8000 3.9000

J. u. yyyo u. yyyy u. yyyy n QQQQ u. yyyy 1 0000 1 0000 1. 0000 1. 0000 1. 0000 1. 0000 XIO . 9995 . 9997 . 9998 . 9999 . yyyy 1 0000 1 0000 1 0000 1 0000 1 0000

1.20 . 9984 . 9990 . 9993 . 9996 9997 . 9998 0V- QQQQ yyyy 0. 9999 1. 0000 1. 0000

0,30 . 9944 . 9960 . 9971 . 9980 9986 9990 . 9993 . 9995 0. 9997 u. yyyo

).40 . 9825 . 9863 . 9894 . 9918 9937 9952 . 9964 . 9973 . 9980 . 9985

J.50 .9515 .9594 .9662 .9720 [9769 !9810 !9844 !9873 !9897 !9917

0.60 .8831 .8965 .9085 .9194 .9292 .9380 .9458 . 9528. .9590 .9645

0.70 . 7556 .7733 .7901 .8058 .8207 ,8347 .8478 .8601 .8716 .8823

0.80 .5550 .5726 .5897 .6064 .6226 .6384 .6538 .6687 .6832 .6972

0.90 .2905 .3015 .3123 .3231 .3338 .3445 .3550 .3655 .3759 .3862

1.00 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

15
Table 6. Fractional temperature excese, <t>2, for a cylinder—Continued

4. OCCO 4. 1000 4. 2000 4.3000 4. 4000 4. 5000 4. 6000 4. 7000 4.8000 4.9000'

0. I. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1 0000' 0.10 1. 0000 1. 0000 1.0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1 0000

0.20 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1 coon

o'so 0. 9999 0. 9999 0. 9999 1.0000 1. 0000 1. COOO 1. 0000 1. ooco 1. 0000 1 0000

0.40 . 9989 . 9992 .9994 0. 9996 0. 9997 0. 9998 0. 9999 0. 9999 0. 9999 1 0000

u'.50 .9933 .9947 .9958 .9967 .9973 .9979 .9984 .9987 !9990 0. 9992

0.60 .9693 .9736 .9773 .9806 .9834 .9859 .9880 .9899 .9914 . 992S

0.70 .8923 .9016 .9103 .9183 .9257 .9325 .9388 .9446 . 9500 . 9549

0.80 .7108 .7240 . 7367 . 7490 . 7609 .7724 .7834 .7941 .8044 .8143

0.90 .3964 .4066 .4166 .4266 .4365 .4463 .4560 .4656 . 4751 .4846

1.00 0 0. 0. 0. 0. 0. 0. 0. 0. 0.

\? 5. 0000 5. 1000 5. 2000 5. 3000 5. 4000 5. 5000 5. 6000 5. 7000 5. 8000 5 9000

0. 1. 0000 1.0000 1. 0000 1, 0000 1.0000 1 . 0000 1 . 0000 1. 0000 1. 0000 1 0000

0.10 1. 0000 1 . 0000 1. 0000 1. COOO 1.0000 1. 0000 1 . 0000 1. 0000 1. 0000 1 0000

o!20 1. 0000 1. 0000 1.0000 1. 0000 1.0000 1.0000 1.0000 1. 0000 1. 0000 1. 0000

0.30 1. 0000 1. ocoo 1.0000 1.0000 l.COOO 1. 0000 1. 0000 1. 0000 1. 0000 1 01)00 0*40 1 0000 1 0000 1.0000 l.OOOU 1.0000 1. 0000 1. 0000 1. 0000 1 . 0000 1. 000(1 o!^ . 9994 0. 9996 0. 9997 0. 9997 0. 9998 0. 9998 0. 9999 0. 9999 1.0000 i.ncdo Q.60- .9939 .9949 .9958 .9965 .9971 .9976 .9980 .9984 0. 9986 0. 998<i Q;70

0.80

. 9594 . 9635 .9672 .9706 .9737 .9765 .9790 .9813 .9834 . 9853 .8238 .8329 . 8416 .8.501 .8581 .8658 .8732 .8803 .8870 .8934

0.90 .4939 .5032 .5123 .5214 .5303 .5392 .5480 .5666 .6662 . 5737

1.00 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. j

y 6. 0000 6. 1000 6. 2000 6. 3000 6. 4000 6. 5000 6. 6000 6. 7000 6. 8000 0. yuuu

0. 1 . 0000 1. 0000 I. 0000 1.0000 1. 0000 1.0000 1. 0000 1. 0000 1. 0000 1. 0000

0.10 1 . 0000 1.0000 1.0000 1.0000 1. 0000 1.0000 1.0000 1.0000 1 . 0000 1. 000(1 0.20 1 . 0000 1. 0000 1. 0000 1.0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0(1(10 0.30 1. 0000 1. 0000 1. 0000 1.0000 1. 0000 1. 0000 1. 0000 1.0000 1. 0000 1. 000(1 0 40 1. 0000 1. 0000 1. 0000 1. 0000 1.0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000

0.50 1. 0000 1.0000 1.0000 1. 0000 1. 0000 1.0000 1. 0000 1. 0000 1! 0000 1.0000

0.60 0. 9991 0. 9993 0. 9994 0. 9995 0. 9996 0. 9997 0. 9998 0. 9998 0. 9999 0. 99!)',i 0.70 .9870 .9884 .9898 .9910 .9921 .9930 .9939 .9946 .9953 . 99,7.1 0.80 .8996 .9054 .9110 .9163 .9213 .9261 .9307 .9350 .9390 .94.'!i 0.90 . 5820 . 5903 .5985 .6065 .6145 .6224 .6301 .6378 .6463 . 65L'S 1.00 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

7. 0000 7. 1000 7. 2000 7. 3000 7. 4000 7. 5000 7. 6000 7. 7000 7. 8000 7. 9000

0, 1. 0000 1. 0000 1. 0000 1.0000 1. 0000 1. 0000 1.0000 1. 0000 1. 0000 1. 0000

'OilO • 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 000(1 0)20 1. 0000 1. 0000 1. 0000 1.0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000

0^30 • 1. 0000 1.0000 1.0000 1. 0000 1. 0000 1. 0000 1 . 0000 1. 0000 1. 0000 1. 0000 0 40- 1 . 0000 1. 0000 1. 0000 1. 0000 1.0000 1.0000 1.0000 1.0000 1. 0000 1. 0000

oiso. LOOOO 1. 0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1. 0000 l.OOOd

'0.60 • 0. 9999 0. 9999 1. 0000 1.0000 1.0000 1.0000 1.0000 1.0000 1. 0000 1.0(100 0.70- .9964 .9969 0. 9973 0. 9977 0. 9980 0. 9982 0. 9985 0, 9987 0. 9989 0. 99',)o 0.80 .9466 .9600 .9533 .9664 .9593 .9621 .9646 .9671 .9693 .9715

0.90 .6601 .6674 .6745 .6816 .6885 .6953 . 7021 .7087 .7162 .7217

1.00 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

8. 0000 8. 1000 8. 2000 8. 3000 8. 4000 8. 5000 8. 6000 8. 7000 8. 8000 8. 9000

0. 1. 0000 1. 0000 1.0000 1. 0000 1. 0000 1. 0000 1. 0000 1 . 0000 1. 0000 1. 0000

0.10 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1 . 0000 1. 0000

0.20 1. 0000 1 . (11100 1. 0000 1. 0000 1. 0000 1. 0000 1 . 0000 1. 0000 1. 0000 1. 0000

0.30 1. 0000 1 0000 1. 0000 1.0000 1. 0000 1 . 0000 1.0000 1. 0000 1. 0000 1. 0000

0.40 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1 . 0000 1. 0000 1. 0000 1. 0000 1. 0000

oiso 1! 0000 1.0000 1.0000 1.0000 1.0000 1.0000 1. 0000 1.0000 i!oooo 1.0000

0.60 1.0000 1.0000 1.0000 1. 0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000

0.70 0. 9991 0. 9993 0. 9994 0. 9996 0. 9996 0. 9996 0. 9997 0. 9997 0. 9998 0. 999S

0.80 .9735 .9754 .9772 .9789 .9804 .9819 .9832 .9845 .9857 .9868

0.90 . 7280 .7342 .7404 .7464 . 7523 .7581 .7639 . 7695 .7750 . 7805

1.00 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

I/Ct \ 9. 0000 9. 1000 9. 2000 9. 3000 9. 4000 9. 5000 9. 6000 9. 70C0 9. 8000

1

9. 9000

Q 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000

0 10 1. 0000 1. 0000 1. 0000 1 . 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000

0.20 1. 0000 1! 0000 1. 0000 1 . 0000 1. 0000 1. 0000 1. 0000 1. 0000 1 . 0000 1 . 0000 0 30 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1 . 0000 1. 0000 1. 0000 1. 0000 1. 0000

0.40 1. 0000 1. 0000 1.0000 1. 0000 1.0000 1 . 0000 1.0000 1. 0000 1. 0000 1 . 0000

oiso lioooo 1! 0000 1. 0000 1.0000 1. 0000 1.0000 1.0000 1.0000 1. 0000 1.0000

0.60 1. 0000 1. 0000 1.0000 1.0000 1.0000 1. 0000 1.0000 1.0000 1.0000 1.0000

0.70 0. 9998 0. 9999 0. 9999 0. 9999 0. 9999 0. 9999 1.0000 1. 0000 1.0000 1.0000

0.80 .9878 .9888 .9897 .9905 .9912 .9920 0. 9926 0. 9932 0. 9938 0. 9943

0.90 .7858 . 7911 .7962 .8013 .8063 .8111 .8159 .8206 .8262 . 8297 ! 1.00 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

'

I 1

1

' 1

! 0

16
Table 6. Fractional temperature excess, (}>2, for a cylinder—Continued

Ui,/

\.9 10. 0000 10. 1000 10. 2000 10. 3000 10. 4000 10. 5000 10. 6000 10. 7000 10. 8000 10. 9000

1. 0000 1.0000 1. 0000 1. 0000 1. 0000 1.0000 1. 0000 1. 0000 1. 0000 1.0000

J. 10 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1.0000 1. 0000 1. 0000 1. 0000 1. 0000

0.20 1. 0000 1. 0000 1. 0000 1. 0000 1.0000 1. 0000 1. 0000 1.0000 1. 0000 1. 0000

0.30 1.0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1.0000 1.0000 1.0000

0.40 1.0000 1. 0000 1. 0000 1. 0000 1.0000 1. 0000 1. 0000 1.0000 1.0000 1. 0000

0.50 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1.0000 1. 0000 1.0000 1. 0000 1. 0000 0 60 1 0000 1 0000 1 0000 1. 0000 1 . 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000

o!70 LOOOO 1.0000 1! 0000 1.0000 1. 0000 1.0000 1. 0000 1. 0000 1.0000 1.0000

0.80 0.9948 0. 9952 0. 9956 0. 9960 0. 9964 0. 9967 0. 9970 0. 9972 0. 9975 0. 9977

0.90 .8341 .8385 .8427 .8469 .8510 .8549 .8588 .8627 .8664 .8701

1.00 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

Table 7. Generalized temperature averages, , for a cylinder

\? 0.4000 0.4200 0.4400 0. 4600 0. 4800 0. 5000 0. 5200 0. 5400 0. 5600 0. 5800

0. 0. 0001 0.0002 0.0005 0. 0009 0. 0015 0. 0025 0. 0038 0. 0056 0. 0080 0.0109

0. 10 .0001 . 0002 . 0005 0009 001*1 0024 0038 0066 .0079 .0108 0.20 .0001 ]0002 !0004 !0008 ]0015 !0024 !0037 '. 0065 .0077 .0106 0.30 .0001 .0002 .0004 .0008 .0014 U023 .0036 .0053 .0075 .0102 0.40 .0001 .0002 0004 .0008 .0013 .0022 0034 .0050 .0071 .0097 0.50 .0001 .0002 .0004 .0007 .0013 .0020 .0032 .0047 .0066 .0090

0. 60 0001 .0002 .0003 . 0007 .0011 . 0019 . 0029 .0043 . 0061 . UuaO 0^70 !oooi .0002 . 0003 .0006 . 0010 . 0017 . 0026 .0039 .0055 . 0075 0.80 .0001 .0001 . 0003 .0005 .0009 . 0015 . 0023 . 0034 .0048 .0066 0.90 .0000 .COOl . 0002 .0004 .0008 . 0CI3 .0020 .0029 . 0041 .0056

1.00 .0000 .0001 . 0002 .0004 . 0007 . 0011 .0016 . 0024 .0034 .0047

? 0. 6000 0. 6200 0. 6400 0. 6600 0.6800 0. 7000 0. 7200 0. 7400 0. 7600 0. 7800

0. 0.0144 0.0186 0. 0235 0. 0290 0. 0351 0.0419 0. 0492 0. 0571 0. 0655 0. 0744

0. 10 .0143 . 0185 . 0233 . 0288 . 0349 . 0416 . 0489 . 0567 .0651 .0739 0.20 .0140 .0181 .0228 .0282 .0341 .0407 .0478 .0555 .0637 .0723 0.30 .0135 .0174 .0220 .0271 .0329 .0392 .0461 .0536 .0614 . 0697 0.40 .0128 .0166 .0209 .0258 .0312 .0372 .0438 .0508 . 0582 .0661 0.50 .0120 .0155 .0195 .0241 .0292 .0348 .0409 .0474 0544 .0617

0. 60 0110 .0142 .0179 .0221 . 0267 .0319 .0375 .0435 0499 . 0566

0^70 !0099 .0128 . 0161 . 0199 .0241 . 0287 . 0337 . 0392 !0449 !0510 0.80 .0087 .0112 .0142 . 0175 . 0212 .0253 .0297 .0345 .0396 .0449 0.90 .0075 .0097 .0122 .0150 .0182 . 0217 .0255 .0296 .0340 .0386

1.00 .0062 .0080 . 0101 .0125 . 0152 . 0181 .0213 . 0247 .0283 .0321

0.8000 0.8200 0.8400 0. 8600 0.8800 0. 9000 0. 9200 0. 9400 0. 9600 0. 9800

0. 0. 0837 0. 0933 0. 1032 0. 1134 0. 1238 0. 1344 0. 1451 0. 1559 0. 1667 0. 1776

0. 10 .0831 . 0926 . 1025 . 1126 . 1229 . 1334 . 1440 1547 . 1655 . 1763 0.20 .0813 .0906 .1002 . 1101 .1203 . 1305 . 1409 !l514 . 1619 . 1725 0.30 .0783 .0873 .0966 .1062 .1159 . 1258 . 1359 .1460 . 1561 . 1663 0.40 .0743 .0829 .0917 .1008 .1100 .1194 . 1290 .1386 . 1482 . 1579 0.50 .0694 .0774 .0856 .0941 .1027 . 1115 . 1204 . 1294 . 1384 .1475 0 60 . 0637 .0710 .0786 .0863 .0943 . 1023 .1105 . 1187 1270 . 1353 o!70 .0573 .0639 .0707 .0777 .0848 .0921 .0995 . 1069 !ll43 !l218 0.80 .0505 .0563 .0623 .0684 .0747 .0811 .0876 .0941 .1007 .1073 0.90 .0434 .0483 .0535 .0588 .0642 .0697 .0752 .0808 .0865 .0921 1.00 .0361 .0403 .0446 .0490 .0535 .0580 . 0627 .0673 .0721 .0768

1.0000 1, 0^0 1. 0400 ncnn

1. ObOO i. UoUU 1 1000 1 1200 1. 1400 1. 1600 1. 1800

0. 0. 1884 0. 1992 0. 2100 0.2206 0. 2311 0. 2415 0. 2517 0. 2618 0. 2717 0. 2813

0. 10 . 1871 . 1978 . 2085 . 2190 . 2295 .2398 .2500 .2600 . 2698 . 2794 0.20 . 1831 .1936 .2040 .2144 .2246 .2347 .2447 .2545 . 2641 . 2735 0.30 .1765 .1866 .1967 .2067 .2166 .2264 .2360 .2455 . 2548 . 2640 0.40 .1675 .1772 .1868 . 1963 .2057 .2150 .2242 .2332 . 2422 . 2509 0.50 .1565 .1655 .1745 .1834 . )922 .2009 .2095 .2180 . 2264 .2346

U. uU . 1436 .1519 .1602 .1683 .1765 .1845 . 1924 .2003 . 2080 .2156 0.70 .1293 .1368 . 1442 .1516 . 1589 . 1662 .1733 .1804 '. 1874 '. 1943 0.80 .1139 . 1205 . 1270 .1335 .1400 .1464 . 1528 .1590 .1652 .1713 0.90 .0978 .1035 .1091 .1147 1203 . 1268 .1313 . 1367 .1420 . 1472 1.00 .0815 .0862 .0909 .0956 . 1002 .1048 .1094 . 1139 . 1183 . 1227

xju 1 nnnn 1. JOOO 1. 2200 1.2400 1.2600 1.2800 1. 3000 1. .3200 1. 3400 1. 3600 1. 3800

0. 0. 2908 0. 3000 0. 3090 0. 3178 0, 3263 0. 3346 0. 3426 0.3504 0. 3579 0. 3652

0.10 .2888 .2980 .3069 .3157 .3241 .3324 .3404 .3481 . 3556 . 3628

0.20 .2828 .2918 .3006 .3092 .3176 .3257 .3336 .3412 . 3487 . 3558

0.30 .2729 .2817 .2903 .2986 .3068 .3147 .3224 .3299 . 3372 , 3443

0.40 .2595 .2679 .2761 .2841 .2920 .2996 .3071 .3144 . 3214 . 3283

0.50 .2427 .2507 .2584 .2660 .2735 .2807 .2878 .2948 .3015 .3081

0.80 . 2231 .2305 .2377 .2447 .2517 . 2585 .2651 .2716 . 2780 . 2842

0.70 .2011 .2078 .2144 .2208 .2271 .2333 .2394 .2454 !2512 !2570

0.80 .1774 .1833 .1891 .1948 .2005 .2060 .2114 .2168 .2220 .2271

0.90 .1524 . 1576 .1626 .1675 . 1724 . 1772 .1819 . 1865 .1911 .1955

1.00 .1270 .1313 .1355 1397 .1437 .1477 . 1517 . 1555 .1593 .1631

17
Table 7. Generalized temperature averages, \p2, for a cylinder—Continued

xia \ 1.4000 1. 4200 1.4400 1. 4600 1. 4800 1. 5O00 1. 5200 1. 5400 1. 5600 1. 5800

0. 0. 3722 0. 3789 0.3854 0. 3916 0. 3976 0. 4033 0. 4088 0. 4141 0. 4191 0. 4239

0.10 .3698 .3765 .3830 .3893 .3952 .4010 .4065 .4117 .4168 .4216

0.20 .3628 .3695 .3759 .3821 .3881 .3939 .3994 .4047 .4098 .4146

0.30 .3511 .3577 .3641 .3703 .3762 .3820 .3875 .3928 .3980 .4029

0.40 .3349 .3414 .3477 .3538 .3596 . 3653 .3708 .3762 .3813 .3863

0.50 .3145 .3208 .3268 .3328 . 3385 .3441 .3495 .3547 .3598 .3648

0.60 .2902 .2961 .3019 .3076 .3130 .3184 . 3236 .3287 3336 .3384

0.70 .2626 .2680 .2734 .2786 .2838 .2888 .2937 .2985 .3032 .3077

0.80 .2322 .2371 .2420 .2467 .2513 .2559 .2604 .2647 . 2090 .2732 0 90 . 1999 2042 2085 . 2126 . 2167 . 2207 . 2246 2284 2322 2359

i!oo !l667 !l703 '. 1739 ] 1773 !l808 !l841 '. 1874 !l906 !l938 !l969

x/a \\3 1. 6O00 1. 6200 1. 6400 1. 6600 1.6800 1. 7000 1.7200 1.7400 1. 7600 1.7800

0. 0. 4285 0. 4328 0. 4370 0.4409 0. 4446 0. 4482 0. 4516 0. 4547 0. 4577 0.4606

0.10 .4262 .4306 .4347 .4387 .4425 .4461 .4495 .4527 .4557 . 4586 ' 0.20 .4193 .4237 .4280 .4320 .4359 .4396 .4431 .4464 .4496 .4526

0.30 .4076 .4122 .4165 .4207 .4247 .4285 . 4322 .4357 .4390 .4422

0.40 .3911 .3957 .4001 .4044 .4085 .4125 .4164 .4200 .4236 .4270

0.50 .3695 .3742 .3787 .3830 .3872 .3913 .3953 .3991 .4028 .4063

0.60 .3431 .3477 .3521 .3564 .3606 .3647 .3687 .3726 .3763 .3800

0.70 .3122 .3165 .3208 .3250 .3290 .3330 . 3369 .3407 .3444 .3480

0.80 .2773 .2814 .2853 .2892 .2930 .2967 .3003 .3039 .3074 .3108

1^ on 2395 2431 2466 2500 2534 2567 2599 2631 2663 2693

1.00 !2000 '.2030 !2059 ^2088 !2117 !2144 !2172 !2199 !2225 !2251

\« 1. 8000 1. 8200 1.8400 1. 8600 1.8800 1.9000 1. 9200 1. 9400 1. 9600 1. 9800

0. 0. 4633 0. 4658 0. 4682 0. 4704 0. 4725 0. 4745 0. 4763 0. 4781 0. 4797 0. 4812

0.10 .4613 .4639 .4663 .4686 .4708 .4728 .4747 .4765 .4782 .4798

0.20 .4555 .4582 .4608 .4632 .4655 .4677 .4697 .4717 .4735 .4752

0.30 .4453 .4482 .4510 .4536 .4561 .4585 .4608 .4630 .4651 .4670

0.40 .4302 .4334 .4364 .4393 .4421 .4447 .4473 .4497 .4520 .4543

0.50 .4098 .4131 .4163 .4195 .4225 .4254 .4282 .4309 .4335 .4360

0.60 .3835 .3870 .3904 .3936 .3968 .3999 .4029 .4C58 .4086 .4113

0.70 .3515 .3550 .3583 .3616 .3648 .3679 .3710 .3740 .3769 .3798

0.80 .3141 .3174 .3206 .3238 .3269 .3299 .3328 .3358 .3386 .3414 0 90 2723 2753 2782 •2811 2839 2866 2893 2920 2946 2972

I'oo !2277 !2302 !2327 !2351 !2375 !2398 !2421 !2444 '. 2466 !2488

x/a \\9 2. 0000 2. 0200 2.0400 2. 0600 2.0800 2. 1000 2. 1200 2. 1400 2. 1600 2.1800

0. 0. 4826 0. 4840 0. 4852 0. 4864 0. 4874 0. 4884 0. 4894 0. 4902 0. 4910 0.4918

0.10 .4812 .4826 .4839 .4851 .4862 .4873 .4883 .4892 .4900 .4908

0.20 .4769 .4784 .4798 .4812 .4825 .4837 .4848 .4858 .4868 .4877

0.30 .4689 .4707 .4723 .4739 .4754 .4769 .4782 .4795 .4807 .4819

0.40 .4564 .4585 .4605 .4623 .4641 .4659 .4675 .4691 .4706 .4720

0.50 .4385 .4408 .4431 .4453 .4474 .4494 .4514 .4533 .4551 .4568

0.60 .4140 .4166 .4191 .4216 .4239 .4263 .4285 .4307 .4328 .4348

0.70 .3825 .3853 .3879 .3905 .3931 .3955 .3979 .4003 .4026 .4049

0.80 .3441 .3468 .3495 .3521 .3546 .3571 .3595 .3619 .3643 .3666 n on OQQ7 . 3022 . oU^O . 3070 . 3094 . 3117 . OilU 3162 3184 3206

1.00 .2510 .2531 .2552 .2572 .2592 .2612 .2632 !2651 !2670 !2689

x/a ^\3 2. 2000 2. 2200 2. 2400 2. 2600 2. 2800 2. 3000 2. 3200 2.3400 2.3600 2. 3800

0. 0. 4924 0. 4931 C. 4937 0.4942 0. 4947 0. 4952 0. 4956 0.4960 0. 4903 0. 4967

0.10 .4915 .4922 .4928 .4934 .4940 .4945 .4949 .4954 .4957 .4961

0.20 .4886 .4894 . 4902 . 4909 .4915 .4922 .4927 .4933 .4938 .4942

0.30 .4829 .4840 .4849 .4859 .4867 .4875 .4883 .4890 .4897 .4904

0.40 .4734 .4747 .4760 .4772 .4783 .4794 .4804 .4814 .4824 .4833

0.50 .4585 .4602 .4617 .4633 .4647 .4661 .4675 .4688 .4700 .4712

0.60 .4368 .4388 .4406 .4425 .4442 .4459 .4476 .4492 .4508 .4523

0.70 .4071 .4092 .4113 .4134 .4154 .4173 .4192 .4211 .4229 .4247

0.80 .3688 .3710 .3732 .3753 .3774 .3795 .3815 .3835 .3854 .3873 0 90 . 3227 .3248 .3269 . 3289 . 3309 . oozy . OO^o 3368 3386 . 3405

i!oo .2707 .2725 .2743 .2760 .2778 .2795 .2812 !2828 !2844 !2860

2. 4000 2. 4200 2. 4400 2.4600 2. 4800 2. 5000 2. 5200 2. 5400 2. 5600 2. 5800

0. 0. 4970 0. 4972 0. 4975 0. 4977 0. 4979 0. 4981 0.4983 0. 4985 0. 4986 0. 4988

0.10 .4964 .4967 .4970 .4973 .4975 .4978 .4980 .4981 .4983 .4985

0.20 .4947 .4951 .4954 .4958 . 4961 .4964 .4967 . 4970 .4972 .4974

0.30 .4910 .4915 .4921 .4926 .4931 .4935 .4939 . 4943 .4947 .4950

0.40 .4841 .4849 .4857 .4864 .4871 .4878 .4884 .4891 .4896 .4902

0.50 .4724 .4735 .4746 .4756 .4766 .4776 .4785 . 4794 .4802 . 4811

0.60 .4538 .4553 .4566 . 4580 .4593 .4606 .4618 .4630 .4642 .4653

0.70 .4265 .4282 .4298 .4315 .4331 .4346 .4361 .4376 .4391 .4405

0.80 .3892 .3911 .3929 .3946 .3964 .3981 .3998 .4014 . 4030 .4046

0.90 .3423 .3441 .3459 .3476 .3493 .3510 . 3527 .3543 .3559 .3575

1.00 .2876 .2892 . 2907 . 2922 .2937 .2952 .2967 .2981 .2995 . 3009

18
Table 7. Generalized temperature averages, for a cylinder—Continued

2. 6000 2. 6200 2. 6400 2.6600 2. 6800 2. 7000 2. 7200 2. 7400 2. 7600 2. 7800

0. 0. 4989 0.4990 0. 4991 0. 4992 0. 4993 0. 4993 0. 4994 0. 4995 0. 4995 0. 4996

0.10 .4986 .4987 .4989 .4990 .4991 .4992 .4992 .4993 .4994 .4994

0.20 .4976 .4978 .4980 .4982 .4983 .4984 .4986 .4987 .4988 .4989

0.30 .4954 .4957 .4960 .4962 .4965 .4967 .4970 .4972 .4973 .4975

0.40 .4907 .4912 .4917 .4921 .4925 .4930 .4933 .4937 .4940 .4944

0.50 .4818 .4826 .4833 .4840 .4847 .4853 .4860 .4866 .4871 .4877

0.60 .4664 .4675 .4685 .4695 .4705 .4714 .4723 .4732 .4741 .4749

0.70 .4419 .4432 .4445 .4458 .4471 .4483 .4495 .4507 .4519 .4530

0.80 .4062 .4077 .4092 .4107 .4122 .4136 .4150 .4164 .4178 .4191

[) 90 3591 3606 . ODZl . 3636 . 3651 . 3666 . 3680 . 3694 . 3708 . 3722

i!oo .'3023 .' 3036 .3050 .3063 .3076 .3089 .3101 .3114 .3126 .3138

2. 8000 2. 8200 2.8400 2.8600 2.8800 2. 9000 2. 9200 2. 9400 2. 9600 2. 9800

0. 0. 4996 0. 4997 0. 4997 0. 4997 0. 4998 0. 4998 0. 4998 0. 4998 0. 4998 0. 4999

0.10 .4995 .4995 .4996 .4996 .4997 .4997 .4997 .4998 .4998 .4998

0.20 .4990 .4991 .4992 .4992 .4993 .4994 .4994 .4995 .4995 .4996

0.30 .4977 .4979 .4980 .4981 .4983 .4984 .4985 .4986 .4987 .4988

0.40 .4947 .4950 .4952 .4955 .4958 .4960 .4962 .4964 .4966 .4968

0.50 .4882 .4887 .4892 .4897 .4901 . 4905 .4909 .4913 .4917 .4921

0.60 .4757 .4765 .4773 .4780 .4787 .4794 .4801 .4807 .4814 .4820

0.70 .4541 .4552 .4562 .4572 .4582 .4592 .4602 .4611 .4620 .4629

0.80 .4204 .4217 .4230 .4242 .4254 .4266 .4278 .4290 .4301 .4312

D.90 3735 3749 3762 3775 . 0/00 3800 3813 3825 3837 . 3849

i!oo !3151 ." 3162 !3174 !3186 .3197 !3209 !3220 !3231 !3242 !3252

3.0000 3. 1000 3.2000 3. 3000 3.4000 3. 5000 3. 6000 3. 7000 3. 8000 3. 9000

0. 0.4999 0.4999 0. 5000 0. 5000 0. 5000 0. 5000 0. 5000 0. 5000 0. 5000 0. 5000

0.10 .4998 .4999 .4999 .5000 .5000 .5000 .5000 .5000 .5000 .5000

0.20 .4996 .4997 .4998 .4999 .4999 .5000 .5000 .5000 .5000 .5000

0.30 .4989 .4992 .4995 .4997 .4998 .4998 .4999 .4999 .5000 .5000

0.40 .4970 .4978 .4984 .4988 .4991 .4994 .4995 .4997 .4998 .4998

0.50 .4924 .4940 .4952 .4962 .4970 .4977 .4982 .4986 .4989 .4991

0.60 .4826 .4853 .4876 .4895 .4912 .4926 .4938 .4948 .4957 .4964

0.70 .4638 .4679 .4715 .4747 .4776 .4802 .4825 .4845 .4863 .4879

0.80 .43-23 .4376 .4424 .4469 .4510 .4547 .4582 .4614 .4644 .4671

0 90 . 3861 . 3918 . 3971 . 4022 . 4069 . 4113 4154 4194 4231 . 4266

i!oo .3263 .3315 .3363 , .3409 .3452 .3493 !3532 [3569 !3604 '. 3638

4. 0000 4. 1000 4.2000 4.3000 4.4000 4. 5000 4. 6000 4. 7000 4. 8000 4. 9000

0. 0.5000 0. 5000 0. 5000 0.5000 0.5000 0.5000 0. 5000 0. 5000 0. 5000 0. 5000

0.10 .5000 . 5000 . 5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000

0.20 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000

0.30 .5000 . 5000 .5000 . 5000 .5000 .5000 .5000 .5000 .5000 .5000

0.40 .4999 .4999 .4999 .5000 .5000 .5000 .5000 .5000 .5000 .5000

0.50 .4993 .4995 .4996 .4997 .4998 .4998 .4999 .4999 .4999 .4999

0.60 .4970 .4975 .4980 .4983 .4986 .4989 .4991 .4992 .4994 .4995

0.70 .4894 .4906 .4917 .4927 .4936 .4944 .4951 .4957 .4963 .4967

0.80 .4696 .4720 .4741 .4761 .4780 .4797 .4812 .4827 .4840 .4853 0 90 . 4299 . 4331 . 4361 . 4389 . 4416 4441 4465 . 4488 . 4510 . 4631

i!oo .3670 .3700 .3729 .3757 .3784 .'3809 .'3834 !3857 .'3880 !3902

5.0000 5. 1000 5.2000 5. 3000 5.4000 5. 5000 5. 6000 5. 7000 5. 8000 6. 9000

0. 0.5000 0. 5000 0. 5000 0. 5000 0. 5000 0. 5000 0.5000 0.5000 0. 5000 0. 5000

0.10 .5000 .5000 .5000 . 5000 .5000 .5000 .5000 .5000 .5000 .5000

0.20 .5000 . 5000 . 5000 . 5000 .5000 .5000 .5000 .5000 .6000 .6000

0.30 .5000 .5000 . 5000 .5000 .5000 .5000 .5000 .5000 .5000 .6000

0.40 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .6000

0.50 .5000 .5000 . 5000 .5000 .5000 .5000 .5000 .5000 .5000 .6000

0.60 .4996 .4997 .4997 .4998 .4998 .4999 .4999 .4999 .4999 .4999

0.70 .4971 .4975 .4978 .4981 .4984 .4986 .4988 .4989 .4991 .4992

0.80 .4864 .4875 .4885 .4894 .4902 .4910 .4917 .4924 .4930 .4936

. 4550 . 4569 . 4587 . 4605 . 4621 4637 4652 . 4666 . 4680 .4693

1.00 .3923 .3943 .3962 .3981 .3999 ."4016 !4033 !4049 !4065 [4080

. q 6.0000 6. 1000 6. 2000 6. 3000 6.4000 6. 5000 6.6000 6. 7000 6. 8000 6. 9000

0. 0. 5000 0.5000 0. 5000 0. 5000 0. 5000 . 05000 0. 5000 0. 5000 0.5000 0. 5000

0.10 . 5000 . 5000 . 5000 . 5000 . 5000 .5000 .5000 .5000 .5000 .5000

0.20 . 5000 . 5000 . 5000 . 5000 .5000 .5000 .5000 .5000 .5000 .5000

0.30 . 5000 . 5000 . 5000 . 5000 .5000 .5000 .5000 .5000 .5000 .5000

0.40 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000

0.50 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000

0.60 .4999 .5000 .5000 .5000 .5000 .5000 . 5000 . 5000 . 5000 . 5000

0.70 .4993 .4994 .4995 .4996 .4997 .4997 .4997 .4998 .4998

.4970 .4998

0.80 .4973 .4941 .4946 .4950 .4954 .4958 .4962 .4965 .4968

0.90 .4705 .4717 .4728 .4739 .4750 .4760 .4769 .4779 .4788 .4796 1X0 .4095 .4109 .4123 .4136 .4149 .4162 .4174 .4186 .4198 .4209

19
Table 7. Generalized temperature averages, ^2, for a cylinder—Continued

7. 0000 7. 1000 7. 2000 7. 3000 7. 4000 7. 5000 7. 6000 7. 7000 7. 8000 7. 9000

0. 0. 5000 0. 5000 0. 5000 0. 5000 0. 5000 0. 5000 0. 5000 0. 5000 0. 5000 0. 5000

0.10 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000

0.20 .5000 .5000 .5000 .5000 . 5000 .5000 . 5000 .5000 .5000 .5000

0.30 .5000 .5000 .5000 .5000 .5000 .5000 . 5000 .5000 .5000 .5000

0.40 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000

0.50 . 5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000

0.60 . 5000 .5000 . 5000 . 5000 . 5000 . 5000 . 5000 .5000 . 5000 . 5000

0.70 .4999 .4999 .4999 .4999 .4999 .4999 . 5000 .5000 .5000 .5000

0.80 .4975 .4977 .4979 .4981 .4983 .4984 .4985 .4987 .4988 .4989

0.90 . 4804 . 4812 . 4819 . 4827 . 4834 4840 . 4847 . 4853 . 4859 . 4864

i!oo !4220 !4230 !4241 [4251 ^4261 !4270 !4279 '4288 ]4297 ]4306

\? . 80000 8. 1000 8. 2000 8. 3000 8. 4000 8. 5000 8. 6000 8. 7000 8. 8000 8. 9000

0. 0. 5000 0. 5000 0. 5000 0. 5000 0. 5000 0. 5000 0. 5000 0. 5000 0. 5000 0. 5000

Olio .5000 .5000 .5000 .5000 .5000 .5000 . .5000 .5000 .5000 . 50011 0;20 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000 . 5000 . 500(1 0:30 .5000 . 5000 . 5000 .5000 .5000 .5000 . 5000 .5000 .5000 .5000

0.40 .5000 . 5000 . 5000 .5000 .5000 .5000 . 5000 .5000 .5000 .5000

O.'SO .5000 . 5000 . 5000 .5000 .5000 .5000 .5000 . .5000 .5000 .5000

O16O . 5000 . 5000 . 5000 . 5000 . 5000 . 5000 . 5000 . 5000 . 5000 . 5000

0:70 . 5000 .5000 .5000 .5000 .5000 .5000 . 5000 . 5000 .5000 .5000

OiSO .4990 .4991 .4992 .4993 .4993 .4994 . 4994 .4995 .4995 .4996 0 90 . 4869 4875 . 4880 . 4884 . 4889 4893 . 4898 . 4902 4906 . 4909

i!oo !4315 .4323 !4331 !4339 !4346 !4354 !4361 !4368 !4375 .4382

9. 0000 9. 1000 9. 2000 9. 3000 9. 4000 9. 5000 9. 6000 9. 7000 9. 8000 9. 9000

0. 0. .5000 0. 5000 0. 5000 0. 5000 0. 5000 0. 5000 0. 5000 0. 5000 0. 5000 0. .5000 0.10 .5000 .5000 .5000 .5000 .5000 . .5000 .5000 .5000 .5000 . .5000 .0.20 .5000 .5000 .5000 .5000 . 5000 .5000 .5000 .5000 .5000 . 5000

0130 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .50011 0.40 . 5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000 . 50110 O.50 .5000 .5000 .5000 .5000 .5000 .5000 . 5000 .5000 .5000 .5000

.0.60. . 5000 . 5000 . 5000 .5000 .5000 .5000 . 5000 . 5000 . 5000 . 5000 0.70 .5000 .5000 .5000 .5000 .5000 . 5000 .5000 .5000 .5000 .5000 0.80 .4996 .4997 .4997 .4997 .4998 .4998 .4998 .4998 .4999 .4999 O;90 . 4913 . 4916 . 4920 . 4923 . 4926 . 4929 . 4932 . 4935 . 4937 . 4940

i!oo !4389 !4395 ;4401 !4408 !4414 !4420 '4426 . 4431 !4437 .4443

10. 0000 10. 1000 10. 2000 10. 3000 10. 4000 10. 5000 10. 6000 10. 7000 10. 8000 10. 9000

0. 0. 5000 0. .5000 0. 5000 0. 5000 0. 5000 0. 5000 0. 5000 0. 5000 0. .5000 0. 5000

,0:10 .5000 .5000 .5000 .5000 .5000 . 5000 .5000 .5000 .5000 .5000 0.20 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000

O.30 .5000 .5000 .5000 .5000 .5000 .5000 . ,5000 .5000 .5000 .5000

Oi40" .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000

0.50 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000 .5000 . 5000

^0:60 . 5000 . 5000 .5000 . 5000 . 5000 . 5000 . 5000 . 5000 .5000 . 5000

0.70 . 5000 .5000 .5000 .5000 .5000 .5000 . ,5000 .5000 .5000 .5000

0.80 .4999 .4999 .4999 .4999 .4999 .5000 .5000 .5000 .5000 .5000

0.90 .4942 .4945 .4947 .4949 .4951 .4953 .4955 .4957 .4958 .4960

1.00 .4448 .4454 .4459 .4464 .4469 .4474 .4479 .4484 .4488 .4493

Table 8. Fractional temperature excess, 0.3, for a sphere

0. 5000 0. 5200 0.5400 0. 5600 0. 5800 0. 6000 0. 6200 0. 6400 0. 6600 0. 6800

0. 0. 0001 0. 0002 0. 0004 0. 0008 0.0013 0. 0021 0. 0033 0. 0048 0. 0069 0. 0096

OiiO .0001 .0002 .0004 .0008 .0013 .0021 .0032 .0048 .0068 .0095

0.20 .0001 .0002 .0004 .0007 .0012 .0020 .0031 .0045 .0065 .0090

0.30 .0001 .0002 .0004 .0007 .0011 .0018 .0028 .0042 .0060 .0083

0.40 .0001 .0002 .0003 .0006 .0010 .0016 .0025 .0037 .0052 .0073

0.50 .0001 .0001 .0003 .0005 .0008 .0013 .0021 .0031 .0044 .0061 0.60 .0001 .0001 .0002 .0004 .0007 .0011 .0016 .0024 .0035 .0049

0.70 .0000 .0001 .0002 .0003 .0005 .0008 .0012 .0018 .0026 .0035

0.80 .0000 .0001 .0001 .0002 .0003 .0005 .0008 .0011 .0016 .0023

0.90 .0000 .0000 .0000 .0001 .0001 .0002 .0004 .0005 .0008 .0011

1.00 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

0. 7000 0. 7200 0. 7400 0. 7600 0. 7800 0. 8000 0. 8200 0. 8400 0. 8600 0. 8800

0. 0. 0130 0.0171 0. 0221 0. 0279 0. 0347 0. 0423 0.0510 0. 0606 0. 0711 0. 0827

0.10 .0128 .0169 .0217 .0275 .0341 .0416 .0501 .0596 .0700 .0813

0.20 .0122 .0160 .0207 .0261 .0324 .0396 .0477 .0567 .0666 .0773

0.30 .0112 .0147 .0190 .0240 .0297 .0363 .0438 .0520 .0611 .0709

0.40 .0098 .0130 .0167 .0211 .0262 .0320 .0386 .0459 .0538 .0626

0.50 .0083 .0109 .0141 .0178 .0221 .0270 .0325 .0386 .0453 .0526

0.60 .0066 .0086 .0111 .0141 .0175 .0214 .0257 .0306 .0359 .0417

0.70 .0048 .0063 .0081 .0103 .0127 .0156 .0188 .0223 .0262 .0304

0.80 .0030 .0040 .0052 .0065 .0081 .0099 .0119 .0142 .0166 .0193

0.90 .0014 .0019 .0024 .0031 .0038 .0046 .0056 .0066 .0078 .0090

1.00 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

)

20
Table 8. Fractional temperature excess, <j>z, for a sphere—Continued

0.9000 0. 9200 0. 9400 0. 9600 0. 9800 1. 0000 1. 0200 1. 0400 1.0600 1. 0800

n. 0.0951 0. 1084 0. 1225 0. 1375 0. 1531 0. 1695 0. 1865 0. 3041 0. 2222 0. 2407

0.10 .0935 .1066 .1205 .1352 .1506 .1667 .1835 .2008 . 2186 . 2368

0.20 .0889 .1014 .1146 .1286 .1433 .1586 . 1745 . 1910 .2079 . 2253

0.30 .0816 .0930 . 1052 .1180 .1315 .1455 . 1602 . 1753 . 1908 . 2068

0.40 .0720 .0820 .0927 .1041 .1159 . 1283 . 1412 . 1546 . 1683 . 1824

0.50 .0605 .0690 .0780 .0875 .0975 . 1080 .1188 .1301 .1416 .1535

0.60 .0480 .0547 .0618 .0694 .0773 .0856 .0942 . 1031 .1123 . 1217

0.70 .0350 .0399 .0451 .0506 .0564 .0624 .0687 .0752 .0819 .0888

0.80 .0222 .0254 .0287 .0322 .0358 .0397 .0437 .0478 .0521 .0565

0.90 .0104 .0118 .0134 .0150 .0168 .0185 .0204 .0224 .0243 .0264

1.00 0. 0. 0. 0. Q Q

A

U. 0. 0. 0.

1. 1000 1. 1200 1. 1400 1. 1600 1. 1800 1. 2000 1.2200 1.2400 1. 2600 1.2800

0. 0. 2597 0. 2790 0. 2986 0. 3183 0. 3383 0. 3584 0. 3785 0. 3986 0. 4187 0. 4388

0.10 .2555 .2745 .2937 .3132 .3328 .3526 .3724 .3923 .4121 .4318

0.20 .2430 .2611 .2795 .2980 .3168 .3356 .3545 .3735 .3924 .4113

0.30 .2231 .2398 .2566 .2737 .2910 .3084 .3258 .3433 .3609 .3783

0.40 .1968 .2115 .2265 .2416 .2569 .2723 .2878 .3034 .3190 .3346

0.50 .1657 .1781 .1907 .2035 .2164 .2295 .2426 .2559 .2691 .2824

0.60 .1314 .1413 .1513 .1615 . 1718 .1822 . 1927 .2033 .2139 .2246

0.70 .0959 .1031 .1104 .1179 . 1254 .1331 . 1408 .1486 . 1564 . 1642

0.80 .0610 .0656 .0702 .0750 .0798 .0847 .0896 .0946 .0996 .1047

0.90 .0285 .0307 .0328 .0351 .0373 .0396 .0419 .0443 .0466 .0490

1.00 0. 0. Q Q Q Q Q Q 0. 0.

1.3000 1. 3200 1. 3400 1. 3600 1.3800 1. 4000 1.4200 1. 4400 1. 4600 1. 4800

0. 0. 4587 0. 4784 0. 4979 0. 5172 0. 5362 0. 5550 0. 5734 0. 5914 0.6091 0. 6263

0. 10 .4514 .4709 .4902 .5092 .5280 .5465 .5647 .5825 .6000 .6171 0.20 .4301 .4488 .4673 .4856 .5036 .5215 .5390 . 5563 .5732 .5899 0.30 .3958 .4131 .4303 .4474 .4643 .4810 .4974 .5137 . 5297 .5454

0.40 .3502 .3657 .3811 .3965 .4117 .4268 .4417 .4565 .4711 .4855 0.50 . 2957 .3090 .3222 .3354 .3485 .3616 .3745 .3874 .4001 .4128

0.60 .2353 .2460 .2566 .2673 .2780 .2886 .2992 .3097 .3202 .3306

0.70 .1721 . 1800 .1880 .1959 .2038 .2117 .2197 .2276 .2354 .2433

0.80 . 1097 . 1148 . 1199 .1250 .1302 . 1353 .1404 .1456 .1507 .1558 0.90 .0514 .0538 .0562 .0586 .0610 .0634 .0659 .0683 .0707 .0732

1. 00 Q Q Q Q Q Q Q Q Q Q

1.5000 1.5200 1.5400 1.5600 1.5800 1.6000 1. 6200 1.6400 1.6600 1. 6800

0. 0.&432 0. 6596 0. 6756 0. 6912 0.7063 0. 7209 0. 7350 0. 7487 0. 7618 0. 7746

0.10 .6339 .6502 . 6661 .6815 .6965 . 7111 .7252 .7388 .7520 .7647 0.20 .6061 .6220 . 6375 . 6527 .6674 . 6818 .6957 .7093 .7224 .7351 0.30 . 5608 . 5760 . 5908 .6054 .6196 .6335 .6471 .6603 .6732 .6857 0.40 .4997 .5137 . 5275 .5410 .5543 .5674 .5802 .5928 .6051 .6172 0.50 .4252 .4376 .4498 .4619 .4738 .4856 .4972 .5086 .5199 .5310 0.60 .3409 .3512 .3614 .3715 .3815 .3915 .4013 .4111 .4208 .4304

0. 70 .2511 .2589 .2667 .2744 .2821 .2898 .2974 .3050 .3126 .3201 0.80 .1609 .1661 .1712 . 1763 .1814 .1865 .1915 . 1966 .2017 .2067 0.90 .0756 .0780 .0805 .0829 .0854 .0878 .0902 .0927 ,0951 .0976

1 00 U. 0. 0. 0. U. Q Q Q Q

1.7000 1. 7200 1.7400 1. 7600 1. 7800 1.8000 1. 8200 1.8400 1.8600 1.880O

0. 0. 7868 0. 7985 0. 8098 0. 8206 0. 8310 0.8409 0.8504 0. 8594 0. 8680 0. 8762 0.10 .7770 .7888 .8001 .8110 .8214 .8315 .8410 .8502 .8590 .8673 0.20 .7474 .7593 . 7707 .7818 .7925 .8027 .8126 .8221 .8312 .8400 0.30 .6979 .7098 . 7213 . 7325 .7434 .7539 .7641 .7740 .7835 .7927 0.40 .6290 . 6406 . 6519 .6629 .6737 .6842 .6945 . 7045 .7143 .7238 0.50 .5420 .5527 .5633 .5738 .5840 .5941 .6040 .6137 .6233 .6327 0.60 .4398 .4492 .4585 .4677 .4768 .4858 .4947 .5035 .5122 .5208 0.70 .3276 .3350 .3424 .3497 .3570 .3642 .3714 .3786 .3857 .3927 0.80 .2117 .2167 .2217 .2267 .2317 .2367 .2416 .2465 .2515 .2564

0. 90 .1000 . 1024 .1049 . 1073 .1097 . 1122 .1146 . 1170 .1194 .1218

1 00 Q

n A

U. u. Q 0. 0. 0. 0. 0.

1.9000 1.9200 1.9400 1. 9600 1. 9800 2.0000 2. 0200 2. 0400 2. 0600 2. 0800

0. 0.8840 0. 8914 0. 8984 0. 9051 0. 9114 0. 9173 0. 9230 0. 9283 0. 9333 0.9380 0.10 .8753 .8828 . 8900 .8969 .9034 .9095 .9154 .9209 .9261 .9310 0.20 .8484 .8564 .8641 . 8715 .8785 .8852 .8916 .8977 .9035 .9090 0.30 .8016 .8102 .8186 .8266 .8343 .8417 .8489 .8558 .8624 .8688 0.40 .7331 .7422 .7510 . 7595 .7679 .7760 . 7838 . 7915 . 7989 .8062 0.50 .6419 .6509 .6598 .6685 .6771 .6854 .6937 .7017 .7096 .7173 0.60 .5293 .5376 .5459 .5541 .5622 .5702 .5781 .5858 . 5935 .6011 0.70 .3997 .4067 .4136 .4205 .4273 .4341 .4408 .4475 . 4541 . 4607

0.80 .2613 .2661 .2710 .2758 .2807 .2855 .2903 .2951 .2998 .3046 0.90 .1243 .1267 .1291 . 1315 .1339 .1363 .1387 . 1412 .1436 .1460 1.00 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

21
Table 8. Fractional temperature excess, <j>z, for a sphere—Continued

2. 1000 2. 1200 2. 1400 2. 1600 2. 1800 2. 2000 2. 2200 2.2400 2. 2600 2, 2800

0. 0. 9424 0. 9466 0. 9541 0 9575 0 9607 0 9637 U- yoDO n QAQ1 u. yoyi U. 9710

0. 10 . 9357 . 9400 . 9442 . 9481 . 9517 9551 9583 9613 9642

0. 20 . 9143 . 9193 . 9240 . 9286 . 9328 9369 9407 9444 0470 . 9511

0. 30 . 8750 . 8808 . 8865 . 8919 8971 9021 9069 9115 Q1 KQ . yioy Q0A1 0 40 . 8132 . 8200 . 8266 . 8330 . 8392 8452 8510 8567 8621 0^50 '. 7249 !7323 !7396 !7467 '. 7536 !7604 !7671 !7736 !7799 .7862

0. 60 .6086 .6160 .6232 .6304 .6375 .6445 .6514 .6582 .6648 .6714 0.70 .4672 .4737 .4801 .4865 .4928 .4991 .5053 .5115 .5177 .5237 0.80 .3093 .3141 .3188 .3235 .3281 .3328 .3374 .3420 .3467 .3512

0. 90 .1484 .1508 . 1532 . 1555 .1579 .1603 .1627 . 1651 .1675 .1699

1. 00 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

^? 2. 3000 2. 3200 2. 3400 2. 3600 2. 3800 2. 4000 2.4200 2. 4400 2. 4600 2.4800

0. 0. 9738 0. 9759 0. 9779 0. 9797 0 9814 n QQOQ u. yo^y 0 9844 0 9857 u. yooy 0 9881

0. 10 . 9693 . 9716 9737 9757 9776 9794 9810 9825 9839 9852

0. 20 . 9542 . 9571 . 9598 . 9624 9648 9671 9693 9713 9732 9750

0. 30 . 9241 . 9279 . 9316 . 9351 9385 9417 9447 9477 9504 9531

0. 40 . 8726 . 8775 . 8823 . 8869 8914 8957 8999 9040 9079 9116 0^50 !7922 !7982 !8040 !8097 !8152 !8206 '. 8259 !8311 !8361 !8410

0. 60 .6779 .6844 .6907 .6969 .7030 .7090 .7150 .7208 .7266 .7323 0.70 . 5298 .5358 .5417 .5476 .5534 .5592 .5649 .5706 .5763 .5818

0. 80 .3558 .3604 .3649 .3694 .3739 .3784 . 3829 .3874 .3918 .3962

0. 90 .1722 .1746 . 1770 .1794 .1817 . 1841 .1865 .1888 .1912 .1936

1. 00 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

2. 5000 2. 5200 2. 5400 2. 5600 2. 5800 2. 6000 2. 6200 2. 6400 2. 6600 2. 6800

Q 0. 9891 0. 9901 0. 9910 0 9918 0 9925 u. yyoz u. yyoo U- yy^^ nu. QQdQ yyiy 0 9954

0.10 . 9864 . 9875 . 9885 . 9895 . 9904 9912 9919 9926 9932 . 9938 0 20 . 9767 . 9783 . 9798 . 9812 9825 9837 9849 9859 9869 9879

0 30 9556 . 9580 . 9603 9624 9645 9665 9684 9701 9718 9734

0.40 . 9153 . 9188 . 9221 . 9254 . 9285 9316 9345 . 9373 9400 . 9426

o!50 !8458 !8505 !8550 ]8595 !8638 !8680 [8721 '. 8761 !8800 !8838

0.60 .7378 .7433 . 7487 . 7540 .7593 .7644 .7695 .7744 .7793 .7841

0.70 .5874 .5929 .5983 .6037 .6090 .6143 .6195 .6247 .6299 .6350

0.80 .4006 .4050 .4094 .4137 .4181 .4224 .4267 .4309 . 4352 .4395

0.90 .1959 .1983 .2006 .2030 .2053 .2077 .2100 .2123 .2147 .2170

1.00 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

^? 2. 7000 2. 7200 2.7400 2. 7600 2. 7800 2. 8000 2. 8200 2. 8400 2. 8600 2. 8800

0. n QQAQ 0 9972 0 9975 0 9978 u. yyou n QQQO 0. 9984

0 10 . 9944 . 9949 9953 9957 9961 9965 9968 9971 9974 . 9976 0 20 9888 . 9896 9903 9910 9917 9923 9929 9934 9939 . 9944

0 30 9749 . 9764 9777 9790 9803 9814 9825 9836 9845 9855

0 40 . 9451 . 9475 . 9498 9520 9542 9562 9582 9601 9619 . 9637

o!50 !8875 !8911 !8946 !8980 !9013 !9046 !9077 '. 9108 !9137 '. 9166

0.60 .7889 .7935 .7981 .8026 . 8070 . 8113 .8156 .8197 .8238 .8279

0.70 .6400 .6450 .6499 .6548 .6597 .6645 .6692 .6739 .6786 .6832

0.80 .4437 .4479 . 4521 .4562 .4604 .4645 .4686 .4727 .4768 .4809

0.90 .2194 .2217 .2240 .2263 .2287 .2310 .2333 .2356 .2379 .2402

1.00 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

\' 2. 9000 2. 9200 2. 9400 2. 9600 2. 9800 3. 0000 3. 0200 3. 0400 3. 0600 3. 0800

Q u. yyoo 0 9987 u. yyoo u. yyyu 0 9991 u. yyyo Ciu. QQQ1 yyyo 0 9994 u. yyyu n in 9978 9980 9982 9984 9985 9987 9988 9989 9990 . 9991

0.20 9948 . 9952 9956 9959 9963 9965 9968 9971 9973 . 9975 0 30 9864 9872 9880 9887 9894 9901 9907 9913 9918 . 9923 0 40 9653 9669 9685 9700 9714 9727 9740 9753 9765 9776

o!50 '. 9194 !9221 !9247 !9273 !9298 !9322 !9346 !9368 !9390 '. 9412

0.60 .8318 .8357 . 8395 . 8432 .8469 .8505 .8540 .8575 .8609 .8642

0.70 .6878 .6923 .6968 .7012 .7055 .7099 .7141 .7184 .7226 .7267

0.80 .4849 .4889 . 4929 .4969 .5009 .5048 .5088 .5127 .5165 .5204

0.90 .2425 .2448 .2471 .2494 .2517 .2540 .2563 .2586 .2609 .2632

1.00 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

3. 0000 3. 1000 3. 2000 3. 3000 3. 4000 3. 5000 3. 6000 3. 7000 3. 8000 3. 9000

Q 0 9992 u. yyyo 0 9997 n QOQQ u. yyyy A QOQQ 1. uuuu 11. nnnn uuuu 1 0000 1 0000 1. 0000 0 10 9987 9992 9995 9997 9999 9999 1 0000 1 0000 1. 0000 1. 0000 0 20 9965 9977 9985 9991 9994 9996 u. yyyo u. yyyy n QQQQ 0. 9999 0 30 9901 9928 9949 9964 9975 9982 9988 9992 ^ 9994 . 9996 0 40 9727 9787 9834 9872 9902 9925 9944 9957 9968 . 9977

o!50 !9322 !9432 !9527 ]9608 !9076 ]9734 !9782 !9822 !9856 !9884

0.60 .8505 .8675 .8829 .8968 .9093 .9205 .9305 .9394 .9474 .9544

0.70 .7099 .7308 .7506 .7693 .7869 .8035 .8190 .8336 .8472 .8600

0.80 .5048 .5243 .5432 .5617 .5797 .5973 .6143 .6309 .6469 .6625

0.90 .2540 .2655 .2768 .2881 2993 .3104 .3215 .3324 .3433 .3542

1.00 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

22
Table 8. Fractional temperature excess, <f>3, for a sphere—Continued

4. 0000 4. 1000 4. 2000 4. 3000 4. 4000 4. 5000 4. 6000 4. 7000 4. 8000 4. 9000

0. oooo 1.0000 1.0000 1.0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1.0000

0.10 i! 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000

0.20 1. 0000 1.0000 1. 0000 1. 0000 1. 0000 1.0000 1. 0000 1. 0000 1. 0000 1.0000

0.30 0. 9997 0. 9998 0. 9999 0. 9999 0. 9999 1. 0000 1. 0000 1. 0000 1. 0000 1.0000

0.40 9983 .9987 .9991 .9993 .9995 0. 9997 0. 9998 0. 9999 0. 9999 0. 9999

0.50 9906 .9925 .9940 .9953 .9963 9971 9977 9982 9986 .9989

0.60 9606 .9660 .9708 .9750 .9787 9818 9846 9869 9890 .9907

0.70 8719 .8829 .8932 .9027 .9115 9197 9272 9341 9404 .9462

0.80 6776 .6923 .7064 .7201 .7334 7461 7585 7703 7818 .7928

0 90 3649 3755 3861 3965 4172 4274 4375 4475 4574

i!oo 0. o! 0. 0. 0. o! 0. o! o] o!

7

5. 0000 5. 1000 5. 2000 5. 3000 5. 4000 5. 5000 5. 6000 5. 7000 5. 8000 5. 9000

0. 1. 0000 1.0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000

0.10 1. 0000 1. 0000 1. 0000 1. 0000 1.0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000

0.20 1. 0000 1. 0000 1.0000 1. 0000 1. 0000 1. 0000 1. 0000 1, 0000 1.0000 1. 0000

0.30 1. 0000 1.0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1.0000 1.0000

0.40 0. 9999 1.0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1.0000

0.50 9992 0. 9994 0. 9995 0. 9996 0. 9997 0. 9998 0. 9999 0. 9999 0. 9999 0. 9999

0.60 9922 .9935 .9946 .9955 .9962 9969 9974 9979 9983 .9986

0.70 9516 .9565 .9609 .9649 .9686 9720 9750 9777 9802 .9824

0.80 8034 .8135 . 8233 .8327 .8417 8503 8585 8663 8739 .8810 4672 . 4769 . 4865 . 4961 . 5055 5148 5240 6331 5421 5510

1.00 o! 0. 0. 0. 0. 0. o' 0. 0. o]

\? 6. 0000 6. 1000 6.2000 6. 3000 6. 4000 6. 5000 6.6000 6. 7000 6. 8000 6. 9000

0. 1, 0000 1.0000 1. 0000 1.0000 1. 0000 1 0000 1 0000 1 0000 1 0000 1. 0000

0.10 1. 0000 1.0000 1. 0000 1. 0000 1. 0000 1 0000 1 0000 1 0000 1 0000 1. 0000

0.20 1 0000 1.0000 1. 0000 1. 0000 1. 0000 1 0000 1 0000 1 0000 1 0000 1. 0000

0.30 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1 0000 1 0000 1 0000 1. 0000 1. 0000

0.40 1 0000 1. 0000 1. 0000 1. 0000 1.0000 1 0000 1 0000 1 0000 1 0000 1. 0000

0.50 1 0000 1. 0000 1.0000 1. 0000 1. 0000 1 0000 1 0000 1 0000 1. 0000 1. 0000

0.60 0 9989 0. 9991 0. 9992 0. 9994 0. 9995 0 9996 0 9997 0 9998 0 9998 0. 9999

0.70 9844 .9862 .9878 .9893 .9905 9917 9927 9936 9944 .9951

0.80 8879 .8944 .9006 .9065 .9122 9175 9226 9274 9319 .9363

0 90 5598 . 5685 . 5771 . 5856 . 5940 6023 6104 6185 6264 . 6343

i!oo 0 0. 0. 0. 0. 0 o] 0 0. 0.

\? 7.0000 7. 1000 7. 2000 7. 3000 7. 4000 7.5000 7. 6000 7. 7000 7. 8000 7. 9000

0. 1 0000 1.0000 1. 0000 1. 0000 1.0000 1 0000 1 0000 1 0000 1 0000 1.0000 o.io 1 0000 1.0000 1. 0000 1. 0000 1. 0000 1 0000 1 0000 1 0000 1 0000 1. 0000

0.20 1 0000 1.0000 1.0000 1. 0000 1. 0000 1 0000 1 0000 1 0000 1 0000 1.0000

0.30 1 0000 1.0000 1. 0000 1. 0000 1. 0000 1 0000 1 0000 1 0000 1 0000 1. 0000

0.40 1 0000 1. 0000 1. 0000 1. 0000 1. 0000 1 0000 1 0000 1 0000 1 0000 1. 0000

0.50 1 0000 1.0000 1.0000 1. 0000 1. 0000 1 0000 1 0000 1 0000 1 0000 1. 0000

0.60 0 9999 0. 9999 0. 9999 1.0000 1.0000 1 0000 1 0000 1 0000 1 0000 1. 0000

0.70 9957 .9963 .9968 .9972 0. 9976 0 9979 0 9982 0 9984 0 9987 0. 9988

0.80 9403 .9442 .9478 .9513 .9546 9576 9605 9632 9658 .9682

0.90 6420 . 6496 . 6572 . 6646 . 6719 6791 6862 6931 7000 . 7068

1.00 0 0. 0. 0. 0. 0 0 0 0 0.

\9 8. 0000 8. 1000 8.2000 8. 3000 8. 4000 8. 5000 8. 6000 8. 7000 8. 8000 8. 900

0. 1 0000 1.0000 1.0000 1.0000 1. 0000 1 0000 1 0000 1 0000 1 0000 1. 0000

0.10 1 0000 1. 0000 1. 0000 1. 0000 1. 0000 1 0000 1 0000 .1 0000 1 0000 1. 0000

0.20 1 0000 1. 0000 1. 0000 1.0000 1. 0000 1 0000 1 0000 1 0000 1 0000 1. 0000

0.30 1 0000 1.0000 1.0000 1.0000 1. 0000 1 0000 1 0000 1 0000 1 0000 1. 0000

0.40 1 0000 1.0000 1. 0000 1. 0000 1. 0000 1 0000 J 0000 1 0000 1 0000 1.0000

0.50 1 0000 1. 0000 1.0000 1. 0000 1. 0000 1 0000 1 0000 1 0000 1 0000 1.0000

0.60 1 0000 1.0000 1. 0000 1. 0000 1.0000 1 0000 1 0000 1 0000 1 0000 1. 0000

8.70 0 9990 0. 9991 0. 9993 0. 9994 0. 9995 0 9996 0 9996 0 9997 0 9997 0. 9998

0.80 9704 .9726 .9745 .9764 .9781 9797 9813 9827 9840 .9852

0.90 7135 . 7200 . 7265 . 7328 . 7391 7452 7512 7572 7630 . 7687

1.00 0 0. 0. 0. 0. 0 0 0 0 0.

\? 9.0000 9. 1000 9. 2000 9. 3000 9. 4000 9. 5000 9. 6000 9. 7000 9. 8000 9. 9000

0. 1 0000 1. 0000 1. 0000 1. 0000 1. 0000 1 0000 1 0000 1 0000 1 0000 1. 0000

0.10 1 0000 1. 0000 1. 0000 1. 0000 1. 0000 1 0000 1 0000 J 0000 1 0000 1. 0000

0.20 1 0000 1.0000 1. 0000 1. 0000 1. 0000 1 0000 1 0000 1 0000 1 0000 1. 0000

0.30 1 0000 1. 0000 1. 0000 1. 0000 1. 0000 1 0000 1 0000 1 0000 1 0000 1. 0000

0.40 1 0000 1.0000 1. 0000 1. 0000 1.0000 1 0000 1 0000 1 0000 1 0000 1. 0000

0.50 1 0000 1. 0000 1. 0000 1. 0000 1. 0000 1 0000 1 0000 1 0000 1 0000 1. 0000

0.60 1 0000 1.0000 1. 0000 1. 0000 1. 0000 1 0000 1 0000 1 0000 1 0000 1. 0000

0.70 0 9998 0. 9998 0. 9999 0. 9999 0. 9999 0 9999 1 0000 1 0000 1 0000 1. 0000

0.85 9864 .9875 .9885 .9894 .9902 9910 9918 9924 9930 .9936

0.90 .8206 7743 .7799 .7853 .7906 .7959 8010 8060 8110 8158

1.00 0 0. 0. 0. 0. 0 0 0 0 0.

23
Table 8. Fractional temperature excess, 4>i, for a sphere—Continued

inJ.U- innn J.UUU in 9nnn in "^nnn in 4000 lu. ouuu 10 fioon in 7000 in Rnnn 10lU. Qnnf> oUUw

0. 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000' 0.10 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000

0.20 1. 0000 1. 0000 1. 0000 1.0000 1. 0000 1.0000 1. 0000 1. 0000 1. 0000 1. 0000

0.30 1. 0000 1.0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1.0000 1. 0000 1. oooo

0.40 1. 0000 1. 0000 1.0000 1. 0000 1. 0000 1. 0000 1. 0000 1.0000 1. 0000 1. 0000

0.50 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000 1. 0000

0.60 1. 0000 1. 0000 1. 0000 1 . 0000 1. 0000 1. 0000 ] . 0000 1. 0000 1. 0000 1. 0000

0.74 1. 0000 1.0000 1. 0000 1. 0000 1.0000 1. 0000 1.0000 1. 0000 1.0000 1. 0000

0.80 0. 9942 0. 9947 0. 9951 0. 9955 0. 9959 0. 9963 0. 9966 0. 9969 0. 9972 0. 9974

0.90 .8252 .8298 .8343 .8387 .8429 .8472 .8513 .8553 .8593 .8631

1.00 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.

Table 9. Generalized temperature averages, i^s, for a sphere

xja '^-^0. 5000 0. 6200 0. 54000 0. 5600 0. 5800 0. 6000 0. 6200 0. 6400 0. 6600 0. 680O

0. 0. 0000 0. 0001 0. 0001 0. 0003 0. 0004 0. 0007 0.0011 0. 0016 0. 0023 0. 0032

1l).10 .0000 .0001 .0001 .0003 .0004 .0007 .0011 .0016 .0023 .0032

();20 . 0000 .0001 .0001 . 0002 .0004 .0007 . 0010 .0016 . 0022 . 0031

0.30 .0000 .0001 .0001 .0002 .0004 .0006 . 0010 .0015 .0021 .0029

0.40 .0000 .0001 .0001 .0002 .0004 .0006 .0009 .0014 .0020 .0027

OlSO- .0000 . 0001 . 0001 0002 . 0003 . 0005 . 00C8 . 0012 .0018 .0025

d;6o !oooo .' 0000 !oooi !0002 !0003 !0005 !0007 !ooii !0016 .0022

0.70 0000 0000 0001 QQQO 0003 0004 0006 0010 0014 . 0019

V.OU 0000 0000 0001 0001 nnno . uuuz . UUUi) 0005 0008 0011 . 001 6^ 0.90 . 0000 0000 0001 0001 0002 0003 0004 0006 . 0009 . 001^

1 00 0000 0000 0000 0001 fifini. UUUi . uuuz 0003 0005 0007 . 001

0

0. 7000 0. 7200 0. 7400 0. 7600 0. 7800 0. 8000 0. 8200 C.8400 0. 8600 0.8800

0. 0. 0043 0. 0057 0. 0074 0. 0093 0. 0116 0. 0141 0. 0170 0. 0202 0. 0237 0. 0276

0.10 .0043 .0057 .0073 .0092 .0114 .0140 .0168 .0200 .0235 .0273

0^20 .0042 . 0055 .0071 . 0089 . 0111 . 0136 . 0163 . 0194 . 0228 . 0265

0^30 .0040 .0052 .0067 .0085 .0106 .0129 .0155 .0185 .0217 .0252

0.40 .0037 .0049 .0063 .0079 .0098 .0120 .0145 .0172 .0202 .0234 0 50 . 0034 . 0044 . 0057 . 0072 . 0089 . 0109 . 0132 . OJ 56 . 0184 .0213

0.60 !0030 !0039 '. 0051 ]0064 ]0079 !0097 !0117 !oi39 !0163 .0189 nU.f70u . UUZD . 0044 . 0055 . 0068 . 0084 . UlUl . 0120 0141 0163 nU.oUsn nnoi . uuzo . 0036 . 0046 . 0057 . 0070 . 0084 . 0100 0117 0136

nu.yuon nni 7 . uui / . 0023 . 0029 . 0037 . 0046 . 0056 . 0068 . 0080 0094 0110

1 nn J .uu . 0013 . 0017 . 0022 . 0028 . 0035 . 0043 . 0052 . 0061 . 00/2 0084

xja '~- 0. 9000 0. 9200 0. 9400 0. 9600 0. 9800 1.0000 1.0200 1.0400 1.0600 1.0800

0. 0.0317 0. 0361 0. 0408 0. 0458 0.0510 0. 0565 0. 0622 0. 0680 0. 0741 0. 0802 OjO .0314 .0358 .0404 .0454 .0505 .0559 .0616 .0674 .0733 .0795 0^0 .0305 .0347 .0392 . 0440 .0491 .0543 .0598 .0654 .0712 .0771 0.30 .0290 .0330 .0373 .0419 .0467 .0516 .0568 .0622 .0677 .0734

0.40 .0270 .0307 .0347 .0390 .0434 .0481 .0529 .0579 .0630 .0683

d;50 .0245 .0280 .0316 .0355 .0395 .0438 .0481 .0527 .0574 .0622

0.60 . .0218 .024^ .0281 .0315 .0351 .0388 .0427 .0468 .0509 .0552

0.70 .0188 .0214 .0242 .0272 .0303 .0335 .0369 .0404 .0440 .0477

0.80 .0157 .0179 .0202 .0227 .0253 .0280 .0308 .0337 .0367 .0398

0.90 .0126 .0144 .0163 .0182 .0203 .0225 .0248 .0271 .0295 .0320

1.00 .0096 .0110 .0124 .0139 .0155 .0172 .0189 .0207 .0226 .0244

xia ^ 1. 1000 1.1200 1. 1400 1. 1600 1.1800 1.2000 1.2200 1. 2400 1.2600 1.2800

0. 0. 0866 0. 0930 0. 0995 0. 1061 0.1128 0. 1195 0. 1262 0 1329 0.]396 0. 1463

0.10 .0857 .0921 .0985 .1051 .1117 .1183 . 1249 .1316 .1382 .1449

0.20 .0832 .0894 .0957 .1020 .1084 .1149 . 1214 . 1278 .1343 .1407 oao .0792 .0851 .0910 .0971 . 1032 . 1093 .1155 . 1217 .1279 .1340

0.40 .0737 .0792 .0848 .0904 .0961 .1019 .1076 .1134 .1192 . 1250

0.50 .0671 .0721 .0772 .0823 .0875 .0928 .0981 .1034 .1087 .1140

0.60 .0596 .0640 .0685 .0731 .0778 .0824 .0871 .0919 .0966 .1014

0.70 .0514 .0553 .0592 .0632 .0672 .0712 .0753 .0794 .0835 .0876

0.80 .0430 .0462 .0495 .0528 .0562 .0596 .0630 .0664 .0699 .0733

0.90 .0345 .0371 .0398 .0424 .0451 .0479 .0506 .0534 .0562 .0590

1.00 .0264 .0284 .0304 .0324 .0345 .0366 .0387 .0408 .0429 .0451

1.3000 1.3200 1.3400 1.3600 1.3800 1.4000 1.4200 1.4400 1.4600 1.4800

0. 0. 1529 0. 1595 0. 1660 0. 1724 0. 1787 0. 1850 0. 1911 0. 1971 0.2030 0.2088

O.IO .1514 . 1580 . 1644 . 1708 . 1771 .1833 . 1894 . 1953 . 2012 .2069

0.20 . 1471 . 1535 . 1698 . 1660 . 1722 .1783 .1842 . 1901 . 1958 .2014 0.30 .1402 . 1463 .1523 .1583 . 1642 . 1700 . 1758 . 1815 .1870 .1925

0;40 .1308 . 1365 .1422 .1478 . 1534 . 1589 . 1644 .1697 .1750 . 1802

0.50 . 1193 . 1245 .1298 .1350 .1401 . 1452 . 1503 . 1553 .1602 . 1650

0.60 . 1061 . 1108 .1155 .1202 . 1248 . 1294 . 1340 . 1385 .1430 . 1474

0.70 .0918 .0959 .1000 . 1041 . 1081 . 1122 .1162 . 1202 . 1241 .1280

0.80 .0768 .0803 .0837 .0872 .0906 .0940 .0974 . 1008 .1042 . 1075

0.90 .0618 .0646 .0674 .0702 . 0729 .0757 .0785 .0812 .0839 .0866

1.00 .0472 .0493 .0515 .0536 .0558 .0579 .0600 .0621 .0642 .0663

24
Table 9. Generalized temperature averages, 4'3, for a sphere—Continued

^9——

>

1.5000 1.5200 1.5400 1.5600 1.5800 l.fiOOO 1 .6200 1 .6400 1 6600 1.6800

0. 0. 2144 0. 2199 0. 2252 0. 2304 0. 2354 0. 2403 0. 2450 0. 2496 0. 2539 0. 2582

0.10 . 2125 . 2180 . 2233 . 2285 . 2335 . 2383 . 2430 . 2476 . 2520 .2562

0.20 . 2070 . 2123 . 2176 . 2227 . 2276 . 2325 . 2371 . 2417 . 2461 .2503

0.30 . 1978 . 2030 . 2082 . 2132 . 2180 . 2228 . 2274 . 2318 . 2362 .2404

0.40 . 1853 . 1903 . 1952 . 2000 . 2047 . 2093 . 2138 . 2182 . 2225 . 2266

0.50 . 1698 . 1745 .1792 .1«!37 .1882 .1925 .1968 .2010 . 2051 .2091 0 60 . 1518 . 1561 . 1604 . 1646 . 1687 . 1727 . 1767 1806 1845 1 flQO o!70 !l319 '. 1357 .1395 !l432 !l469 !l505 ] 1541 '. 1577 !l612 . 1646

0.80 .1108 .1140 . 1173 .1205 .1237 .1268 .1299 .1329 . 1360 . 1390

0.90 .0893 .0920 .0946 .0972 .0998 . 1024 . 1049 . 1074 . 1099 .1124

l.OO .0683 .0704 .0724 .0744 .0764 .0784 .0803 .0823 .0842 .0861

1.7000 1.7200 1 . 7400 1.7600 1.7800 1.8000 1.8200 1.8400 1 8600 1 8800

0. 0. 2623 0. 2662 0. 2699 0. 2735 0. 2770 0. 2803 0. 2835 0. 2865 0. 2893 0. 2921

0.10 . 2603 . 2642 . 2680 . 2716 . 2751 . 2784 . 2816 . 2846 . 2875 . 2903 O.20 . 2544 . 2583 . 2621 . 2658 . 2693 . 2727 . 2759 . 2790 . 2820 . 2849

0.30 . 2445 . 2485 . 2523 . 2560 . 2596 . 2630 . 2663 . 2695 . 2726 . 2756

0.40 . 2307 . 2346 . 2384 . 2421 . 2457 . 2492 . 2526 . 2559 . 2591 . 2621 0.50 . 2131 .2169 .2206 .2243 .2279 .2313 .2347 .2380 .2412 . 2443 0 60 . 1920 1956 . 1991 . 2026 . 2061 . 2094 . 2127 . 2159 2190 2221 0.70 !l680 !l713 ] 1746 '. 1778 !l809 '. 1840 '. 1871 '. 1901 !l931 '. I960

0.80 .1419 .1448 .1477 . 1505 .1533 .1561 .1588 . 1615 .1641 .1667

0.90 .1148 .1172 .1196 . 1220 .1243 .1266 .1289 .1311 .1333 .1355

1.00 .0880 .0898 .0916 .0935 .0953 .0971 .0988 . 1006 .1023 .1040

r/u 1.9000 1.9200 1.9400 1.9600 1.9800 2.0000 2.0200 2.0400

0. 0. 2947 0. 2971 0. 2995 0. 3017 0. 3038 0. 3058 0. 3077 0. 3094 0. 3111 0. 3127

0.10 . 2929 . 2954 . 2978 . 3001 . 3022 . 3042 . 3061 . 3079 . 3097 .3113

0.20 . 2876 . 2902 . 2927 . 2950 . 2973 . 2994 . 3015 . 3034 . 3052 .3070

0.30 . 2784 . 2811 . 2838 . 2863 . 2887 . 2910 . 2932 . 2953 . 2973 . 2992

0.40 . 2651 . 2679 . 2707 . 2734 . 2759 . 2784 . 2808 . 2831 . 2853 . 2875

0.50 .2473 .2503 .2531 .2559 .2586 .2612 .2638 .2662 .2686 . 2709

0.60 . 2251 . 2280 . 2309 . 2337 .2364 . 2391 2417 . 2443 . 2468 . 2492

0.70 '. 1988 .' 2016 !2044 '.2071 .2097 !2123 12149 !2174 !2198 !2222

0.80 .1693 . 1718 . 1743 . 1767 .1791 .1815 . 1839 . 1862 .1884 .1907

0.90 .1377 .1398 . 1419 . 1440 .1460 . 1481 .1501 . 1520 . 1540 . 1559

1.00 . 1056 . 1073 .1089 .1106 .1122 . 1137 . 1153 . 1168 .1184 .1199

^s,.^ xia 2.1000 2.1200 2.1400 2.1600 2.1800 2.2000 2.2200 2,2400 2.2600

0. 0. 3141 0. 3155 0. 3168 0. 3180 0. 3192 0. 3202 0. 3212 0. 3222 0. 3230 0. 3239

0.10 . 3128 .3142 . 3156 .3168 . 3180 . 3191 . 3202 . 3211 . 3221 . 3229

0.20 . 3086 .3102 . 3116 .3130 . 3143 . 3156 . 3168 . 3179 . 3189 . 3199

0.30 . 3011 .3028 . 3045 . 3061 . 3076 . 3090 . 3104 . 3117 . 3129 .3141

0.40 . 2895 . 2915 . 2934 . 2952 . 2969 . 2986 . 3002 . 3018 . 3033 . 3047

0.50 .2732 .2753 .2775 .2795 .2815 .2834 .2852 .2870 .2888 .2904 0 60 2515 . 2539 2561 2583 2604 2625 . 2645 . 2665 . 2685 . 2703

o!70 !2246 '. 2269 !2292 !2314 !2336 !2357 !2378 !2399 !2419 !a438

0.80 .1929 . 1950 . 1972 .1993 .2014 .2034 .2054 .2074 .2093 .2112

0.90 . 1578 .1597 .1615 .1634 .1652 .1670 .1687 . 1704 . 1722 . 1738

1.00 .1214 .1228 .1243 . 1257 .1271 . 1285 .1299 .1313 . 1326 .1340

2.3000 2.3200 2.3400 2.3600 2.3800 2.4000 2.4200 2.4400 2.4600 2,4800

0. 0. 3246 0. 3253 0. 3260 0. 3266 0. 3271 0. 3276 0. 3281 0. 3286 0. 3290 0. 3294

O.IC .3237 .3244 .3251 .3258 .3264 .3269 .3274 .3279 .3284 .3288

0.20 .3208 .3216 .3225 .3232 .3239 .3246 .3252 .3258 .3263 .3269

0.30 .3152 .3163 .3173 .3182 .3191 .3200 .3208 .3215 .3223 .3229

0.40 .3060 .3073 .3086 .3098 .3109 .3120 .3131 .3141 .3150 .3159

0.50 .2920 .2936 .2951 .2966 .2980 .2994 .3007 .3019 .3032 .3043

0.60 .2722 .2739 .2757 .2774 .2790 .2806 .2822 .2837 .2852 .2866

0.70 .2458 .2477 .2495 .2514 .2531 .2549 .2566 .2583 ,2599 .2615

0.80 .2131 .2150 .2168 .2186 .2204 .2221 .2238 . 2255 .2272 .2288

0.90 . 1755 .1772 . 1788 .1804 . 1820 .1835 . 1851 . 1866 ,1881 .1896

1.00 . 1353 .1366 .1379 .1392 . 1404 .1417 . 1429 .1441 ,1453 .1465

2.5000 2.5200 2. 5400 2. 5600 2. 5800 2. 6000 2. 6200 2. 6400 2.6600 2.6800

0. 0. 3297 0. 3300 0. 3303 0. 3306 0. 3308 0. 3311 0. 3313 0. 3315 0. 3316 0. 3318

0.10 .3292 .3295 .3298 .3301 .3304 .3307 .3309 .3311 .3313 .3315

0.20 .3273 .3278 .3282 .3286 .3289 .3293 .3296 .3299 .3301 .3304

0.30 .3236 .3242 .3248 . 3253 .3258 .3263 .3267 .3272 .3276 .3279

0.40 .3168 .3176 .3184 .3192 .3199 .3206 .3212 .3219 .3225 .3230

0.50 .3055 .3066 .3076 .3087 .3096 .3106 .3115 .3124 .3132 .3141

0.60 .2880 .2894 .2907 .2920 .2932 .2944 .2956 .2968 .2979 .2990

0.70 .2631 .2646 .2662 .2676 .2691 .2705 .2719 .2733 .2746 .2759

0.80 .2304 .2320 .2336 .2351 2366 .2381 .2396 .2410 .2424 .2438

0.90 .1911 . 1925 . 1940 .1954 .1968 .1981 .1995 .2008 .2022 .2035

1.00 .1477 .1488 . 1500 .1511 .1522 .1533 .1544 .1555 .1566 .1576

25
Table 9. Generalized temperature averages, ^3, for a sphere—Continued

g 2. 7000 2. 7200 2. 7400 2. 7600 2. 7800 2. 8000 2. 8200 2. 8400 9 Rfino

1

0. 0. 3319 0. 3321 U. 00/2 U. OOZO 0. 3324 0. 3325 0. 3326 0. 3327 0. 3327 0. 3328

0.10 . 3317 . 3318 . 3320 . 3321 . 3322 . 3323 . 3324 . 3325 .3326 .3326

0.20 . 3306 . 3308 . 3310 . 3312 . 3314 . 3315 . 3317 . 3318 .3320 .3321

0.30 . 3283 . 3286 . 3289 . 3292 . 3295 . 3298 . 3300 . 3302 .3304 .3306

0.40 . 3236 . 3241 . 3246 . 3250 . 3255 . 3259 . 3263 . 3267 .3270 .3274

0.50 .3148 .3156 .3163 .3170 .3177 .3184 .3190 .3196 .3202 .3207

0.60 . 3000 . 3011 . 3021 . 3030 . 3040 . 3049 . 3058 . 3066 3075 . OUoO

o!70 '. 2772 .2784 .2797 .2809 .2820 !2832 !2843 ,' 2854 !2865 .2876

0.80 .2452 .2466 .2479 .2492 .2505 .2518 . 2530 .2543 .2555 .2567

0.90 .2048 .2061 .2073 .2086 .2098 .2110 .2122 .2134 .2146 .2157

1.00 .1587 . 1597 .1607 . 1617 . 1627 .1637 .1647 .1657 .1666 .1676

g 2. 9000 2. 9200 2. 9400 2. 9600 2. 9800 3. 0000 3. 0200 3. 0400 3 ofino

0. 0. 3328 U. 3329 0. 3o2y 0. 3330 0. 3330 0. 3331 0. 3.331 0. 3331 0. 3331 0. 3332

0.10 . 3327 . 3328 . 3328 . 3329 . 3329 . 3330 . 3330 . 3330 .3331 .3331

0.20 . 3322 . 3323 . 3324 . 3324 . 3325 . 3326 . 3327 . 3327 . 3328 .3328

0.30 . 3308 . 3310 . 3312 . 3313 . 3315 . 3316 .3317 . 3318 .3319 .3320

0.40 . 3277 . 3280 . 3283 . 3286 . 3289 . 3291 . 3293 . 3296 .3298 .3300

0.50 .3212 .3218 .3223 .3227 .3232 .3236 .3240 .3244 .3248 .3252

0.60 . 3091 . 3098 .3106 .3113 . 3120 . 3127 . 3134 . 3140 . 3146 . 3152 070 .2886 .2896 .2906 .2916 .2925 .2934 .2944 .' 2952 .'2961 !2970

0.80 .2578 .2590 .2602 .2613 .2624 .2635 . 2645 .2656 .2666 .2677

0.90 .2169 .2180 .2191 .2203 .2213 .2224 . 2235 .2245 .2256 .2266

1.00 . 1685 .1694 .1703 .1713 . 1722 . 1730 .1739 .1748 .1756 .1765

^.^ n 3. 0000 3. 1000 3. 2000 3. 3000 3. 4000 3. 5000 3. 6000 3. 7000 0. ouuu 0. yUULT

0. 0. 3331 0. 3332 0. 3332 n 0000 n 0000

0. 00000. 0. 3333 0. 3333 0. 3333 0. 3333

0.10 . 3330 . 3331 . 3332 . 3333 . 3333 . 3333 . 3333 .3333 .3333 .3333

0.20 . 3326 . 3329 . 3330 . 3332 . 3332 . 3333 . 3333 . 3333 . 3333 .3333

0.30 . 3316 . 3321 . 3325 . 3328 . 3330 . 3331 - 3332 . 33.32 .3333 .3333

0.40 . 3291 . 3302 . 3310 . 3316 . 3321 . 3324 . 3327 . 3328 . 3330 .3331

0.50 .3236 .3256 .3271 .3284 .3294 . 3303 .3309 .3314 .3319 ,3322

0.60 . 3127 . 3158 . 3185 . 3208 . 3228 . 3244 . 3259 . 3271 . 3281 3290

o!70 .' 2934 .2978 .3017 . 3052 .3083 .3112 .3137 !3159 !3179 !3197

0.80 .2635 .2687 .2735 .2779 .2820 .2859 . 2894 .2927 .29.57 .2985

0.90 .2224 .2276 .2326 .2372 .2416 . 24.57 .2497 . 2534 .2569 .2603

1.00 . 1730 . 1773 . 1814 . 1853 .1890 . 1925 . 1959 .1991 .2022 .2051 \-

4. 0000 4. 1000 4. 2000 4. 3000 4. 4000 4. 5000 4. 6000 4. 7000 4. 8000 4 9000

0. 0. 3333 0. 3333 n 0000

0. 3333 n 0000

0. -iSSi n 00000. S6sS n 00000. 3333 0. 3333 0. 3333 0. 3333 0. 3333

0.10 . 3333 . 3333 - 3333 . 3333 . 3333 . 3333 . 3333 . 3333 .3333 .3333

0.20 . 3333 . 3333 . 3333 . 3333 . 3333 . 3333 . 3333 . 3333 . 3333 .3333

0.30 .3333 . 3333 . 3333 . 3333 . 3333 . 3333 . 3S33 . 3333 . 3333 .3333

0.40 . 3331 . 3332 . 3332 . 3333 . 3333 . 3333 . 3333 . 3333 . 3333 .3333

0.50 .3325 .3327 .3328 .3329 .3330 .3331 .3332 .3332 .3332 .3333

0.60 . 3297 . 3303 3308 . 3313 . 3316 . 3319 3332 . 3324 . 3326 3327

o!70 !3213 !3227 [3239 '3251 !326I '. 3270 !3277 !3284 !3290 !3296

0.80 .3011 .3036 .3058 .3079 .3098 .3116 .3133 . 3148 .3162 .3175

0.90 .2634 .2665 .2693 .2721 .2747 .2771 .2795 . 2818 .2839 .2859

1.00 .2079 .2106 .2132 .2156 .2180 .2203 .2225 .2246 .2266 .2286

-

^ \. S 0000 5. 1000 5. 2000 5. 3000 5. 4000 5. 5000 5. 6000 5. 7000 0.^ 8000 ouuu 0. auuu

0. 0. 3333 0. 3333 0. 3333 0. 3333 0. 3333 0. 3333 0. 3333 0. 3333 0. 3333 0. 3333

0.10 . 3333 . 3333 . 3333 . 3333 . 3333 . 3333 . 3333 . 3333 . 3333 . 3333

0.20 . 3333 . 3333 . 3333 . 3333 . 3333 . 3333 . 3333 . 3333 . 3333 .3333

0.30 . 3333 .3333 . 3333 . 3333 . 3333 . 3333 . 3333 . 3333 .3333 . 3333

0.40 . 3333 . 3333 . 3333 . 3333 . 3333 . 3333 . 3333 . 3333 . 3333 .3333

0.50 .3333 .3333 .3333 .3333 .3333 .3333 . 3333 .3333 .3333 .3333

0 60 . 3328 . 3329 3330 . 3331 3331 3332 3332 3332 . 3332 . oooo

0.70 .' 3300 .' 3305 .' 3308 !3312 .' 3314 .' 3317 .' 3319 !3321 .'3323 . 3324

0.80 .3188 .3199 .3209 .3219 .3228 . 3236 . 3244 .3251 .3257 .3263

0.90 .2879 .2897 .2915 .2932 .2949 .2964 .2979 .2993 .3007 .3020

1.00 .2305 .2323 .2341 .2358 .2374 .2390 .2406 .2420 .2435 .2449 \

fi 0000 6. 1000 6. 2000 6. 3000 6. 4000 6. 5000 6. 6000 6. 7000 6. 8000 f»u. Qnnn yuliu

0. 0. 3333 0. 3333 0. 3333 0. 3333 0. 3333 0. 3333 0. 3333 0. 3333 0. 3333 0. 3333

0.10 .3333 . 3333 .3333 . 3333 . 3333 . 3333 . 3333 . 3333 . 3333 .3333

0.20 .3333 . 3333 . 3333 . 3333 .3333 . 3333 . 3333 . 3333 . 3333 . 3333

0.30 .3333 .3333 . 3333 . 3333 .3333 . 3333 . 3333 . 3333 . 3333 .3333

0.40 .3333 .3333 .3333 . 3333 .3333 . 3333 . 3333 . 3333 .3333 .3333

0.50 .3333 .3333 .3333 .3333 .3333 .3333 .3333 .3333 .3333 .3333

0 60 . 3333 t'i'i'i. ^000 3333 3333 3333 . 3333 . 3333 3333 3333 . 3333

o!70 .3326 .3327 !3328 !3329 '. 3329 !3330 !3330 '3331 !3331 .3331

0.80 .3269 .3274 .3279 .3283 .3288 .3291 .3295 .3298 .3301 .3304

0.90 .3033 .3045 .3056 .3067 .3077 .3088 .3097 .3107 .3116 .3124

1.00 .2462 .2475 .2488 .2501 .2513 .2525 .2536 .2547 .2558 .2568

26

1

I
Table 9. Generalized temperature averages, \p3, for a sphere—Continued

\5 7.0000 7. 1000 7.2000 7.3000 7.4000 7.5000 7.6000 7. 7000 7.8000 7. 9000

0. u. aooo \J. OOOO U. OOOO U. OOOO 0 3333 U. OOOO U. OOOO U. OOOO U. OOOO U. OOOO

0-10 3333 3333 . 3333 3333 3333 . OOOo . 3333 . 3.333 . 3333 . 3333 0.^ . oooo 3333 3333 3333 3333 . OOOO . 3333 . 3333 . 3333 . 3333

0-30 . oooo 3333 . 3333 3333 3333 . OOOO . 3333 . 3333 . 3333 . 3333

0.40 . oooo 3333 3333 3333 . OOOO . 3333 . 3333 . 3333 . 3333 . 3333

0.50 .3333 !3333 !3333 13333 .3333 .3333 .3333 .3333 .3333 .3333

0.60 .3333 .3333 .3333 .3333 .3333 .3333 .3333 .3333 .3333 .3333

0.70 .3332 .3332 .3332 .3332 .3333 .3333 .3333 .3333 .3333 .3333

0.80 .3306 .3309 .3311 .3313 .3314 .3316 .3317 .3319 .3320 .3321

0.90 .3132 .3140 .3148 .3155 .3162 .3169 .3175 .3182 .3188 .3193

1.00 .2578 .2588 .2598 .2607 .2616 .2625 .2634 .2643 .2651 .2659

\? 8.0000 8. 1000 8.2000 8.3000 8.4000 8.5000 8.6000 8. 7000 8.8000 8.9000

0. U- OOOO U. OOOO U- OOOO U. OOOO U. OOOO nU. ^^^^ OOOO U- OOOO U. 0060 U. OOOO U. OOOO

0.10 . OOOO 3333 3333 3333 3333 3333 . OOOO . 3333 . 3333 . 3333

0.20 . OOOO . OOOO 3333 3333 3333 . OOOO . OOOO . 3333 . 3333 . 3333

0.30 . OOOO . oooo 3333 3333 . OOOO . OOOO . OOOO . 3333 . 3333 . 3333

0.40 . OOOO . OOOO 3333 3333 . OOOo . OOOO . OOOO . 3333 . 3333 . 3333

0.50 .3333 .3333 !3333 !3333 .3333 .3333 .3333 .3333 .3333 .3333

0.60 .3333 .3333 .3333 .3333 .3333 .3333 .3333 .3333 .3333 .3333

0.70 .3333 .3333 .3333 .3333 .3333 .3333 .3333 .3333 .3333 .3333

0.80 .3322 .3323 .3324 .3325 .3326 .3327 .3327 .3328 .3328 .3329

0.90 .3199 .3204 .3209 .3214 .3219 .3223 .3227 .3232 .3236 .3240

1.00 .2667 .2675 ,2682 .2690 .2697 .2704 .2711 .2718 .2724 .2731

9.0000 9.1000 9.2000 9.3000 9.4000 9.5000 9.6000 9. 7000 9.8000 9.9000

u.

ft QQQ^ U. OOOO U. OOOO U. OOOO U. OOOO U. oOt^W U. OOOO U. OOOO U. OoOO U. ooOo

0.10 . 3333 . OOOO . OOOO . OOOO . OOOO . OOOO . OOOO . OOOO . OOOO . 3333

0.20 . 3333 . OOOO . OOOO . OOOO . OOOO . OOOO . OOOO . OOOO . 3333 . 3333

0.30 . 3333 . 3333 . OoOO . OOOO . oOOo . OOOO . OOOO . 3333 . 3333 . 3333- 0.40 - 3333 . 3333 . Oooo . OOOO . OOOO . 3333 . 3333 . OOOO . 3333 . 3333

0..50 .3333 .3333 .3333 .3333 .3333 .3333 .3333 .3333 .3333 .3333

0.60 .3333 .3333 .3333 .3333 .3333 .3333 . 3333 .3333 . 3333 .3333

0.70 .3333 .3333 .3333 .3333 .3333 .3333 .3333 .3333 .3333 .3333

0.80 .3329 .3330 .3330 .3330 .3331 .3331 .3331 .3331 .3332 .3332

0.90 .3243 .3247 .3250 .3254 .3257 .3260 .3263 .3266 .3268 .3271

1.00 .2737 .2743 .2749 .2755 .2761 .2767 .2773 .2778 .2783 .2789

10.0000 10. 1000 10.2000 10.3000 10.4000 10.5000 10. 6000 10. 7000 10. 8000 10. 9000

0. 0.3333 0.3333 0.3333 0.3333 0.3333 0.3333 0. 3333 0.3333 0. 3333 0. 3333

0.10 .3333 .3333 .3333 .3333 .3333 .3333 .3333 .3333 .3-333 .3333

0.20 .3333 .3333 .3333 .3333 .3333 .3333 .3333 .3333 .3333 .3333

0.30 .3333 .3333 .3333 .3333 .3333 .3333 .3333 .3333 .3333 .3333

0.40 .3333 .3333 .3333 .3333 .3333 .3333 .3333 .3333 .3333 .3333

0.50 .3333 .3333 .3333 .3333 .3333 .3333 -3333 .3333 .3333 .3333

0.60 .3333 .3333 .3333 .3333 .3333 .3333 .3333 .3333 .3333 .3333

0.70 .3333 .3333 .3333 .3333 .3333 .3333 .3333 .3333 .3333 .3333

0.80 .3332 .3332 .3332 .3333 .3333 .33.33 .3333 .3333 .3333 .3333

0.90 .3273 .3276 .3278 .3280 .3283 .3284 .3286 .3288 .3290 .3292

1.00 .2794 .2799 .2804 .2809 .2814 .2819 .2823 .2828 .2832 .2837

4. References

[1] J. Hopkinson, Messenger of Math. 8, 168 (1879).

[2] E. D. WUliamson, J. Wash. Acad. 9, 209 (1919).

[3] E. D. WiUiamson and L. H. Adams, Phys. Rev. 14, 99 (1919).

[4] L. H. Adams and E. D. Williamson, J. Frank. Inst. 190, 597, 8.35 (1920).

[5] R. S. Woodward, Phys. Rev. 16, 176 (1903).

[6] A. L. Queneau, School of Mines Quart, of Columbia

Univ. 23, 181 (1902).

[7] C. F. Lucks et al., A. F. Technical Report Xo. 6145, Part I (February 1951); Part II (Julv 1952); Part

III (March 1954).

[8] S. Spinner, J. Am. Ceram. Soc. 39, 113 (1956).

[9] C. J. Smithells, Metals Reference Book, 2 (Inter- science Publishers Inc., New York, N.Y., 1955).

Washington, D.C., April 21, 1959.

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