Tables of Derivatives for
Damped Vibrations
W. E. MILNE, Ph.D.
Professor of Mathematics
Oregon State College
Oregon State College
CORVALLIS
OREGON STATE MONOGRAPHS
Mathematics and Statistics Series
Number 1. December 1935
Published by Oregon State College, Oregon
State System of Higher Education
Corvallis, Oregon
TABLE OF CONTENTS
Page
Introduction
1. Relation to Tables of Damped Vibrations
2. Necessity for Tables of Derivatives
3. Two-Way Interpolation
4. Simultaneous Inverse Interpolation
5. Table XVII
Tables
Table XV
Table XVI
Table XVII
5
3
5
6
7
12
17
17
40
45
Tables of Derivatives for
Damped Vibrations*
By W. E. MILNE
INTRODUCTION
1. Relation to Tables of Damped Vibrations. The tables herewith presented are designed to supplement the Tables of Damped
Vibrations which appeared as University of Oregon Publication,
Mathematics Series, vol. 1, no. 1, March, 1929. The latter tables give
the values of u in terms of x for certain solutions of the differential
equation
(1)
d2u
dx2
4-2 ( dudx)2
1
2b du
dx
u = 0,
together with special tables giving the values of u and x at the maxima, minima, and points of inflection. The purposes and use of the
tables are discussed in the introduction to that publication, and need
not be treated again here.
The present set of supplementary tables gives the values of
du/dx for the same solutions of (1) as are tabulated in Tables of
Damped Vibrations, so that Table XV of this publication is to be
used with Table IV of the former, and Table XVI with Table V.
2. Necessity for Tables of Derivatives. The need for a table of
derivatives to be used in conjunction with the tables already published arises from the fact that each particular solution of (1) requires two conditions for its determination. Thus for example if
the initial values of it and du /dx are given, the corresponding solution is uniquely determined but cannot in fact be found directly from
tables that give the value of u alone, since any given value of u will
occur in many places in the table. This difficulty is not encountered if
it is known that the initial point is maximum, minimum, or point of
inflection, as the special tables already published enable one to locate
the proper particular solution. The specific need for the tables of
derivatives was encountered in an attempt to apply the Tables of
* The preparation and publication of these tables is made possible through grants from
the Joseph Henry Fund of the National Academy of Sciences, and from the General
Research Council of the Oregon State System of Higher Education.
I am happy to express to Professor W. F. Durand my sincere thanks for his assistance
and encouragement.
[5]
OREGON STATE MONOGRAPHS
Damped Vibrations to the problem of the surge tank with a throttling ring, where the differential equation controlling the water level
in the tank can be reduced to equation (1) by appropriate transformations, but where the initial point is not the point of inflection, as is
the case in the absence of the throttling ring. It is hoped that the
publication of the Tables of Derivatives will greatly extend the value
and usefulness of the Tables of Damped Vibrations.
3. Two-Way Interpolation. Since problems involving the combined use of Tables IV and XV or of Tables V and XVI will necessarily lead to a considerable amount of two-way interpolation, it
will be convenient to use a systematic method for performing such
interpolations. Among the many formulas available the one given
below is simple in operation and sufficiently accurate for most of the
tables.*
In the diagram (2) below is shown a section of a table (for
definiteness consider Table IV) in which the independent variables
are x and a, and the dependent variable is u. Let the interval between
successive entries of x be denoted by h, and the interval between suc-
cessive entries of a be denoted by k, so that x1xox2 x1 =h,
thaoa2a1 =k, etc.
(2)
(to
a1
a2
xo
U00
1101
UO2
x1
U10
U11
X2
U20
The entries in the table which are not used in the interpolation are
indicated by dashes. From the values above we form the following
differences :
D
uto
/too
Da =it 01 ttoo
(3)
D xx= u
2u10 + u00
Dxa
uji+ u
D act
u 02
uo
2u o1+ u
* In Tables V and XVI interpolation will not be accurate when the value of u at the
head of the column is nearly equal to 21)=. In such cases graphic methods are probably to be
preferred.
TABLES OF DERIVATIVES FOR DAMPED VIBRATIONS
7
Now if u represents the desired value corresponding to a pair of
values x and a, where ..1-0 <x<
mula may be written in the form
(4)
u = uoo
ao < a <ai, the interpolation f or-
Dx(x-x0)/h+ Da(a-ao)/k
Dxx(x-x0)(x-xi)/2h2+D.(x-x0)(a-a0)/hk
D.(a-ao)(a-a1)/2k2.
EXAMPLE 1. From Table IV let us find u corresponding to b=0.5,
x=1.357, a=-1.568. The values taken from Table IV are
a0=1.5
a1=1.6
a2=1.7
.8112
.7424
.6753
.8692
.7965
.9277
x0 =1.3
x1 =1.4
x2=1.5
Then we have from (3)
D =-.0688
Da= .0580
Dxx= .0017
D
xa
(x-xo)/11
(a-a0)/k
(x-x) (x-x1)/2h2
(x-x) (a-ao)/hk
=-.0039
Das= .0005
(a-a0) (a-a1)/2k2
=
=
.57
.68
-.123
=
=
.388
-.109,
whence by direct substitution into (4) we obtain u = 0.8096. ANS.
4. Simultaneous Inverse Interpolation. When values of 14, and
du/dx are given and we desire to find the corresponding values of
x and a from Tables IV and XV we encounter a sort of simultaneous
inverse interpolation which is not generally familiar. We shall therefore explain the procedure in considerable detail by means of numerical examples.
du/dx=-0.6; to find a and x.
EXAMPLE 2. Given b=0.4,
It is convenient to divide the work into steps, as follows :
First Step. Approximate location of a and x by graphic method. Turning
to Table IV for b=-0.4, we take a value of x, say x=0.7, and by ordinary linear
interpolation we find the approximate value of a at which u=0.8. In this case
we find a=0.989. Next we take x=0.8 and again find the value of a for which
u=0.8, and proceeding in this manner we obtain a table of pairs of values of x
and a for which u=0.8, thus
x=
a=
0.7
.989
0.8
0.9
1.0
1.1
1.2
1.051
1.125
1.210
1.309
1.424
Next turning to Table XV we find a similar set of values of x and a which
give du/d1=-0.6. thus
x=
a=
0.7
0.8
0.9
1.0
1.1
1.2
1.308
1.231
1.181
1.151
1.138
1.138
From these data we readily plot the two curves shown in Figure 1.
OREGON STATE MONOGRAPHS
8
a
1.4
1.3
1.2
1.1
10
0.9
0.8
0.7
0.8
1.0
0.9
1.1
1.2
1.3
FIGURE 1
The coordinates of the point of intersection, which are of course the desired
values of x and a, are seen to be approximately x=0.945, a=1.165. In carrying
out the above work it is obviously desirable to do some preliminary scouting
in Tables IV and XV to see where the values of x should be chosen so that a
small number of points will locate the intersection. It is not really necessary to
take as many as were taken in the example above.
If we are not much concerned with accuracy in the third decimal place, the
values already obtained may be regarded as satisfactory and we need go no
farther. If we desire greater accuracy we may proceed as follows :
Second Step. Direct check of first approximation. We now calculate the
values of u and du/dx corresponding to x-=0.945, a=1.165, that is, corresponding to the values given by the first approximation. The discrepancy between
the values of u and du/dx thus found and the desired values u=0.8, dit/dx=0.6
will serve to indicate the error of the first approximation. The calculation is
performed by means of (4), and appears as follows :
From Table IV
a=1.1
a=1.2
a=1.3
x= .9
x=1.0
x=1.1
whence by (4)
.7822
.7253
.6673
u=
.8547
.7931
.7822X
.0569X
+.0725 X
1
.45
.65
-.0011 X -.123
-.0047 X
and u=-0.8024.
.293
+.0003 X -.113
.9275
TABLES OF DERIVATIVES FOR DAMPED VIBRATIONS
From Table XV
du
.5622X
+.0136 X
+.0465 X
dx
.6546
.6087
.6228
.5622
.5758
.5823
x=1.0
x=1.1
whence by (4)
a=1.3
a=1.2
a=1.1
x= .9
9
1
.45
.65
-.0071 X -.123
+.0005 X
.293
-.0006X -.113
and du/dx-=-.5996. Thus the values x=0.945, a=1.165 give u=0.8024,
du/dx=-0.5996 instead of u=0.8000 and du/dx=---- 0.6000, as desired.
Third Step. The second approximation. We now use the first three
terms only of formula (2), taking xo = 0.945, a1 = 1.165, u, = 0.8024,
-(du/dx)o=0.5996, u=0.8, -du/dx=0.6, regarding x and a as unknowns to
be determined. For the differences u10-u00, etc., we use the same values as in
the second step. In this manner we obtain the two simultaneous equations
a-1.165
x-0.945
),
)+.0725(
0.8 = 0.8024-.0569(
0.1
0.1
0.6
= 0.5996-.0136(
or
x-0.945
569X
)+.0465(
a-1.165
),
0.1
0.1
725A = 24,
136X + 465A = 4,
where X =
x-0.945
0.1
,A -=
a-1.165
Solving for X and A we obtain
0.1
X = 0.387, A --= -0.0027.
Then
x = 0.945 + 0.1X = 0.94887,
a = 1.165 + 0.1A = 1.16473.
Inasmuch as the figures in the fifth decimal place are unreliable, owing to the
fact that the tables themselves are good to four places only, we take as the
final solution of the problem
x = 0.9489,
a = 1.1647.
To show how the work may be systematically and compactly
arranged we present a second example, omitting the detailed explanations.
EXAMPLE 3. Given b = 0.6, u = 1.00, du/dx = -0.7; to find a and x:
From Table IV
x = 1.0
a ---= 1.436
From Table XV
x = 1.0
a = 1.605
1.1
1.530
1.2
1.3
1.634
1.750
1.1
1.2
1.3
1.605
1.622
1.654
OREGON STATE MONOGRAPHS
10
a
= 1.0
1.7.
du
dx
= -0.7
1.6.
1.5
1.4
1.0
1.1
1.2
1. 3
FIGURE 2
From the graph a=1.620, x=--1.187.
From Table IV
x=1.1
x=1.2
x=1.3
a=1.6
a=1.7
1.0474
.9778
.9092
a=1.8
1.1153
1.0420
1.1835
From (4) for x=1.187, a=1.620, we have
= 1.0474-00696) (.87) +(.0679) (.20) +(.0010) (-.056)
(.0037) (.174) +(.0003) (-.08) = 0.99970.
From Table XV
a=1.6
a=-1.7
a-=1.8
x=1.2
x=-.1.3
.6989
.6918
.6802
.7364
.7294
.7742
From (4) for x=1.187, a=1.620, we have
du/dx = . 6989 (.0071) (.87)+(.0375) (.20)- (.0045) (-.056)
+(.0001) (.174) + (.0003) (-.08) = .70047.
TABLES OF DERIVATIVES FOR DAMPED VIBRATIONS
Then
11
696X + 679A = 3.0,
71X + 375A = 4.7,
whence X = .0203, A = .0164, and x = 1.1850, a = 1.6184.
Checking these values in (4) we get
u = 1.00001,
du
dx
=.70001.
The inverse interpolation in Tables V and XVI is quite similar,
but it must be remembered that the values of x start at the point of
inflection and that the value of u given at the head of each column
is uo, the ordinate at the point of inflection. Thus uo plays the same
role in Tables V and XVI as a does in Tables IV and XV.
EXAMPLE 4. Given b=0.7, u=0.03, du/dx = --0.17 to find u0 and x:
From Table V
u0= .2
.3
.4
.6
.5
1.377
1.449
1.493
1.497
x =1.423
Note that we select values of u0 and interpolate to find x which gives us
u=0.03. This is done because here the value of x is changing much more slowly
than the value of u0.
From Table XVI
x = 1.3
1.4
1.5
u0=0.406
0.446
0.498
0
.6.
.5.
.4
.3,
.2.
1.2
1.3
1.4
FIGURE 3
1.5
12
OREGON STATE MONOGRAPHS
From the figure we have x=1.463,240=0.475.
From Table V
x=1.4
x=1.5
x=1.6
u=.4
u=.5
u=.6
.0434
.0290
.0162
.0389
.0208
.0244
From (4) for x=1.463, u=0.475, we have
u = .0434+(- .0144) (.63) + (-.0045) (.75) + (.0016) (-.116)
+ (-.0037) (.473) + (.0100) (-.093)
= 0.02995.
From Table XVI
x=1.4
x=1.5
x=-1.6
u=.4
u=.5
u=.6
.1515
.1361
.1211
.1913
.1706
.2319
From (4) for x=1.463, u=0.475, we have
-du /dx = 0.1515+(- .0154) (.63) + (.0398) (.75) + (.0004) (-.116)
+ (-.0053) (.473) + (.0008) (-.093)
= 0.16902.
Then
+.0144 X + .0045 U = -.00005,
+.0154 X
.0398 U = -.00098.
whence X = -.0100, U = .0208, and we have
x = 1.4620,
u = 0.4771.
5. Table XVII. In order to provide a complete solution of the
surge tank problem the additional Table XVII has been calculated.
This table supplies values of u for the case not covered by Table V ;
namely, the case in which the level in the surge tank is falling from a
maximum following an upward surge. The initial values of u chosen
are the final values of u in Table IV, since Table XVII will always
immediately follow Table IV in the construction of a curve showing
water level plotted against time. The following examples are given
to illustrate the use of the tables in tracing the course of the surges.
In order not to divert the reader's attention from essential principles,
the numbers have been chosen so as to avoid cumbersome interpolation
rather than to represent actual engineering problems.
TABLES OF DERIVATIVES FOR DAMPED VIBRATIONS
13
Let H =head at any time (measured from static level
to level in surge tank, and positive when level
in surge tank is below static level).
Hoinitial head.
Hifinal head (after oscillations subside).
P
LA
. (See introduction to Tables of Damped
2gkF
Vibrations.)
Then we have the following relationships:
(1) b2---H1 /2P.
(2) uo
HoH, , in case of decreased demand.
(3) uo
HiHo , in case of increased demand.
P
(4) H=Hi+Pu, on rising surge. (u taken from Table IV.)
(5) H= H1 Pu, on falling surge. (u taken from Table V or
XVII.)
Let us assume that in a problem to which the tables are to be applied
we know both the initial level and the final level. It will now be
shown how the complete course of the oscillations can be traced by
means of the tables.
EXAMPLE 5. Given a surge tank installation such that P=30.5. Suppose
Ho=27.20 ft., H1 =9.75 ft.; to determine the course of the oscillations.
9.75
From equation (1) we have b2 = , whence b = 0.39975. We may take
61
b = 0.4 with sufficient accuracy for our purpose.
From equation (2) we have uo =
27.2-9.75
= .5721. Entering Table VI
30.5
with b=0.4, u0=.5721, we find a=1.0005. For simplicity let us take a=1.0.
We now turn to Table IV for b=0.4, a=1.0, and plot the value of H, using
equation (4), as shown in Figure 4. The portion of the curve AB represents the
rising surge, starting from the point of inflection A. Upon reaching the point B
we continue the curve by the use of Table XVII, which we enter with the values
b=0.4, a=1.0. The value of x in Table XVII is of course measured from E. Upon
reaching the point C we return to Table IV, which we enter with the values b=0.4,
OREGON STATE MONOGRAPHS
14
CF=a=0.045. The section CD is thus obtained from Table IV by interpolation.
The time t corresponding to any point on the graph may be obtained from the
formula
t=
L.P gA x.
Figure 5 gives the graph of a similar problem with final level close to static
level, showing that the surges are much intensified.
EXAMPLE 6. Let P=40, Ho=0, H1 =20 ft. Determine the motion.
Here we find
b2 =
20
1
=
u=
20
= 0.5.
4
2.40
= 0.5.
40
We therefore enter Table V with b=0.5, uo=0.5 to plot the initial downward
surge, which appears as AB in Figure 6. From B to C we use Table IV, and from
C to D Table XVII just as in the previous examples.
Statio Level
Final Level
C
Initial Level
/
Table IV
Table XVII
Table IV
FIGURE 4
P=30.5, H0=27.20 ft., H1=9.75 ft.
TABLES OF DERIVATIVES FOR DAMPED VIBRATIONS
B
Static Level
Final Level
Initial Level
V
Table IV
Table IVII
Table IV
'
FIGURE 5
P=100, 1-10=41 ft., H1 =2 ft.
15
16
OREGON STATE MONOGRAPHS
Static, Level
Table IV
1
Table XVII
FIGURE 6
P=40, Ho=0, H1 =20
ft.
TABLES
Table XV
Values of du /dx
b=.1
b=.1
a=.1
a=.2
a=.3
a=.4
a=.5
a.=.6
a=.7
a=.8
a=.9 a=1.0
.0000
.0099
.0196
.0286
.0373
.0454
.0000
.0198
.0389
.0572
.0745
.0906
.0000
.0296
.0583
.0857
.1114
.1353
.0000
.0395
.0777
.1141
.1482
.1797
.0000
.0494
.0971
.1424
.1848
.2238
.0000
.0593
.1164
.1706
.2212
.2675
.0000
.0691
.1357
.1988
.2574
.3110
.0000
.0790
.1550
.2269
.2935
.3641
.0000
.0888
.1743
.2548
.3294
.3969
.0000
.0987
.1935
.2828
.3650
.4393
.7
.8
.9
1.0
.0529
.0597
.0657
.0709
.0752
.1053
.1185
.1302
.1402
.1485
.1571
.1765
.1935
.2080
.2198
.2083
.2337
.2557
.2743
.2893
.2590
.2901
.3169
.3392
.3571
.3092
.3457
.3769
.4028
.4232
.3585
.4006
.4360
.4651
.4877
.4080
.4547
.4941
.5261
.5508
.4566
.5081
.5512
.5859
.6124
.5047
.5608
.6075
.6446
.6726
1.1
1.2
1.3
1.4
1.5
.0787
.0814
.0831
.0840
.0840
.1550
.1598
.1629
.1642
.1639
.2290
.2356
.2396
.2410
.2401
.3008
.3088
.3133
.3147
.3128
.3705
.3796
.3845
.3854
.3825
.4383
.4482
.4531
.4534
.4493
.5042
.5147
.5195
.5190
.5135
.5684
.5792
.5837
.5822
.5752
.6309
.6419
.6459
.6433
.6348
.6919
.7028
.7061
.7024
.6923
1.6
1.7
1.8
1.9
2.0
.0832
.0816
.0793
.0762
.0725
.1620
.1586
.1538
.1475
.1401
.2368
.2314
.2240
.2146
.2036
.3081
.3005
.2904
.2780
.2634
.3760
.3663
.3535
.3380
.3200
.4410
.4290
.4136
.3951
.3739
,5034
.4891
.4711
.4497
.4253
.5632
.5467
.5261
.5019
.4745
.6209
.6021
.5790
.5520
.5217
.6764
.6554
.6298
.6003
.5672
2.1
.0681
.0632
.0578
.0520
.0458
.1315
.1219
.1115
.1002
.0883
.1909
.1769
.1617
.1454
.1283
.2469
.2287
.2090
.1881
.1661
.2998
.2777
.2539
.2286
.2022
.3501
.3243
.2966
.2674
.2368
.3982
.3689
.3375
.3046
.2703
.4442
.4116
.3769
.3405
.3027
.4885
.4527
.4149
.3752
.3342
.5311
.4924
.4516
.4090
.3649
.0393
.0325
.0256
.0186
.0116
.0759
.0630
.0499
.0366
.0232
.1104
.0920
.0733
.0543
.0352
.1433
.1199
.0960
.0719
.0477
.1749
.1468
.1183
.0894
.0606
.2053
.1730
.1402
.1071
.0740
.2349
.1988
.1621
.1251
.0880
.2638
.2240
.1838
.1432
.1026
.2920
.2490
.2054
.1616
.1177
.3197
.2737
.2271
.1803
.1334
.0236
.0318
.0034
.0410
.0084
0
.1
.2
.3
.4
.5
.6
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
.0046
-.0023
.0099
-.0032
.0163
-.0025
-.0002
-.0235
-.0244
-.0236
.0512
.0148
-.0211
Table XV
Values of dii/dx
b=.1
.0622
.0223
-.0170
.0741
.0309
-.0116
.0867
.0406
-.0049
b=.1
a=1.6 a=1.7
a=1.8 a=1.9 a=2.0
x
a=1.1
a=1.2
a=-1.3
a=1.4
a=1.5
0
.1
.2
.3
.4
.5
.0000
.1085
.2127
.3106
.4005
.4815
.0000
.1184
.2319
.3384
.4360
.5234
.0000
.1282
.2511
.3660
.4712
.5649
.0000
.1381
.2702
.3936
.5062
.6062
.0000
.1479
.2893
.4212
.5410
.6471
.0000
.1577
.3084
.4486
.5757
.6878
.0000
.1676
.3275
.4760
.6102
.7282
.0000
.1774
.3465
.5033
.6446
.7683
.0000
.1872
.3655
.5305
.6788
.8082
.0000
.1970
.3845
.5577
.7129
.8478
.6
.7
.8
.9
1.0
.5524
.6130
.6628
.7023
.7316
.5996
.6643
.7173
.7588
.7892
.6464
.7150
.7709
.8142
.8457
.6926
.7651
.8237
.8687
.9010
.7385
.8146
.8757
.9222
.9551
.7839
.8635
.9269
.9748
1.0083
.8288
.9118
.9774
1.0265
1.0604
.8734
.9595
1.0272
1.0774
1.1115
.9175
1.0067
1.0762
1.1274
1.1617
.9612
1.0533
1.1246
1.1766
1.2109
1.1
1.2
1.3
.7513
.7621
.7647
.7597
.7479
.8093
.8198
.8215
.8152
.8018
.8660
.8760
.8768
.8691
.8540
.9213
.9308
.9305
.9215
.9047
.9755
.9843
.9829
.9724
.9539
1.0284
1.0366
1.0341
1.0221
1.0019
1.0803
1.0876
1.0039
1.0704
1.0485
1.1310
1.1375
1.1325
1.1176
1.0940
1.1807
1.1863
1.1801
1.1637
1.1385
1.2294
1.2341
1.2266
1.2086
1.1818
1.9
2.0
.7301
.7069
.6789
.6468
.6111
.7820
.7566
.7264
.6918
.6536
.8323
.8048
.7723
.7355
.6949
.8810
.8515
.8168
.7777
.7349
.9284
.8969
.8601
.8189
.7738
.9745
.9410
.9022
.8589
.8118
1.0194
.9839
.9432
.8979
.8489
1.0631
1.0258
.9832
.9361
.8851
1.1058
1.0667
1.0223
.9733
.9206
1.1474
1.1066
1.0605
1.0098
.9554
2.1
2.2
2.3
2.4
2.5
.5723
.5309
.4873
.4419
.3950
.6123
.5683
.5221
.4741
.4246
.6511
.6047
.5560
.5056
.4537
.6898
.6401
.5892
.5365
.4823
.7256
.6747
.6217
.5668
.5105
.7616
.7086
.6535
.5967
.5384
.7967
.7418
.6848
.6261
.5660
.8311
.7744
.7156
.6551
.5933
.8648
.8064
.7459
.6838
.6204
.8979
.8379
.7758
.7121
.6472
2.6
2.7
2.8
.3470
.2982
.2488
.1992
.1496
.3740
.3226
.2706
.2185
.1663
.4007
.3469
.2926
.2380
.1834
.4271
.3710
.3145
.2577
.2010
.4532
.3951
.3365
.2777
.2190
.4791
.4191
.3586
.2978
.2373
.5049
.4431
.3808
.3184
.2560
.5305
.4671
.4031
.3391
.2751
.5560
.4910
.4255
.3599
.2944
.5813
.5148
.4479
.3809
.3140
.1002
.0513
.0030
.1143
.0629
.0121
.1291
.0753
.0222
.1445
.0885
.0333
.1605
.1025
.0452
.1770
.1171
.0579
.0003
.1939
.1323
.0714
.0115
.2113
.1481
.0856
.0240
.2291
.1644
.1004
.0373
.2473
.1812
.1157
.0512
1.4
1.5
1.6
1.7
1.8
2.9
3.0
3.1
3.2
3.3
3.4
3.5
-.0977
-.0299
-.0210
-.0111
[17]
-.0365
-.0247
-.0122
Table XV
Values of du/dx
b=.1
b=.1
a=2.1
a=2.2
a=2.3
a=2.4
a=2.5
a=2.6 0=2.7
a=2.8
a=2.9 a=3.0
.1
.2
.3
.4
.5
.0000
.2068
.4035
.5847
.7467
.8870
.0000
.2167
.4224
.6117
.7804
.9261
.0000
.2265
.4413
.6386
.8139
.9648
.0000
.2363
.4602
.6655
.8474
1.0033
.0000
.2461
.4791
.6923
.8807
1.0416
.0000
.2559
.4979
.7190
.9138
1.0796
.0000
.2657
.5168
.7456
.9468
1.1173
.0000
.2755
.5356
.7721
.9796
1.1548
.0000
.2853
.5543
.7986
1.0122
1.1920
.0000
.2951
.5731
.8250
1.0447
1.2290
.6
.7
.8
.9
1.0
1.0045
1.0993
1.1723
1.2250
1.2593
1.0475
1.1449
1.2193
1.2726
1.3068
1.0900
1.1899
1.2658
1.3195
1.3535
1.1322
1.2344
1.3116
1.3657
1.3995
1.1740
1.2785
1.3568
1.4113
1.4446
1.2154
1,3220
1.4014
1.4561
1.4891
1.2564
1,3651
1.4454
1.5003
1.5328
1.2971
1.4077
1.4889
1.5439
1.5759
1.3375
1.4499
1.5319
1.5868
1.6183
1.3775
1.4916
1.5743
1.6292
1.6600
1.1
1.2
1.3
1.4
1.5
1.2772
1.2808
1.2720
1.2526
1.2242
1.3241
1.3267
1.3166
1.2957
1.2657
1.3701
1.3716
1.3602
1.3379
1.3063
1.4153
1.4157
1.4030
1.3792
1.3461
1.4597
1.4589
1.4449
1.4197
1.3851
1.5033
1.6014
1.4861
1.4594
1.4234
1.5461
1.5431
1.5264
1.4984
1.4609
1.5883
1.5841
1.5661
1.5367
1.4979
1.6297
1.6244
1.6051
1.5743
1.5341
1.6705
1.6640
1.6434
1.6113
1.5698
1.6
1.7
1.8
1.9
2.0
1.1882
1.1457
1.0978
1.0455
.9895
1.2280
1.1839
1.1344
1.0805
1.0230
1.2670
1.2214
1.1703
1.1149
1.0559
1.3053
1.2581
1.2056
1.1487
1.0882
1.3428
1.2941
1.2402
1.1819
1.1201
1.3796
1.3295
1.2742
1.2145
1.1505
1.4158
1.3643
1.3076
1.2467
1.1824
1.4513
1.3985
1.3405
1.2784
1.2129
1.4862
1.4323
1.3729
1.3096
1.2431
1.5206
1.4658
1.4048
1.3404
1.2728
2.1
2.2
2.3
2.4
2.5
.9304
.8689
.8053
.7402
.6738
.9624
.8994
.8344
.7679
.7002
.9939
.9295
.8632
.7954
.7264
1,0249
.9592
.8916
.8226
.7525
1.0554
.9885
.9198
.8496
.7784
1.0856
1.0175
.9476
.8764
.8041
1.1154
1.0462
.9752
.9029
.8297
1.1448
1.0745
1.0026
.9293
.8551
1.1739
1.1026
1.0296
.9555
.8804
1.2026
1.1303
1.0565
.9815
.9055
2.6
2.7
2.8
2.9
3.0
.6066
.5387
.4705
.4021
.3339
.6317
.5625
.4931
.4234
.3540
.6567
.5863
.5157
.4449
.3743
.6815
.6101
.5383
.4665
.3947
.7063
.6338
.5610
.4881
.4154
.7310
.6575
.5837
.5099
.4362
.7557
.6812
.6065
.5318
.4572
.7802
.7049
.6293
.5537
.4783
.8046
.7285
.6521
.5757
.4995
.8290
.7520
.6749
.5978
.5208
3.1
3.2
3.3
3.4
3.5
.2659
.1984
.1317
.0658
.0010
.2848
.2161
.1481
.0809
.0148
.3039
.2341
.1649
.0966
.0293
.3233
.2524
.1821
.1127
.0444
.3430
.2710
.1998
.1293
.0599
.3628
.2900
.2177
.1463
.0759
.3829
.3091
.2360
.1637
.0924
.4032
.3286
.2546
.1814
.1092
.4236
.3482
.2735
.1995
.1264
.4442
.3680
.2925
.2178
.1439
3.6
-.0626
-.0500
-.0367
-.0228
-.0083
-.0067
.0222
.0381
.0545
.0712
0
Table XV
Values of du/da-
b=.2
b=.2
x
a=.1
a=.2
a=.3
a=.4
a=.5
a=.6
a=.7
a=.8
a=.9
0
.0000
.0098
.0191
.0273
.0359
.0433
.0000
.0196
.0381
.0555
.0716
.0863
.0000
.0293
.0572
.0832
.1072
.1289
.0000
.0391
.0762
.1108
.1426
.1713
.0000
.0489
.0952
.1383
.1778
.2133
.0000
.0684
.1331
.1931
.2477
.2966
.0000
.0782
.1520
.2203
.2824
.3377
.0000
.0880
.1709
.2475
.3170
.3786
.0000
.0977
.1897
.2746
.2514
.4192
.0499
.0558
.0609
.0651
.0685
.0711
.0729
.0739
.0741
.0736
.0724
.0705
.0680
.0650
.0615
.0994
.1109
.1207
.1289
.1354
.1403
.1435
.1451
.1453
.1440
.1414
.1376
.1326
.1266
.1197
.1120
.1035
.0945
.0850
.0751
.0649
.0545
.0441
.0337
.0234
.0133
.0033
.1483
.1652
.1796
.1914
.2007
.2075
.2119
.2140
.2138
.2117
.2075
.2017
.1942
.1852
.1750
.1968
.2189
.2375
.2527
.2645
.2730
.2783
.2805
.2799
.2767
.2709
.2630
.2529
.2411
.2277
.2447
.2718
.2945
.3128
.3268
.0000
.0587
.1141
.1657
.2128
.2551
.2922
.3241
.3506
.3718
.3878
.3393
.3757
.4057
.4296
.4474
.3859
.4266
.4601
.4864
.5058
.4777
.5267
.5663
.5969
.6189
.3368
.3427
.3450
.3437
.3392
.3989
.4053
.4074
.4053
.3996
.5187
.5255
.5268
.5228
.5143
.3318
.3217
.3092
.2946
.2781
.3904
.3782
.3633
.3459
.3265
.4595
.4662
.4679
.4650
.4579
.4469
.4326
.4153
.3953
.3731
.1636
.1513
.1382
.1244
.1101
.2129
.1969
.1799
.1622
.1438
.2600
.2406
.2200
.1986
.1764
.3053
.2826
.2586
.2337
.2081
.3489
.3231
.2960
.2679
.2390
.3910
.3623
.3323
.3011
.2692
.0954
.0805
.0655
.0506
.0358
.1250
.1059
.0868
.0677
.0488
.1538
.1309
.1079
.0850
.0624
.1819
.1555
.1289
.1025
.0764
.2095
.1798
.1500
.1203
.0910
.0401
.0185
.0508
.0257
.0015
.0622
.0341
.0068
.2367
.2039
.1711
.1384
.1061
.0744
.0434
.0133
.4320
.4770
.5136
.5421
.5630
.5765
.5833
.5839
.5789
.5688
.5543
.5358
.5139
.4889
.4614
.4318
.4003
.3675
.3335
.2987
.2634
.2278
.1922
.1567
.1217
.0872
.0536
.0209
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
.0576
.0532
.0485
.0436
.0384
.0331
.0277
.0223
.0168
.0115
.0062
.0011
-.0039
-.0060
.0214
.0073
-.0064
.0303
.0123
-.0052
-.0025
[ 18 ]
-.0218
-.0194
.5015
.4851
.4654
.4429
.4180
-.0157
-.0107
a=1.0
.6330
.6396
.6395
.6334
.6218
.6055
.5850
.5607
.5334
5035
.4713
.4373
.4018
.3652
.3278
.2898
.2516
.2133
.1753
.1377
.1007
.0645
.0294
-.0046
Table XV
a =1.8
.3024
.4358
.5544
.6570
.0000
.1659
.3211
.4624
.5877
.6957
.0000
.1756
.3398
.4890
.6209
.7341
.0000
.1854
.3584
.5154
.6539
.7723
.0000
.1951
.3771
.5418
.6867
.8102
.6998
.7665
.8186
.8571
.8829
.7430
.8128
.8670
.9066
.9328
.7858
.8586
.9147
.9553
.9818
.8283
.9038
.9617
1.0032
1.0300
.8704
.9486
1.0081
1.0505
1.0773
.9121
.9929
1.0540
1.0970
1.1239
.8467
.8517
.8480
.8369
.8191
.8974
.9017
.8971
.8846
.8653
.9471
.9507
.9450
.9312
.9103
.9958
.9987
.9919
.9767
.9543
1.0436
1.0457
1.0377
1.0212
.9973
1.0905
1.0917
1.0826
1.0647
1.0394
1.1365
1.1369
1.1266
1.1074
1.0806
.7501
.7237
.6932
.6594
.6227
.7957
.7674
.7350
.6992
.6604
.8402
.8101
.7758
.7380
.6973
.8835
.8517
.8156
.7759
.7333
.9259
.8923
.8544
.8130
.7686
.9673
.9320
.8924
.8493
.8032
1.0078
.9709
.9296
.8849
.8372
1.0474
1.0090
.9662
.9198
.8706
.5471
.5083
.4681
.4267
.3845
.5836
.5427
.5002
.4566
.4123
.6193
.5763
.5317
.4861
.4397
.6542
.6092
.5628
.5152
.4669
.6884
.6415
.5933
.5439
.4938
.7219
.6733
.6233
.5722
.5204
.7548
.7046
.6529
.6002
.5468
.7872
.7353
.6821
.6279
.5730
.8190
.7657
.7110
.6553
.5990
.3159
.2752
.2345
.1941
.1541
.3417
.2988
.2559
.2132
.1710
.3674
.3223
.2773
.2325
.1882
.3928
.3457
.2987
.2520
.2057
.4181
.3692
.3203
.2717
.2236
.4432
.3925
.3419
.2916
.2418
.4682
.4158
.3636
.3117
.2603
.4930
.4391
.3853
.3319
.2791
.5177
.4624
.4071
.3523
.2980
.5423
.4856
.4290
.3728
.3172
.1148
.0763
.0389
.0027
.1294
.0888
.0492
.0110
.1446
.1019
.0603
.0203
.1602
.1157
.0722
.0303
.1763
.1300
.0847
.0408
.1929
.1448
.0979
.0526
.0083
.2097
.1601
.1116
.0649
.0189
.2270
.1759
.1259
.0774
.0303
.2446
.1920
.1407
.0907
.0422
.2624
.2086
.1559
.1046
.0547
a=1.1
a=1.2
a=1.3
0
.1
.2
.3
.4
.5
.0000
.1075
.2085
.3017
.3856
.4595
.0000
.1172
.2274
.3287
.4197
.4995
.0000
.1270
.2462
.3556
.4536
.5393
.0000
.1367
.2649
.3824
.4874
.5788
.0000
.1464
.2837
.4091
.5210
.6180
.6
.7
.8
.9
1.0
.5229
.5758
.6183
.6508
.6738
.5678
.6243
.6694
.7036
.7276
.6122
.6722
.7199
.7556
.7804
.6562
.7196
.7696
.8067
.8321
1.1
1.2
1.3
1.4
1.5
.6881
.6945
.6936
.6863
.6731
.7421
.7481
.7464
.7378
.7231
.7950
.8005
.7978
.7880
.7718
1.6
1.7
1.8
1.9
.6550
.6325
.6062
.5766
.5443
.7032
.6787
.6503
.6185
.5840
.5097
.4732
.4353
.3962
.3563
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
3.5
a=1.9 (z=2.0
a=1.7
a=1.4 a=1.5
x
2.0
b=.2
Values of - du/dx
b=.2
-.0016
a = 1.6
.0000
.1562
Table XV
Values of du/dx
b =.2
b=.2
x
a=2.1
a =2.2
a=2.3
a= 2.4
a=2.5
a=2.6
a=2.7
a =2.8
a=2.9
a=3.0
0
.3
.4
.5
.0000
.2048
.3957
.5682
.7194
.8479
.9534
1.0367
1.0992
1.1428
1.1696
.0000
.2242
.4328
.6206
.7844
.9226
1.0350
1.1229
1.1879
1.2325
1.2590
.0000
.2339
.4513
.6467
.8167
.9595
1.0753
1.1653
1.2315
1.2764
1.3026
.0000
.2436
.4699
.6728
.8488
.9962
1.1153
1.2072
1.2745
1.3197
1.3456
1.1818
1.1812
1.1698
1.1492
1.1209
1.0862
1.0463
1.0019
.9540
.9033
1.2262
1.2246
1.2120
1.1901
1.1604
1.2699
1.2674
1.2536
1.2303
1.1992
1.3128
1.3093
1.2943
1.2698
1.2373
1.3550
1.3506
1.3344
1.3086
1.2748
.0000
.2631
.5068
.7247
.9126
1.0690
1.1942
1.2899
1.3590
1.4045
1.4296
1.4375
1.4310
1.4124
1.3841
1.3477
.0000
.2728
.5253
.7505
.9443
1.1050
1.2331
1.3306
1.4005
1.4460
1.4707
1.4778
1.4702
1.4505
1.4209
1.3832
.0000
.2824
.5437
.7763
.9759
1.1408
1.2718
1.3709
1.4415
1.4870
1.5112
1.1
1.2
1.3
.0000
.2534
.4884
.6988
.8808
1.0327
1.1549
1.2488
1.3170
1.3624
1.3879
1.3966
1.3911
1.3738
1.3467
1.3115
.0000
.2921
.5621
.8020
1.0073
1.1764
.6
.7
.8
.9
1.0
.0000
.2145
.4142
.5944
.7520
.8854
.9944
1.0800
1.1438
1.1880
1.2147
1.5175
1.5088
1.4879
1.4571
1.4182
1.1243
1.0828
1.0370
.9877
.9355
1.1617
1.1188
1.0715
1.0208
.9673
1.1984
1.1541
1.1055
1.0534
.9986
1.2345
1.1888
1.1389
1.0856
1.0295
1.2699
1.2230
1.1718
1.1172
1.0599
1.3047
1.2565
1.2041
1.1483
1.0899
1.3390
1.2896
1.2360
1.1791
1,1196
1.3728
1.3222
1.2674
1.2094
1.1489
2.4
2.5
.8503
.7956
.7359
.6824
.6248
.8811
.8250
.7676
.7093
.6504
.9116
.8542
.7955
.7359
.6758
.9416
.8830
.8231
.7624
.7011
.9713
.9115
.8505
.7887
.7263
1.0006
.9396
.8776
.8147
.7514
1.0295
.9675
.9044
.8405
.7762
1.0581
.9951
.9310
.8662
.8009
1.0864
1.0224
.9574
.8917
.8256
1.5565
1.5468
1.5248
1.4928
1.4526
1.4061
1.3543
1.2985
1.2394
1.1778
1.1144
1.0495
.9836
.9170
.8501
2.6
2.7
2.8
2.9
3.0
.5668
.5088
.4509
.3935
.3366
.5911
.5319
.4728
.4142
.3562
.6154
.5550
.4948
.4351
.3759
.6396
.5781
.5168
.4560
.3959
.6638
.6012
.5389
.4771
.4159
.6878
.6243
.5610
.4982
.4360
.7116
.6473
.5831
.5194
.4563
.7355
.6702
.6052
.5406
.4766
.7593
.6931
.6273
.5619
.4971
.7830
.7160
.6494
.5832
.5176
3.1
3.2
3.3
.2806
.2255
.1716
.1189
.0679
.2989
.2427
.1876
.1338
.0815
.3176
.2602
.2040
.1490
.0956
.3364
.2780
.2207
.1647
.1102
.3554
.2961
.2377
.1807
.1251
.3746
.3142
.2550
.1970
.1405
.3940
.3327
.2725
.2136
.1562
.4135
.3513
.2903
.2305
.1722
.4331
.3701
.3083
.2477
.1884
.4529
.3891
.3264
.2649
.2049
.1
.2
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
3.4
3.5
[ 19 ]
1.3101
1.4108
1.4820
1.5275
1.5511
Table XV
Values of - du/dx
b=.3
x
a=.1
a=.2
b=.3
a=.4
a=.5
a=.6
a=.7
a=.8
a=.9
a=1.0
0
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.1
.2
.3
.4
.5
.0097
.0187
.0270
.0345
.0413
.0194
.0374
.0539
.0689
.0823
.0291
.0561
.0808
.1032
.1230
.0387
.0747
.1076
.1372
.1634
.0484
.0933
.1343
.1711
.2036
.0581
.1119
.1610
.2049
.2434
.0678
.1305
.1875
.2385
.2830
.0774
.1490
.2140
.2719
.3224
.0000
.0871
.1676
.2405
.3052
.3616
.0006
.0967
.1861
.2668
.3384
.4003
.6
.7
.8
.9
1.0
.0472
.0523
.0566
.0610
.0626
.0939
.1039
.1122
.1189
.1239
.1402
.1549
.1670
.1767
.1839
.1861
.2058
.2210
.2334
.2426
.2315
.2551
.2742
.2891
.3000
.2765
.3042
.3266
.3439
.3564
.3211
.3528
.3782
.3971
.4116
.3653
.4008
.4292
.4506
.4657
.4091
.4483
.4794
.5027
.5188
.4525
.4952
.5288
.5539
.5710
1.1
1.2
1.3
1.4
1.5
.0645
.0656
.0661
.0658
.0649
.1274
.1294
.1301
.1294
.1275
.1888
.1916
.1921
.1909
.1879
.2486
.2519
.2524
.2505
.2463
.3071
.3107
.3109
.3082
.3028
.3643
.3680
.3679
.3643
.3575
.4202
.4240
.4233
.4188
.4106
.4748
.4786
.4774
.4718
.4623
.5284
.5319
.5301
.5233
.5124
.5808
.5840
.5814
.5736
.6612
1.6
1.7
1.8
1.9
2.0
.0635
.0615
.0591
.0563
.0531
.1246
.1206
.1158
.1101
.1038
.1834
.1774
.1702
.1618
.1526
.2401
.2321
.2225
.2116
.1994
.2949
.2849
.2731
.2596
.2447
.3480
.3360
.3219
.3059
.2884
.3994
.3854
.3691
.3508
.3308
.4493
.4334
.4150
.3943
.3719
.4978
.4800
.4595
.4366
.4119
.5449
.5253
.5027
.4778
.4508
2.1
2.2
2.3
2.4
2.5
.0495
.0458
.0418
.0376
.0334
.0970
.0896
.0819
.0739
.0657
.1425
.1318
.1205
.1089
.0970
.1863
.1724
.1579
.1429
.1276
.2287
.2117
.1941
.1759
.1574
.2697
.2499
.2293
.2082
.1867
.3094
.2869
.2636
.2397
.2154
.3480
.3230
.2971
.2705
.2436
.3856
.3582
.3298
.3008
.2714
.4223
.3925
.3619
.3305
.2989
2.6
2.7
2.8
2.9
3.0
.0291
.0247
.0204
.0162
.0120
.0574
.0490
.0408
.0326
.0246
.0850
.0730
.0610
.0492
.0377
.1121
.0966
.0813
.0661
.0513
.1388
.1201
.1016
.0833
.0654
.1650
.1434
.1219
.1007
.0800
.1909
.1665
.1422
.1184
.0950
.2165
.1895
.1627
.1363
.1105
.2419
.2126
.1833
.1545
.1263
.2670
.2353
.2038
.1729
.1425
3.1
.0080
.0042
.0005
.0169
.0095
.0024
.0265
.0158
.0055
.0369
.0231
.0099
-.0043
-.0042
.0481
.0314
.0154
.0002
-.0141
.0599
.0405
.0219
.0042
.0723
.0504
.0294
.0073
.0854
.0611
.0378
.0155
.0989
.0724
.0469
.0226
-.0096
-.0056
-.0005
.1130
.0844
.0569
.0306
.0056
a_=1.8
n000
.1739
a=1.9
a=2.0
.0000
.1835
.000^
.1932
.3515
.3698
3.2
3.3
3.4
3.5
-.0030
-.0027
-.0125
Table XV
Values of du/dx
b =.3
b=.3
x
a=1.1
a.=1.2
a=1.3
a=1.4
a=1.5
a=1.6
a=1.7
0
.1
.2
.3
.4
.0000
.1064
.2045
.2931
.3714
.0000
.1160
.2230
.3193
.4042
.0000
.1257
.0000
.1354
.0000
.1450
.0000
.1546
.0000
.1643
.2414
.2598
.2782
.2966
.3149
.3332
.4494
.5663
.6650
.4752
.5010
.6302
.4388
.4236
.5342
.6279
.5983
.5
.3976
.5019
.5906
.7019
.7385
.7749
.6
.5267
.4771
.3455
.4369
.5152
.3716
.4695
.5530
.8
.9
1.0
.4955
.5416
.5776
.6042
.6221
.5381
.5875
.6258
.6538
.6724
.5804
.6329
.6733
.7026
.7218
.6223
.6777
.7202
.7506
.7702
.6638
.7221
.7664
.7980
.8179
.7050
.7660
.8121
.8446
.8648
.7458
.8094
.8572
.8905
.9110
.7863
.8524
.9017
.9358
.9564
.8264
.8949
.9457
.9804
1.0010
.8662
.9370
.9891
1.0244
1.0450
1.1
1.2
1.3
1.4
1.5
.6321
.6350
.6316
.6226
.6088
.6824
.6848
.6806
.6701
.6552
.7318
.7337
.7285
.7172
.7005
.7805
.7815
.7754
.7628
.7447
.8277
.8284
.8213
.8075
.7880
.8743
.8743
.8663
.8512
.8303
.9201
.9194
.9103
.8940
.8717
.9651
.9637
.9535
.9360
.9123
1.0094
1.0071
.9959
.9772
.9521
1.0529
1.0498
1.0375
1.0176
.9912
1.6
1.7
1.8
1.9
2.0
.5909
.5694
.5449
.5179
.4888
.6356
.6123
.5859
.5570
.5259
.6793
.6543
.6260
.5952
.5622
.7219
.6952
.6652
.6325
.5977
.7636
.7352
.7035
.6691
.6325
.8044
.7744
.7411
.7050
.6667
.8444
.8128
.7778
.7401
.7002
.8835
.8504
.8139
.7746
.7331
.9219
.8873
.8493
.8085
.7656
.9595
.9236
.8841
.8419
.7975
2.1
2.2
2.3
2.4
2.5
.4582
.4262
.3933
.3598
.3260
.4932
.4592
.4243
.3887
.3528
.5275
.4916
.4547
.4171
.3793
.5612
.5233
.4846
.4452
.4055
.5942
.5546
.5141
.4730
.4316
.6267
.5854
.5432
.5004
.4574
.6586
.6157
.5719
.5276
.4830
.5900
.6456
.6003
.5545
.5084
.7209
.6751
.6284
.5811
.5337
.7514
.7042
.6561
.6075
.5588
2.6
2.7
2.8
2.9
3.0
.2920
.2581
.2245
.1915
.1591
.3167
.2808
.2453
.2103
.1760
.3414
.3036
.2661
.2293
.1932
.3658
.3262
.2870
.2484
.2106
.3901
.3489
.3080
.2678
.2283
.4143
.3715
.3291
.2873
.2463
.4384
.3941
.3502
.3069
.2645
.4624
.4166
.3713
.3267
.2829
.4863
.4392
.3925
.3466
.3014
.5101
.4617
.4138
.3666
.3202
3.1
3.2
3.3
.1425
.1101
.0788
.0488
.1579
.1237
.0908
.0591
.1737
.1379
.1032
.0700
.1898
.1524
.1162
.0815
.2063
.1674
.1297
.0935
.2230
.1827
.1436
.1060
3.5
.0125
.0203
-.0067
.0289
.0382
.0482
.0588
.0699
.2400
.1984
.1580
.1190
.0816
.2573
.2144
.1727
.1325
.0938
.2748
.2307
.1878
3.4
.1275
.0970
.0675
.0393
.0001
.0079
.0165
.0257
-.0134
-.0056
.0355
.0029
.7
3.6
3.7
3.8
[ 20 ]
.0459
.0120
.0569
.0217
-.0200
-.0115
.6619
.1464
.1065
.0684
.0320
-.0024
Table XV
b=.3
Values of - du/dx
b=.3
x
a=2.1
a X2.2
a=2.3
a=2.4
a=2.5
a=2.6
a=2.7
a=2.8
a=2.9
a=3.0
o
.4
.5
.0000
.2028
.3881
.5523
.6934
.8111
.0000
.2124
.4063
.5778
.7249
.8470
.0000
.2220
.4245
.6033
.7562
.8827
.0000
.2317
.4427
.6287
.7874
.9182
.0000
.2413
.4608
.6540
.8184
.9535
.0000
.2509
.4790
.6793
.8493
.9886
.0000
.2605
.4971
.7045
.8801
1.0234
.0000
.2701
.5153
.7297
.9108
1.0581
.0000
.2797
.5333
.7548
.9413
1.0926
.0000
.2893
.5513
.7798
.9717
1.1268
.6
.7
.8
.9
1.0
.9057
.9787
1.0320
1.0678
1.0883
.9448
1.0199
1.0744
1.1107
1.1310
.9837
1.0608
1.1163
1.1529
1.1731
1.0222
1.1012
1.1578
1.1946
1.2145
1.0604
1.1412
1.1987
1.2358
1.2553
1.0983
1.1809
1.2392
1.2764
1.2956
1.1359
1.2201
1.2792
1.3165
1.3354
1.1733
1.2590
1.3188
1.3561
1.3746
1.2103
1.2975
1.3579
1.3953
1.4133
1.2470
1.3357
1.3966
1.4339
1.4514
1.1
1.2
1.0956
1.0917
1.0784
1.0572
1.0295
1.1377
1.1330
1.1186
1.0962
1.0672
1.1792
1.1735
1.1580
1.1344
1.1041
1.2200
1.2134
1.1968
1.1720
1.1405
1.2602
1.2527
1.2350
1.2091
1.1763
1.2998
1.2913
1.2726
1.2455
1.2116
1.3388
1.3294
1.3096
1.2813
1.2462
1.3772
1.3669
1.3461
1.3167
1.2804
1.4152
1.4039
1.3820
1.3515
1.3141
1.4526
1.4403
1.4174
1.3858
1.3473
.9965
.9591
.9183
.8746
.8289
1.0328
.9941
.9519
.9069
.8599
1.0685
1.0285
.9850
.9387
.8904
1.1036
1.0623
1.0175
.9701
.9205
1.1382
1.0956
1.0497
1.0010
.9503
1.1732
1.1285
1.0813
1.0315
.9797
1.2057
1.1608
1.1125
1.0616
1.0087
1.2387
1.1927
1.1433
1.0913
1.0374
1.2713
1.2242
1.1737
1.1207
1.0658
1.3034
1.2552
1.2037
1.1497
1.0939
.7815
.7329
.6836
.6337
.5837
.8112
.7614
.7107
.6597
.6085
.8405
.7895
.7377
.6855
.6331
.8694
.8172
.7643
.7110
.6576
.8981
.8448
.7908
.7364
.6820
.9264
.8720
.8170
.7616
.7062
.9544
.8990
.8430
.7867
.7303
.9821
.9258
.8688
.8116
.7543
1.0095
.9523
.8944
.8363
.7782
1.0367
.9785
.9198
.8609
.8019
2.9
3.0
.5338
.4841
.4350
.3866
.3391
.5574
.5066
.4564
.4068
.3581
.5809
.5290
.4777
.4270
.3773
.6043
.5514
.4990
.4474
.3966
.6277
.5738
.5204
.4678
.4160
.6509
.5961
.5418
.4882
.4355
.6741
.6183
.5631
.5087
.4551
.6973
.6406
.5845
.5292
.4748
.7203
.6628
.6059
.5497
.4945
.7432
.6849
.6272
.5703
.5143
3.1
3.2
3.3
3.4
3.5
.2926
.2472
.2032
.1606
.1196
.3105
.2640
.2189
.1752
.1331
.3286
.2811
.2349
.1901
.1470
.3469
.2984
.2511
.2054
.1612
.3653
.3158
.2676
.2209
.1757
.3839
.3335
.2843
.2367
.1905
.4026
.3513
.3012
.2527
.2056
.4214
.3692
.3183
.2689
.2210
.4403
.3873
.3356
.2853
.2366
.4593
.4055
.3533
.3019
.2523
3.6
3.7
3.8
3.9
4.0
.0803
.0428
.0072
.0927
.0541
.0174
.1055
.0658
.0281
.1187
.0780
.0392
.0024
.1323
.0905
.0507
.0129
.1461
.1034
.0627
.0239
.1602
.1167
.0750
.0353
.1747
.1302
.0876
.0470
.0084
.1894
.1441
.1006
.0591
.0195
.2044
.1582
.1138
.0714
.0311
.1
.2
.3
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
-.0263
-.0172
-.0077
-.0322
[ 21 ]
-.0023
Table XV
b=.4
b=.4
Values of - du/dx
x
a=.1
a=.2
a=.3
a=.4
a=.5
a=.6
a=.7
a=.8
a=.9 a=1.0
.1
.2
.3
.4
.5
.0000
.0096
.0184
.0262
.0333
.0394
.0000
.0192
.0367
.0524
.0664
.0785
.0000
.0288
.0550
.0785
.0993
.1174
.0000
.0384
.0733
.1046
.1321
.1560
.0000
.0479
.0915
.1305
.1648
.1944
.0000
.0575
.1098
.1564
.1973
.2325
.0000
.0671
.1280
.1827
.2297
.2704
.0000
.0767
.1461
.2080
.2619
.3080
.0000
.0862
.1643
.2337
2941
.3454
.0000
.0958
.1825
.2593
.3260
.3825
.6
.7
.8
.9
1.0
.0447
.0491
.0526
.0554
.0575
.0889
.0975
.1045
.1099
.1138
.1328
.1455
.1557
.1635
.1690
.1762
.1929
.2061
.2161
.2231
.2193
.2397
.2558
.2679
.2763
.2620
.2860
.3049
.3189
.3284
.3043
.3319
.3533
.3691
.3797
.3463
.3772
.4011
.4185
.4300
.3879
.4220
.4482
.4671
.4794
.4291
.4663
.4947
.5150
.5280
1.1
1.2
1.3
1.4
1.5
.0588
.0594
.0594
.0589
.0578
.1162
.1173
.1172
.1160
.1138
.1724
.1738
.1735
.1715
.1681
.2273
.2290
.2282
.2254
.2208
.2811
.2828
.2817
.2780
.2720
.3338
.3355
.3338
.3292
.3219
.3855
.3870
.3847
.3791
.3705
.4361
.4374
.4344
.4278
.4178
.4858
.4868
.4831
.4753
.4640
.5345
.5351
.5306
.5218
.5091
1.6
1.7
1.8
1.9
2.0
.0563
.0543
.0520
.0494
.0465
.1107
.1068
.1022
.0970
.0914
.1634
.1575
.1507
.1431
.1348
.2145
.2067
.1978
.1878
.1769
.2641
.2545
.2434
.2311
.2178
.3123
.3009
.2877
.2732
.2576
.3593
.3460
.3309
.3142
.2964
.4051
.3900
.3729
.3542
.3342
.4497
.4329
.4139
.3932
.3711
.4933
.4747
.4539
.4313
.4072
2.1
2.2
2.3
2.4
2.5
.0434
.0401
.0368
.0333
.0298
.0854
.0790
.0725
.0658
.0590
.1260
.1167
.1072
.0974
.0875
.1654
.1534
.1410
.1283
.1166
.2038
.1891
.1740
.1587
.1432
.2411
.2239
.2063
.1884
.1704
.2775
.2580
.2380
.2177
.1973
.3131
.2914
.2691
.2465
.2238
.3480
.3241
.2996
.2749
.2501
.3821
.3562
.3297
.3029
.2761
2.6
2.7
2.8
2.9
.0263
.0228
.0194
.0160
.0128
.0530
.0777
.0454
.0388
.0324
.0261
.0679
.0583
.0489
.0398
.1029
.0903
.0779
.0658
.0540
.1278
.1125
.0975
.0828
.0686
.1525
.1347
.1172
.1002
.0836
.1770
.1568
.1371
.1177
.0990
.2013
.1789
.1569
.1355
.1146
.2254
.2009
.1769
.1534
.1306
.2494
.2230
.1970
.1716
.1470
.0097
.0067
.0039
.0013
.0201
.0144
.0089
.0039
.0311
.0228
.0149
.0075
.0006
.0427
.0320
.0217
.0121
.0031
.0550
.0419
.0294
.0177
.0067
.0677
.0524
.0379
.0242
.0113
.0809
.0635
.0470
.0314
.0167
.0945
.0752
.0568
.0394
.0230
.1086
.0875
.0673
.0481
.0301
.1231
.1002
.0783
.0575
.0379
-.0058
-.0052
-.0035
-.0007
.0077
.0132
.0195
.0023
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
-.0011
-.0009
[ 22 ]
.0030
-.0096
-.0065 -..0025
-.0136
Table XV
Values of du/dx
b =.4
x
a=1.1
a=1.2
.1
.2
.3
.4
.5
.0000
.1054
.2006
.2849
.3578
.4194
.0000
.1149
.2187
.3104
.3895
.4561
.0000
.1245
.2368
.3358
.4211
.4926
.6
.7
.8
.9
1.0
.4700
.5102
.5406
.5622
.5758
.5106
.5536
.5860
.6087
.6228
1.1
1.2
1.3
1.4
1.5
.5823
.5825
.5772
.5672
.5532
1.6
1.7
1.8
1.9
2.0
a=1.5
a=1.6
a=1.7
a=1.8
a=1.9
a=2.0
.0000
.1340
.2548
.3612
.4525
.5288
.0000
.1436
.2729
.3865
.4838
.5648
.0000
.1531
.2909
.4117
.5149
.6006
.0000
.1627
.3089
.4369
.5459
.6362
.0000
.1722
.3269
.4620
.5768
.6715
.0000
.1817
.3448
.4870
.6076
.7067
.0000
.1913
.3627
.5120
.6382
.7416
.5508
.5965
.6308
.6546
.6691
.5907
.6391
.6750
.6998
.7147
.6303
.6811
.7187
.7443
.7594
.6695
.7228
.7619
.7883
.8035
.7084
.7640
.8045
.8316
.8470
.7471
.8049
.8467
.8744
.8898
.7854
.8453
.8884
.9166
.9320
.8234
.8853
.9296
.9583
.9736
.6292
.6289
.6228
.6117
.5964
.6754
.6745
.6675
.6553
.6386
.7207
.7193
.7114
.6980
.6800
.7653
.7632
.7544
.7398
.7205
.8091
.8063
.7966
.7809
.7603
.8522
.8487
.8380
.8212
.7993
.8946
.8904
.8787
.8608
.8376
.9363
.9314
.9188
.8996
.8752
.9775
.9718
.9581
.9379
.9122
.5358
.5156
.4930
.4686
.4426
.5775
.5556
.5313
.5051
.4773
.6182
.5948
.5688
.5409
.5114
.6581
.6331
.6055
.5760
.5448
.6972
.6707
.6416
.6104
.5777
.7356
.7076
.6770
.6443
.6100
.7732
.7438
.7118
.6776
.6418
.8102
.7794
.7459
.7103
.6731
.8465
.8144
.7795
.7426
.7040
.8823
.8488
.8126
.7743
.7345
2.1
2.2
2.3
2.4
2.5
.4156
.3877
.3593
.3306
.3018
.4484
.4187
.3884
.3579
.3273
.4807
.4492
.4172
.3849
.3527
.5124
.4793
.4456
.4117
.3778
.5437
.5089
.4737
.4382
.4027
.5745
.5382
.5014
.4644
.4275
.6048
.5671
.5289
.4904
.4521
.6348
.5956
.5560
.5162
.4765
.6643
.6238
.5829
.5418
.5008
.6934
.6517
.6095
.5672
.5250
2.6
2.9
3.0
.2732
.2449
.2171
.1900
.1635
.2969
.2669
.2373
.2085
.1804
.3205
.2888
.2576
.2271
.1974
.3440
.3107
.2780
.2459
.2147
.3674
.3326
.2984
.2649
.2322
.3907
.3545
.3188
.2839
.2499
.4140
.3763
.3393
.3031
.2678
.4371
.3981
.3598
.3223
.2858
.4601
.4199
.3804
.3417
.3039
.4831
.4417
.4010
.3611
.3222
3.1
3.2
3.3
3.4
3.5
.1380
.1134
.0899
.0675
.0464
.1532
.1270
.1019
.0781
.0554
.1687
.1410
.1144
.0891
.0651
.1845
.1554
.1274
.1007
.0753
.2006
.1700
.1407
.1127
.0860
2169
.1851
.1544
.1251
.0972
.2335
.2004
.1685
.1380
.1088
.2503
.2159
.1829
.1511
.1209
.2673
.2318
.1975
.1647
.1333
.2844
.2478
.2125
.1785
.1460
3.6
3.7
3.8
3.9
4.0
-.0092
.0341
.0143
.0424
.0213
.0015
.0513
.0289
.0079
.0608
.0371
.0149
.0707
.0459
.0225
.0008
.0812
.0552
.0307
.0079
.0921
.0649
.0394
.0155
.1034
.0751
.0485
.0236
.1151
.0858
.0581
.0321
.0080
2.7
2.8
.0265
.0080
-.0042
a=1.3 a=1.4
-.0057
[ 23 ]
Table XV
b=.4
Values of - du/dx
b=.4
a=2.9
a=3.0
a=2.1
a=2.2
a=2.3
a=2.4
a=2.5
a=2.6
a=2.7
a=2.8
.0000
.2008
.3806
.5369
.6687
.7763
.0000
.2104
.3985
.5618
.6991
.8108
.0000
.2199
.4164
.5866
.7294
.8452
.0000
.2294
.4342
.6113
.7595
.8793
.0000
.2389
.4621
.6360
.7895
.9132
.0000
.2484
.4699
.6606
.8194
.9469
.0000
.2579
.4876
.6851
.8491
.9804
.0000
.2674
.5054
.7096
.8788
1.0138
.0000
.2769
.6231
.7340
.9083
1.0469
.0000
.2864
.5409
.7584
.9377
1.0799
.8611
.9250
.9703
.9994
1.0146
.8985
.9643
1.0106
1.0400
1.0550
.9357
1.0032
1.0504
1.0801
1.0949
.9725
1.0417
1.0898
1.1198
1.1343
1.0091
1.0799
1.1288
1.1589
1.1731
1.0454
1.1178
1.1674
1.1976
1.2115
1.0814
1.1553
1.2056
1.2358
1.2493
1.1172
1.1924
1.2434
1.2736
1.2867
1.1527
1.2293
1.2807
1.3110
1.3237
1.1879
1.2658
1.3177
1.3479
1.3601
1.2
1.3
1.4
1.5
1.0180
1.0115
.9968
.9754
.9486
1.0578
1.0505
1.0349
1.0124
.9844
1.0972
1.0890
1.0724
1.0488
1.0197
1.1359
1.1270
1.1093
1.0847
1.0544
1.1742
1.1644
1.1458
1.1200
1.0886
1.2119
1.2012
1.1816
1.1548
1.1223
1.2491
1.2376
1.2170
1.1891
1.1555
1.2858
1.2734
1.2519
1.2230
1.1883
1.3220
1.3088
1.2862
1.2564
1.2207
1.3578
1.3437
1.3202
1.2893
1.2526
1.6
1.7
1.8
1.9
2.0
.9174
.8827
.8452
.8057
.7645
.9520
.9160
.8773
.8365
.7941
.9860
.9489
.9090
.8670
.8284
1.0196
.9812
.9402
.8970
.8524
1.0527
1.0132
.9710
.9268
.8811
1.0853
1.0447
1.0014
.9561
.9094
1.1174
1.0758
1.0315
.9851
.9374
1.1492
1.1065
1.0611
1.0138
.9651
1.1805
1.1368
1.0905
1.0422
.9925
1.2114
1.1667
1.1194
1.0702
1.0197
2.1
2.2
2.3
2.4
2.5
.7222
.6793
.6359
.5924
.5490
.7507
.7065
.6620
.6174
.5729
.7788
.7336
.6879
.6422
.5967
.8067
.7604
.7136
.6669
.6203
.8343
.7869
.7392
.6914
.6439
.8616
.8132
.7645
.7158
.6673
.8886
.8393
.7896
.7400
.6906
.9154
.8651
.8146
.7641
.7138
.9419
.8907
.8393
.7880
.7369
.9682
.9162
.8640
.8118
.7599
2.6
2.7
2.8
2.9
3.0
.5060
.4635
.4216
.3807
.3407
.5287
.4852
.4423
.4002
.3592
.5515
.5068
.4629
.4199
.3778
.5741
.5285
.4836
.4396
.3966
.5967
.5501
.5043
.4593
.4154
.6192
.5717
.5249
.4791
.4343
.6416
.5932
.5456
.4989
.4532
.6640
.6147
.5663
.5187
.4722
.6862
.6362
.5870
.5386
.4913
.7085
.6577
.6076
.5585
.5104
3.1
3.2
3.3
3.4
3.5
.3018
.2641
.2277
.1927
.1591
.3193
.2805
.2431
.2071
.1726
.3369
.2972
.2588
.2218
.1863
.3547
.3140
.2746
.2367
.2002
.3725
.3309
.2906
.2518
.2144
.3905
.3480
.3069
.2671
.2288
.4086
.3653
.3232
.2826
.2435
.4268
.3826
.3398
.2983
.2584
.4451
.4001
.3564
.3142
.2734
.4634
.4177
.3732
.3302
.2886
3.6
3.7
3.8
3.9
4.0
.1272
.0968
.0681
.0412
.1396
.1082
.0785
.0506
.1523
.1199
.0893
.0604
.1653
.1320
.1005
.0707
.1786
.1444
.1119
.0812
.0523
.1921
.1571
.1237
.0921
.2059
.1700
.1358
.1034
.2200
.1832
.1482
.1149
.2342
.1966
.1607
.1266
.2486
.2102
.1735
.1386
.1056
x
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
1.1
[ 24 ]
Table XV
b=.5
Values of - du/dx
b=.5
a=.1
a=.2
a=.3
a=.4
a =.5
a=.6
a=.7
a=.8
a=.9
a=1.0
.0000
.0095
.0000
.0285
.0539
.0763
.0957
.1121
.0000
.0380
.0718
.1016
.1273
.1490
.0000
.0475
.0898
.1268
.1588
.1857
.0000
.0570
.1077
.1520
.1901
.2222
.0000
.0664
.1255
.1771
.2214
.2584
.0000
.0759
.1434
.2022
.2525
.2944
.0000
.0854
.1812
.2272
.2834
.3302
.0000
.0949
.1790
.2521
.3143
.3658
.5
.0320
.0376
.0000
.0190
.0360
.0509
.0639
.0750
.6
.7
.8
.9
1.0
.0423
.0461
.0491
.0513
.0529
.0842
.0917
.0976
.1019
.1048
.1258
.1368
.1454
.1516
.1558
.1670
.1815
.1926
.2006
.2059
.2079
.2256
.2392
.2489
.2551
.2485
.2693
.2852
.2964
.3036
.2887
.3126
.3306
.3433
.3512
.3286
.3554
.3755
.3895
.3981
.3682
.3978
.4198
.4351
.4442
.4074
.4397
.4636
.4800
.4896
1.1
1.2
1.3
1.4
1.5
.0537
.0540
.0537
.0530
.0518
.1064
.1068
.1062
.1046
.1023
.1580
.1585
.1574
.1550
.1513
.2086
.2090
.2075
.2041
.1992
.2582
.2585
.2564
.2521
.2459
.3070
.3071
.3043
.2990
.2916
.3548
.3547
.3512
.3449
.3362
.4018
.4013
.3971
.3898
.3798
.4480
.4471
.4421
.4338
.4224
.4934
.4920
.4863
.4768
.4642
1.6
1,7
1.8
1.9
2.0
.0503
.0484
.0462
.0439
.0413
.0991
.0954
.0912
.0865
.0815
.1467
.1411
.1349
.1280
.1206
.1930
.1857
.1774
.1684
.1588
.2382
.2291
.2189
.2078
.1961
.2823
.2715
.2594
.2463
.2325
.3254
.3129
.2990
.2840
.2681
.3675
.3534
.3377
.3209
.3031
.4087
.3930
.3756
.3570
.3373
.4491
.4317
.4127
.3924
.3710
2.1
2.2
.0763
.0708
.0652
.0596
.0540
.1129
.1049
.0968
.0886
.0804
.1488
.1384
.1278
.1172
.1066
.1838
.1711
.1583
.1453
.1324
.2180
.2032
.1882
.1730
.1579
.2517
2.3
2.4
2.5
.0386
.0358
.0330
.0301
.0272
.2348
.2176
.2004
.1832
.2847
.2658
.2466
.2274
.2083
.3170
.2963
.2752
.2541
.2331
.3489
.3263
.3035
.2806
.2578
2.6
2.7
2.8
2.9
3.0
.0243
.0215
.0188
.0161
.0135
.0484
.0429
.0376
.0324
.0275
.0723
.0643
.0566
.0490
.0418
.0960
.0857
.0756
.0658
.0564
.1196
.1070
.0948
.0829
.0714
..1430
.1283
.1140
.1001
.0867
.1662
.1495
.1333
.1175
.1022
.1894
.1708
.1527
.1351
.1181
.2124
.1920
.1721
.1528
.1341
.2354
.2133
.1917
.1707
.1505
3.1
3.2
3.3
.0111
.0088
.0066
.0046
.0027
.0227
.0182
.0140
.0100
.0348
.0283
.0221
.0162
.0108
.0474
.0389
.0308
.0232
.0161
.0605
.0500
.0401
.0309
.0222
.0738
.0616
.0500
.0391
.0289
.0876
.0737
.0605
.0480
.0364
.1018
.0862
.0714
.0575
.0444
.1162
.0991
.0828
.0674
.0529
.1310
.1124
.0947
.0779
.0621
.0059
.0013
.0096
.0036
.0142
.0068
.0000
.0195
.0107
.0027
.0255
.0154
.0062
-.0061
-.0045
.0321
.0208
.0104
.0009
.0394
.0268
.0152
.0046
.0473
.0335
.0207
.0090
o
.1
.2
.3
.4
3.4
3.5
3.6
3.7
3.8
3.9
4.0
.C180
.0255
.0009
-.0006
.0063
.0029
-.0002
-.0029
-.0019
[ 25
-.0022
-.0078
-.0051
-.0016
Table XV
Values of du/dx
b=.5
a=1.1
a:.---1.2
.2
.3
.4
.5
.0000
.1043
.1968
.2770
.3450
.4012
.0000
.1138
.2145
.3018
.3755
.4363
.0000
.1233
.2323
.3265
.4060
.4713
.0000
.1327
.2500
.3512
.4363
.5059
.0000
.1422
.6
.7
.8
.9
1.0
.4463
.4812
.5069
.5242
.5344
.4849
.5224
.5496
.5679
.5783
.5233
.5631
.6919
.6111
.6217
1.1
1.2
1.3
1.4
1.5
.5380
.5761
.5296
.5191
.5052
.5819
.5795
.5721
.5605
.5454
1.6
1.7
1.8
1.9
.4886
.4698
.4492
.4271
.4040
2.1
x
a=1.3 a=1.4 a=1.5
b=.5
a=1.6
a=1.7
.2677
.3758
.4665
.5405
.0000
.1516
.2854
.4003
.4966
.5749
.0000
.1611
.3030
.4248
.5265
.6090
.0000
.1705
.3206
.4493
.5664
.6429
.0000
.1800
.3383
.4736
.5861
.6767
.0000
.1894
.3559
.4980
.6167
.7103
.5613
.6034
.6337
.6536
.6645
.6990
.6433
.6750
.6956
.7066
.6365
.6829
.7159
.7371
.7482
.6736
.7221
.7563
.7780
.7891
.7105
.7609
.7962
.8185
.8296
.7471
.7994
.8358
.8584
.8694
.7834
.8375
.8749
.8979
.9088
.6250
.6221
.6139
.6012
.5848
.6675
.6640
.6549
.6411
.6236
.7094
.7052
.6952
.6804
.6616
.7506
.7458
.7349
.7190
.6990
.7912
.7357
.7739
.7569
.7357
.8312
.8250
.8123
.7942
.7719
.8706
.8637
.8601
.8310
.8075
.9095
.9019
.8873
.8672
.8426
.5274
.5071
.4849
.4613
.4365
.5655
.6437
.5200
.4948
.4685
.6029
.5797
.5545
.5278
.5000
.6396
.6150
.5884
.5603
.5310
.6757
.6498
.6218
,5923
.5616
.7112
.6840
.6546
.6238
.5918
.7461
.7176
.6870
.6549
.6216
.7805
.7508
.7189
.6855
.6510
.8141
.7835
.7504
.7158
2.2
2.3
2.4
2.5
.3802
.3559
.3314
.3068
.2823
.4111
.3861
.3589
.3327
.3067
.4415
.4140
.3862
.3584
.3309
.4715
.4424
.4132
.3839
.3549
.6010
.4705
.4399
.4092
.3787
.5302
.4983
.4663
.4343
.4025
.5590
.5258
.4925
.4592
.4261
.5875
.5531
.5184
.4839
.4496
.6157
.5800
.5442
.5084
.4730
.6436
.6067
.5697
.5328
.4962
2.6
2.7
2.8
2.9
3.0
.2582
.2345
.2113
.1888
.1670
.2809
.2557
.2310
.2070
.1837
.3036
.2769
.2507
.2203
.2006
.3262
.2980
.2705
.2437
.2177
.3487
.3192
.2903
.2622
.2350
.3711
.3403
.3101
.2808
.2524
.3935
.3614
.3300
.2995
.2699
.4157
.3825
.3500
.3183
.2875
.4379
.4036
.3699
.3371
.3053
.4601
.4246
.3899
.3560
.3231
3.1
3.2
3.3
.1461
.1260
.1069
.0888
.0717
.1614
.1399
.1195
.1001
.0817
.1769
.1542
.1324
.1118
.0922
.1927
.1687
.1457
.1239
.1032
.2087
.1834
.1593
.1363
.1144
.2249
.1985
.1732
.1490
.1261
.2412
.2137
.1873
.1621
.1381
.2578
.2291
.2015
.1754
.1604
.2745
.2448
.2162
.1890
.1630
.2913
.2606
.2311
.2028
.1758
.0556
.0407
.0268
.0141
.0024
.0645
.0484
.0335
.0197
.0070
.0739
.0566
.0406
.0258
.0122
.0836
.0653
.0483
.0324
.0179
.0938
.0745
.0396
.0240
.1044
.0840
.0649
.0472
.0307
.1154
.0940
.0739
.0552
.0378
.1267
.1043
.0832
.0636
.0453
.1383
.1149
.0930
.0724
.0532
.1502
.1269
.1030
.0815
.0615
-.0032
.0017
.0071
.0129
.0191
.0257
1
2.0
3.4
3.5
3.6
3.7
3.8
3.9
4.0
4.2
.0564
[ 26 ]
a=1.8 a=1.9
a=2.0
.6800
Table XV
1.0988
1.1658
1.2094
1.2335
1.2417
1.1326
1.2008
1.2448
1.2688
1.2766
1.1676
1.2026
1.2372
1.2714
1.1545
1.1887
.8908
.9498
.9898
1.0136
1.0239
.9261
.9866
1.0273
1.0513
1.0613
.9611
1.0231
1.0645
1.0886
1.0983
.9959
1.0592
1.1013
1.1254
1.1348
1.0304
1.0951
1.1376
1.1618
1.1708
.9857
1.0230
1.0599
1.0962
1.1321
.9765
.9601
1.0131
.9957
1.0491
1.0847
1.1198
.8195
.8763
.9136
.9369
.9476
.8553
.9127
.9519
.9755
.9860
.9479
.9394
.9240
1.4
1.5
.9028
.8771
.9379
.9111
1.6
1.7
.8478
.8157
1.8
1.9
2.0
1.2225
1.2559
1.1335
1.1064
1.0745
1.1669
1.1388
1.1059
1.1998
1.1708
1.1369
1.2323
1.2024
1.1676
1.0080
1.0388
1.0692
1.0993
1.1291
.9705
.9309
.8898
1.0003
.9597
.9177
1.0298
.9883
.9453
1.0590
1.0165
.9726
1.0878
1.0445
.9997
.8204
.8477
.8746
.9013
.9278
.9540
.7787
.7365
.6944
.6524
.6108
.8049
.7618
.7188
.6759
.6334
.8310
.7870
.7430
.6993
.6559
.8568
.8119
.7671
.7225
.6784
.8824
.8367
.7911
.7457
.7007
.9077
.8613
.8148
.7686
.7229
.5697
.5915
.6132
.6348
.6563
.5919
.6127
.6778
.5711
.4321
.3954
.4513
.4137
.5099
.4704
.4320
.5299
.4896
.4503
.5499
.5088
.4688
.5699
.5280
.4872
.5898
.5472
.5057
.3425
.3089
.2766
.2455
.2158
.3598
.3253
.2921
.2602
.2295
.3772
.3419
.3078
.2750
.2435
.3946
.3585
.3236
.2900
.2577
.4122
.3752
.3395
.3051
.2720
.4298
.3921
.3556
.3204
.2865
.4475
.4090
.3717
.3358
.3012
.4652
.4260
.3880
.3513
.3159
.1748
.1487
.1240
.1008
.0791
.1875
.1605
.1350
.1110
.0884
.2004
.1726
.1462
.1214
.0980
.2135
.1849
.1577
.1320
.1079
.2268
.1974
.1694
.1430
.1180
.2404
.2101
.1814
.1541
.1284
.2541
.2231
.1935
.1655
.1390
.2680
.2362
.2059
.1771
.1499
.2820
.2495
.2184
.1889
.1609
.0401
.0075
.0477
.0021
.0640
.0097
.0726
.0161
.0004
.0815
.0228
.0063
.0906
.0297
.0126
.1000
.0370
.0191
.1096
.0444
.0259
.9726
.9447
1.0309
1.0067
.9778
1.0656
1.0404
1.0104
1.0998
1.0736
1.0427
.8807
.8474
.9131
.8788
.9452
.9097
.9768
.7814
.7456
.7087
.8121
.7751
.7371
.8423
.8043
.8722
.8331
.9403
.9017
.8616
.7652
.7930
2.1
2.2
2.3
2.4
2.5
.6711
.6331
.5950
.5570
.5193
.6984
.6593
.6201
.5811
.5424
.7254
.6853
.6451
.6050
.5653
.7522
.7110
.6698
.6287
.5881
2.6
2.7
.4822
.4456
.5042
.4666
.5261
.4876
.5479
2.8
.4099
.4299
.4499
2.9
3.0
.3750
.3411
.3940
.3591
.4130
.3772
3.1
3.2
3.3
3.6
.3082
.2765
.2461
.2168
.1889
.3253
.2927
.2612
.2311
.2023
3.6
3.7
3.8
3.9
4.0
.1623
.1371
.1134
.0910
.0701
4.2
4.4
4.6
.0327
.0125
3.4
1.0647
1.1306
1.1737
1.1978
1.2065
.0000
.2554
.4784
.6665
.8196
.9399
.0000
.2177
.4085
.5705
.7088
.8098
1.3
.0000
.2837
.5307
.7378
.9053
1.0356
.0000
.2460
.4610
.6426
.7908
.9076
.0000
.2083
.3910
.5464
.6745
.7768
1.1
1.2
.0000
.2743
.5133
.7141
.8769
1.0039
.0000
.2366
.4435
.6186
.7619
.8751
.0000
.1988
.3734
.5222
.6452
.7436
.9
1.0
.0000
.2648
.4959
.6903
.8483
.9719
.0000
.2271
.4260
.5946
.7329
.8425
.1
.2
.3
.4
.5
.8
a=3.0
a=2.6
a=2.1
.7
a=2.7 a=2.8 a=2.9
a=2.4 a=2.5
a=2.2 a=2.3
x
.6
b=.5
Values of - du/dx
b =.5
.5085
.4699
.0551
.0037
-.0107
.5294
4899
-.0053
[27]
.5503
.6334
Table XV
b=.6
b=.6
Values of - du/War
x
a=.1
a=.2
a=.3
a=.4
a=.5
a=.6
a=.7
a=.8
a=.9
a=1.0
0
.0000
.0094
.0177
.0248
.0309
.0359
.0000
.0188
.0353
.0495
.0616
.0717
.0000
.0282
.0529
.0742
.0922
.1072
.0000
.0376
.0705
.0988
.1227
.1425
.0000
.0470
.0881
.1233
.1531
.1776
.0000
.0564
.1056
.1478
.1833
.2125
.0000
.0658
.1231
.1722
.2134
.2472
.0000
.0752
.1406
.1966
.2434
.2817
.0000
.0845
.1581
.2209
.2733
.3160
.0000
.0939
.1756
.2451
.3031
.3501
.0401
.0434
.0459
.0477
.0488
.0799
.0864
.0912
.0947
.0968
.1194
.1289
.1360
.1410
.1440
.1585
.1710
.1803
.1867
.1905
.1974
.2127
.2240
.2317
.2363
.2359
.2540
.2672
.2762
.2814
.2742
.2949
.3100
.3201
.3258
.3121
.3354
.3522
.3633
.3695
.3498
.3755
.3939
.4060
.4126
.3871
.4152
.4352
.4482
.4551
1.2
1.3
1.4
1.5
.0493
.0493
.0489
.0480
.0468
.0978
.0977
.0967
.0949
.0925
.1453
.1451
.1436
.1409
.1372
.1921
.1916
.1895
.1858
.1809
.2380
.2373
.2345
.2299
.2237
.2832
.2822
.2787
.2731
.2657
.3277
.3262
.3220
.3154
.3067
.3714
.3695
.3645
.3569
.3471
.4144
.4121
.4063
.3977
.3866
.4568
.4540
.4474
.4377
.4255
1.6
1.7
1.8
1.9
2.0
.0453
.0435
.0416
.0394
.0372
.0895
.0860
.0821
.0780
.0736
.1327
.1275
.1218
.1157
.1092
.1750
.1681
.1606
.1525
.1441
.2163
.2079
.1986
.1887
.1783
.2568
.2468
.2358
.2241
.2118
.2965
.2849
.2723
.2588
.2448
.3354
.3223
.3081
.2930
.2773
.3736
.3590
.3433
.3266
.3092
.4111
.3951
.3778
.3596
.3406
2.1
2.2
2.3
2.4
2.6
.0349
.0326
.0301
.0276
.0252
.0690
.0643
.0596
.0549
.0502
.1025
.0956
.0887
.0818
.0749
.1353
.1264
.1174
.1084
.0995
.1675
.1566
.1456
.1347
.1238
.1992
.1864
.1735
.1606
.1479
.2304
.2158
.2011
.1864
.1719
.2611
.2447
.2283
.2119
.1957
.2914
.2733
.2552
.2372
.2194
.3212
.3016
.2819
.2623
.2429
2.6
2.7
2.8
2.9
3.0
.0229
.0206
.0183
.0162
.0141
.0456
.0411
.0368
.0326
.0286
.0682
.0617
.0553
.0492
.0433
.0907
.0822
.0739
.0659
.0583
.1131
.1027
.0926
.0828
.0735
.1354
.1232
.1114
.0999
.0890
.1576
.1437
.1302
.1172
.1046
.1798
.1642
.1491
.1346
.1205
.2018
.1847
.1681
.1521
.1366
.2238
.2053
.1872
.1697
.1529
3.1
3.2
3.3
3.4
3.6
.0122
.0103
.0086
.0070
.0056
.0247
.0211
.0177
.0146
.0116
.0377
.0324
.0274
.0227
.0184
.0510
.0440
.0376
.0314
.0257
.0646
.0561
.0481
.0406
.0336
.0785
.0685
.0591
.0502
.0420
.0927
.0813
.0705
.0603
.0508
.1071
.0943
.0822
.0708
.0601
.1218
.1077
.0943
.0817
.0698
.1368
.1214
.1068
.0929
.0799
3.6
3.7
3.8
3.9
4.0
.0041
.0028
.0017
.0007
-.0003
.0089
.0064
.0041
.0021
.0002
.0144
.0107
.0073
.0042
.0015
.0204
.0155
.0111
.0070
.0034
.0271
.0210
.0155
.0106
.0059
.0342
.0271
.0205
.0145
.0090
.0419
.0337
.0261
.0191
.0128
.0501
.0408
.0322
.0243
.0170
.0587
.0483
.0387
.0299
.0218
.0677
.0563
.0458
.0360
.0270
4.2
4.4
-.0018
-.0028
-.0032
-.0028
-.0018
-.0003
.0019
.0046
.0077
.0114
-.0012
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
1.1
[ 28 ]
Table XV
x
b=.6
Values of - du/dx
b=.6
a=1.1
a=1.2 a=1.3
a=1.4 a=1.5
a=1.6
a=1.7 a=1.8
a=1,9 a=.2.0
.1
.2
.3
.4
.5
.0000
.1033
.1930
.2693
.3327
.3840
.0000
.1127
.2105
.2934
.3622
.4177
.0000
.1221
.2279
.3175
.3916
.4512
.0000
.1314
.2452
.3415
.4209
.4845
.0000
.1408
.2626
.3655
.4501
.5177
.0000
.1501
.2800
.3894
.4791
.5506
,0000
.1595
.2973
.4132
.5081
.5834
.0000
.1689
.3146
.4370
.5369
.6160
.0000
.1782
.3319
.4607
.5656
.6484
.0000
.1876
.3491
.4844
.5942
.6807
.6
.7
.8
.9
1.0
.4242
.4546
.4760
.4899
.4970
.4611
.4935
.5164
.5310
.5383
.4976
.5322
.5564
.5716
.5791
.5339
.5705
.5959
.6117
.6193
.5699
.6084
.6350
.6514
.6590
.6056
.6460
.6737
.6905
.6982
.6411
.6833
.7120
.7293
.7369
.6764
.7203
.7499
.7676
.7751
.7114
.7569
.7875
.8054
.8128
.7461
.7932
.8246
.8428
.8501
1.1
1.2
1.3
1.4
1.5
.4985
.4952
.4878
.4771
.4636
.5396
.5357
.5275
.5158
.5011
.5801
.5756
.5666
.5538
.5380
.6200
.6149
.6050
.5912
.5743
.6593
.6536
.6429
.6281
.6101
.6982
.6918
.6802
.6644
.6453
.7364
.7294
.7170
.7002
.6799
.7742
.7665
.7532
.7354
.7141
.8115
.8031
.7889
.7701
.7477
.8483
.8392
.8241
.8044
.7810
1.6
1.7
1.8
1.9
2.0
.4480
.4306
.4118
.3921
.3716
.4842
.4654
.4453
.4240
.4021
.5198
.4997
.4782
.4556
.4322
.5549
.5335
.5106
.4866
.4619
.5894
.5668
.5426
.5173
.4913
.6234
.5995
.5741
.5476
.5202
.6569
.6319
.6052
.5774
.5489
.6900
.6637
.6359
.6069
.5772
.7225
.6951
.6662
.6360
.6052
.7547
.7262
.6961
.6648
.6329
2.1
2.2
2.3
2.4
2.5
.3506
.3295
.3082
.2871
.2663
.3797
.3570
.3344
.3118
.2895
.4084
.3843
.3602
.3363
.3126
.4367
.4113
.3859
.3606
.3356
.4647
.4380
.4113
.3848
.3585
.4925
.4645
.4365
.4057
.3813
.5199
.4907
.4615
.4326
.4040
.5470
.5166
.4863
.4563
.4266
.5739
.5424
.5110
.4798
.4490
.6005
.5679
.5354
.5032
.4714
2.6
2.7
2.8
2.9
3.0
.2458
.2258
.2063
.1875
.1693
.2676
.2462
.2254
.2053
.1859
.2894
.2667
.2446
.2232
.2026
.3111
.2872
.2639
.2413
.2195
.3328
.3076
.2831
.2594
.2364
.3544
.3281
.3024
.2775
.2535
.3759
.3485
.3217
.2958
.2707
.3974
.3689
.3411
.3141
.2880
.4188
.3892
.3604
.3324
.3053
.4401
.4096
.3798
.3508
.3227
3.1
3.2
3.3
3.4
3.5
.1519
.1353
.1195
.1045
.0904
.1673
.1494
.1325
.1164
.1012
.1828
.1638
.1458
.1286
.1124
.1985
.1784
.1593
.1410
.1238
.2144
.1932
.1730
.1538
.1355
.2304
.2082
.1869
.1667
.1475
.2465
.2233
.2011
.1799
.1598
.2627
.2386
.2154
.1933
.1723
.2792
.2540
.2299
.2069
.1850
.2957
.2696
.2446
.2207
.1979
3.6
.0771
.0648
.0532
.0426
.0327
.0869
.0736
.0611
.0495
.0389
.0971
.0827
.0694
.0569
.0454
.1075
.0923
.0780
.0647
.0523
.1183
.1021
.0869
.0728
.0596
.1294
.1123
.0962
.0812
.0673
.1407
.1227
.1058
.0900
.0752
.1523
.1334
.1157
.0991
.0835
.1641
.1444
.1259
.1084
.0921
.1762
.1557
.1363
.1180
.1009
.0156
.0016
.0201
.0048
.0251
.0084
.0305
.0124
.0363
.0168
.0424
.0215
.0044
.0489
.0266
.0083
.0557
.0321
.0125
.0628
.0378
.0170
.0000
.0702
.0438
.0218
.0037
3.7
3.8
3.9
4.0
4.2
4.4
4.6
4.8
5.0
-.0009
[ 29 ]
-.0092
-.0064
-.0033
-.0106
Table XV
Values of du/dx
b=.6
x
a=2.1 a=2.2
a=2.3 a=2.4
b=.6
a=2.5 a=2.6
a=2.7
a=2.8 a=2.9
a=3.0
.5
.0000
.1969
.3664
.5080
.6227
.7127
.0000
.2062
.3836
.5315
.6511
.7446
.0000
.2156
.4008
.5550
.6794
.7763
.0000
.2249
.4180
.5785
.7076
.8079
.0000
.2343
.4352
.6019
.7356
.8393
.0000
.2486
.4523
.6252
.7636
.8705
.0000
.2629
.4694
.6485
.7914
.9015
.0000
.2623
.4865
.6717
.8192
.9324
.0000
.2716
.5036
.6949
.8468
.9631
.0000
.2809
.5207
.7180
.8744
.9937
.6
.7
.8
.9
1.0
.7806
.8292
.8614
.8799
.8869
.8149
.8649
.8979
.9165
.9233
.8489
.9003
.9340
.9527
.9593
.8827
.9355
.9697
.9886
.9948
.9162
.9703
1.0051
1.0240
1.0300
.9496
1.0048
1.0402
1.0591
1.0648
.9827
1.0391
1.0749
1.0939
1.0992
1.0156
1.0731
1.1094
1.1283
1.1332
1.0482
1.1068
1.1435
1.1623
1.1668
1.0807
1.1403
1.1773
1.1961
1.2002
1.1
1.2
1.3
1.4
1.5
.8846
.8748
.8589
.8382
.8137
.9205
.9099
.8932
.8716
.8461
.9559
.9447
.9271
.9045
.8780
.9909
.9789
.9605
.9370
.9096
1.0255
1.0128
.9935
.9691
.9407
1.0597
1.0463
1.0261
1.0008
.9715
1.0935
1.0793
1.0584
1.0321
1.0020
1.1270
1.1120
1.0902
1.0631
1.0320
1.1600
1.1443
1.1217
1.0938
1.0618
1.1928
1.1763
1.1529
1.1241
1.0912
1.6
1.7
1.8
1.9
.7864
.7568
.7256
.6933
.6603
.8177
.7871
.7548
.7215
.6874
.8486
.8170
.7837
.7494
.7143
.8792
.8465
.8123
.7769
.7409
.9094
.8758
.8406
.8043
.7673
.9392
.9047
.8685
.8313
.7934
.9687
.9333
.8962
.8581
.8193
.9979
.9616
.9236
.8847
.8450
1.0268
.9896
.9508
.9110
.8705
1.0654
1.0173
.9777
.9370
.8958
.6268
.5932
.5597
.5265
.4937
.6529
.6183
.5838
.5496
.5158
.6788
.6432
.6078
.5726
.5379
.7045
.6679
.6316
.5955
.5599
.7299
.6925
.6552
.6183
.5818
.7552
.7169
.6787
.6409
.6036
.7802
.7411
.7021
.6635
.6254
.8061
.7651
.7253
.6859
.6470
.8297
.7890
.7484
.7082
.6686
.8542
.8127
.7714
.7304
.6901
.4614
.4299
.3991
.3692
.3402
.4826
.4502
.4185
.3877
.3578
.5038
.4704
.4379
.4062
.3754
.5249
.4907
.4572
.4247
.3931
.5460
.5109
.4766
.4433
.4109
.5670
.5311
.4960
.4618
.4287
.5879
.5512
.5154
.4804
.4465
.6088
.5713
.5347
.4990
.4643
.6296
.5914
.5540
.5176
.4822
.6503
.6114
.5733
.5362
.5001
3.5
.3123
.2853
.2594
.2346
.2110
.3289
.3011
.2744
.2488
.2243
.3457
.3171
.2895
.2630
.2377
.3626
.3331
.3047
.2775
.2514
.3795
.3492
.3200
.2920
.2651
.3965
.3654
.3355
.3067
.2791
.4135
.3817
.3510
.3215
.2931
.4307
.3981
.3667
.3364
.3073
.4478
.4145
.3824
.3514
.3216
.4650
.4310
.3981
.3665
.3360
3.6
3.7
3.8
3.9
4.0
.1885
.1671
.1469
.1279
.1101
.2009
.1788
.1578
.1381
.1195
.2136
.1907
.1690
.1484
.1291
.2265
.2028
.1803
.1590
.1390
.2395
.2150
.1918
.1698
.1491
.2527
.2275
.2035
.1808
.1594
.2660
.2401
.2154
.1920
.1698
.2795
.2528
.2275
.2034
.1805
.2931
.2657
.2397
.2149
.1914
.3068
.2788
.2520
.2266
.2024
4.2
4.4
4.6
4.8
.0779
.0502
.0268
.0076
.0858
.0568
.0322
.0119
.0940
.0637
.0378
.0163
.1025
.0708
.0437
.0211
.1112
.0782
.0498
.0260
.1201
.0858
.0562
.0312
.1292
.0936
.0628
.0367
.1386
.1017
.0696
.0424
.1481
.1099
.0767
.0482
.1578
.1184
.0839
.0543
0
.1
.2
.3
.4
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
[ 30 ]
Table XV
b=.7
Values of - du/dx
b=.7
a=.1
a=.2
a=.3
a=.4
a=.5
a=.6
a=.7
a=.8
a=.9 a=.1.0
.0000
.0093
.0173
.0241
.0298
.0344
.0000
.0186
.0346
.0481
.0594
.0686
.0000
.0279
.0519
.0721
.0889
.1026
.0000
.0372
.0691
.0961
.1183
.1364
.0000
.0465
.0864
.1199
.1476
.1700
.0000
.0558
.1036
.1437
.1768
.2034
.0000
.0651
.1208
.1675
.2059
.2367
.0000
.0744
.1380
.1912
.2349
.2697
.0000
.0837
.1552
.2149
.2637
.3026
.0000
.0930
.1723
.2384
.2924
.3353
.9
1.0
.0381
.0409
.0430
.0444
.0452
.0759
.0815
.0855
.0882
.0897
.1134
.1216
.1276
.1315
.1336
,1506
.1614
.1691
.1742
.1769
.1876
.2003
.2102
.2163
.2195
.2242
.2399
.2509
.2579
.2615
.2607
.2786
.2911
.2991
.3030
.2968
.3169
.3309
.3397
.3439
.3327
.3549
.8703
.3798
.3843
.3683
.3926
.4093
.4195
.4242
1.1
1.2
1.3
1.4
1.5
.0455
.0453
.0447
.0438
.0426
.0902
.0898
.0886
.0867
.0843
.1342
.1335
.1316
.1288
.1252
.1775
.1765
.1739
.1701
.1653
.2202
.2187
.2155
.2107
.2047
.2622
.2603
.2563
.2506
.2434
.3036
.3012
.2965
.2898
.2815
.3443
.3415
.3360
.3283
.3189
.3846
.3812
.3750
.3663
.3557
.4242
.4203
.4133
.4036
.3918
1.6
2.0
.0411
.0395
.0377
.0358
.0339
.0814
.0782
.0747
.0710
.0671
.1210
.1162
.1110
.1055
.0998
.1597
.1534
.1466
.1394
.1319
.1977
.1899
.1816
.1727
.1636
.2351
.2259
.2159
.2055
.1947
.2718
.2612
.2497
.2377
.2254
.3079
.2959
.2830
.2695
.2556
.3435
.3301
.3158
.3009
.2855
.3784
.3637
.3481
.3317
.3150
2.1
2.2
2.3
2.4
2.5
.0318
.0298
.0277
.0257
.0237
.0631
.0591
.0551
.0511
.0472
.0940
.0881
.0822
.0763
.0706
.1243
,1166
.1089
.1013
.0938
.1542
.1448
.1353
.1260
.1169
.1837
.1726
.1615
.1506
.1398
.2128
.2001
.1874
.1749
.1626
.2415
.2273
.2131
.1991
.1853
.2699
.2542
.2386
.2231
.2079
.2979
.2808
.2638
.2469
.2304
2.6
2.7
2.8
2.9
3.0
.0217
.0198
.0180
.0162
.0145
.0434
.0396
.0361
.0326
.0293
.0649
.0595
.0542
.0492
.0443
.0864
.0793
.0724
.0658
.0595
.1079
.0992
.0907
.0826
.0749
.1292
.1190
.1091
.0996
.0905
.1506
.1389
.1275
.1166
.1062
.1718
.1587
.1460
.1338
.1221
.1930
.1786
.1645
.1511
.1381
.2142
.1984
.1831
.1684
.1542
3.1
3.2
3.3
3.4
3.5
.0129
.0114
.0100
.0087
.0075
.0262
.0232
.0205
.0179
.0154
.0397
.0354
.0313
.0274
.0238
.0535
.0478
.0424
.0374
.0326
.0675
.0605
.0539
.0477
.0419
.0818
.0735
.0657
.0583
.0514
.0962
.0867
.0778
.0693
.0613
.1109
.1002
.0901
.0805
.0716
.1257
.1139
.1027
.0921
.0821
.1407
.1278
.1155
.1039
.0930
3.6
3.7
3.8
3.9
4.0
.0064
.0053
.0043
.0035
.0027
.0132
.0111
.0092
.0075
.0059
.0205
.0174
.0146
.0120
.0096
.0282
.0242
.0204
.0170
.0138
.0364
.0314
.0267
.0224
.0185
.0450
.0390
.0334
.0283
.0236
.0539
.0470
.0405
.0346
.0291
.0632
.0553
.0480
.0412
.0350
.0728
.0640
.0559
.0483
.0413
.0827
.0730
.0640
.0557
.0479
4.2
4.4
4.6
.0013
.0003
.0032
.0010
.0055
.0023
.0084
.0039
.0005
.0116
.0061
.0017
.0154
.0086
.0033
.0195
.0116
.0052
.0003
.0240
.0149
.0076
.0017
.0289
.0187
.0103
.0036
.0342
.0227
.0133
.0057
x
.1
.2
.3
.4
.5
.6
.7
8
1.7
1.8
1.9
-.0007
-.0003
4.8
5.0
[ 31 ]
-.0009
-.0002
Table XV
Values of du/dx
b=.7
a=1.2
a=1.3 a=.1.4
a=1.5 a=1.6
b=.7
a=1.9 a=2.0
x
a--1.1
a=1.7
a=1.8
o
.1
.2
.3
.4
.5
.0000
.1023
.1894
.2620
.3210
.3678
.0000
.1116
.2065
.2854
.3495
.4002
.0000
.1209
.2236
.3089
.3780
.4324
.0000
.1301
.2407
.3323
.4063
.4643
.0000
.1394
.2577
.3556
.4345
.4962
.0000
.1487
.2747
.3788
.4625
.5278
.0000
.1580
.2917
.4020
.4905
.5593
.0000
.1672
.3087
.4252
.5184
.5906
.0000
.1765
.3257
.4483
.5461
.6218
.0000
.1857
.3426
.4713
.5738
.6528
.6
.7
.8
.9
1.0
.4037
.4299
.4479
.4587
.4635
.4388
.4669
.4860
.4974
.5023
.4737
.5037
.5239
.5358
.5407
.5083
.5401
.5613
.5737
.5786
.5427
.5761
.5984
.6111
.6160
.5769
.6119
.6351
.6482
.6530
.6108
.6474
.6714
.6848
.6896
.6445
.6826
.7074
.7211
.7257
.6780
.7175
.7431
.7570
.7614
.7112
.7521
.7784
.7925
.7968
1.1
1.2
1.3
1.4
1.5
.4633
.4588
.4510
.4403
.4275
.5018
.4968
.4882
.4766
.4626
.5399
.5343
.5248
.5123
.4972
.5774
.5712
.5609
.5474
.5313
.6145
.6076
.5965
.5821
.5649
.6511
.6436
.6317
.6162
.5980
.6872
.6791
.6663
.6500
.6308
.7229
.7141
.7006
.6833
.6631
.7581
.7487
.7343
.7161
.6950
.7930
.7829
.7677
.7486
.7265
1.6
1.7
1.8
1.9
2.0
.4126
.3969
.3799
.3622
.3441
.4468
.4296
.4113
.3923
.3728
.4802
.4618
.4422
.4220
.4012
.5132
.4935
.4728
.4513
.4293
.5457
.5249
.5030
.4803
.4571
.5777
.5558
.5328
.5089
.4846
.6094
.5864
.5622
.5372
.5118
.6406
.6166
.5913
.5652
.5387
.6715
.6464
.6201
.5930
.5654
.7020
.6759
.6486
.6204
.5918
2.1
2.2
2.3
2.4
2.5
.3257
.3072
.2888
.2706
.2527
.3531
.3333
.3136
.2941
.2750
.3802
.3592
.3382
.3175
.2972
.4071
.3848
.3626
.3407
.3192
.4337
.4102
.3869
.3638
.3412
.4600
.4364
.4110
.3868
.3631
.4861
.4604
.4349
.4097
.3849
.5120
.4852
.4586
.4324
.4066
.5376
.5098
.4822
.4550
.4282
.5630
.5342
.5057
.4775
.4497
2.6
2.7
2.8
2.9
3.0
.2353
.2182
.2017
.1858
.1705
.2563
.2380
.2204
.2033
.1869
.2773
.2579
.2391
.2209
.2034
.2982
.2777
.2578
.2385
.2200
.3190
.2974
.2765
.2562
.2367
.3398
.3172
.2952
.2740
.2535
.3605
.3370
.3140
.2918
.2703
.3813
.3567
.3327
.3096
.2872
.4020
.3764
.3515
.3274
.3042
.4226
.3961
.3703
.3453
.3212
3.1
3.2
3.3
3.4
3.5
.1559
.1419
.1286
.1160
.1041
.1712
.1562
.1419
.1283
.1155
.1867
.1706
.1553
.1408
.1271
.2022
.1852
.1690
.1535
.1389
.2179
.2000
.1828
.1665
.1509
.2337
.2149
.1968
.1796
.1632
.2497
.2299
.2109
.1928
.1756
.2657
.2450
.2252
.2063
.1883
.2818
.2602
.2396
.2199
.2011
.2979
.2756
.2541
.2336
.2140
3.6
3.7
3.8
3.9
4.0
.0929
.0824
.0725
.0634
.0549
.1034
.0920
.0813
.0714
.0622
.1141
.1019
.0904
.0797
.0697
.1251
.1120
.0998
.0883
.0776
.1363
.1224
.1094
.0972
.0857
.1477
.1331
.1193
.1063
.0942
.1593
.1439
.1294
.1157
.1028
.1712
.1550
.1397
.1252
.1117
.1832
.1662
.1502
.1351
.1208
.1954
.1777
.1609
.1451
.1302
4.2
4.4
4.6
4.8
5.0
.0398
.0271
.0166
.0081
.0017
.0457
.0318
.0202
.0108
.0034
.0519
.0368
.0242
.0138
.0056
.0584
.0421
.0284
.0171
.0080
.0652
.0477
.0329
.0206
.0107
.0723
.0535
.0377
.0244
.0136
.0796
.0597
.0427
.0285
.0168
.0872
.0660
.0479
.0327
.0202
.0951
.0726
.0534
.0372
.0238
.1031
.0795
.0592
.0420
.0276
-.0024
-.0009
.0009
.0028
.0050
.0074
.0100
.0020
.0129
.0042
.0159
.0065
5.2
6.4
[ 32 ]
Table XV
x
b=.7
Values of - du/dx
b=.7
a=2.1
a=2.2
a=2.3 a=2.4
a=2.5
a=2.6 a=2.7 a=2.8
a=2.9
a=3.0
.0000
.2597
.4775
.6538
.7915
.8951
.0000
.2689
.4943
.6763
.8183
.9247
.0000
,2782
.5110
.6989
.8449
.9542
.1
.2
.3
.4
.5
.0000
.1950
.3595
.4943
.6014
.6836
.0000
.2043
.3765
.5173
.6288
.7143
.0000
.2135
.3933
.5401
.6562
.7448
.0000
.2228
.4102
.5630
.6835
.7752
.0000
.2320
.4271
.5857
.7106
.8054
.0000
.2412
.4439
.6085
.7377
.8354
.0000
.2505
.4607
.6311
.7646
.8653
.6
.7
.8
.9
1.0
.7442
.7865
.8134
.8277
.8317
.7771
.8206
.8481
.8625
.8663
.8097
.8544
.8825
.8970
.9005
.8420
.8880
.9166
.9311
.9343
.8742
.9213
.9503
.9649
.9678
.9062
.9543
.9838
.9984
1.0009
.9379
.9871
1.0170
1.0315
1.0337
.9695
1.0196
1.0499
1.0644
1.0662
1.0009
1.0519
1.0826
1.0969
1.0984
1.0320
1.0840
1.1149
1.1291
1.1302
1.1
1.2
1.3
1.4
1.5
.8275
.8167
.8007
.7807
.7576
.8615
.8501
.8333
.8124
.7884
.8952
.8831
.8655
.8437
.8188
.9285
.9156
.8973
.8747
.8488
.9614
.9479
.9287
.9053
.8785
.9940
.9798
.9598
.9356
.9080
1.0263
1.0113
.9906
.9655
.9371
1.0582
1.0426
1.0211
.9952
.9659
1.0898
1.0735
1.0512
1.0245
.9944
1.1211
1.1040
1.0810
1.0535
1.0226
1.6
1.8
1.9
2.0
.7322
.7050
.6767
.6476
.6180
.7620
.7339
.7046
.6745
.6439
.7915
.7624
.7322
.7011
.6697
.8206
.7906
.7595
.7275
.6952
.8494
.8186
.7865
.7537
.7205
.8780
.8462
.8133
.7796
.7455
.9062
.8736
.8399
.8054
.7704
.9342
.9008
.8662
.8309
.7952
.9619
.9277
.8923
.8562
.8197
.9893
.9543
.9181
.8813
.8440
2.1
2.2
2.3
2.4
2.5
.5882
.5585
.5290
.4998
.4712
.6132
.5825
.5521
.5221
.4925
.6380
.6065
.5751
.5442
.5138
.6626
.6302
.5980
.5663
.5350
.6871
.6538
.6208
.5882
.5561
.7114
.6772
.6434
.6100
.5772
.7354
.7005
.6659
.6318
.5982
.7593
.7237
.6883
.6534
.6191
.7831
.7467
.7106
.6749
.6399
.8067
.7696
.7327
.6964
.6606
2.6
2.7
.4431
.4157
.3891
.3633
.3383
.4636
.4354
.4079
.3812
.3554
.4840
.4550
.4267
.3992
.3726
.5044
.4745
.4454
.4172
.3898
.5247
.4941
.4642
.4352
.4070
.5450
.5136
.4830
.4632
.4243
.5652
.5331
.5017
.4712
.4416
.5854
.5525
.5204
.4892
.4589
.6055
.5719
.5391
.5072
.4763
.6256
.6913
.5578
.5253
.4936
.3142
.2910
.2687
.2474
.2271
.3305
.3065
.2835
.2614
.2403
.3469
.3221
.2983
.2755
.2537
.3633
.3378
.3133
.2897
.2672
.3798
.3536
.3283
.3040
.2808
.3964
.3694
.3434
.3185
.2945
.4130
.3853
.3586
.3330
.3083
.4296
.4012
.3739
.3476
.3223
.4463
.4172
.3892
.3622
.3363
.4629
.4333
.4046
.3770
.3504
3.9
4.0
.2077
.1893
.1718
.1553
.1397
.2202
.2010
.1829
.1657
.1494
.2328
.2130
.1941
.1762
.1594
.2456
.2251
.2055
.1870
.1695
.2586
.2373
.2171
.1979
.1797
.2716
.2497
.2288
.2090
.1902
.2847
.2622
.2407
.2202
.2008
.2980
.2748
.2527
.2316
.2115
.3114
.2875
.2648
.2430
.2224
.3248
.3004
.2770
.2646
.2334
4.2
4.4
4.6
4.8
5.0
.1113
.0865
.0651
.0469
.0316
.1195
.0938
.0713
,0520
.0359
.1283
.1013
.0777
.0574
.0403
.1373
.1089
.0842
.0630
.0449
.1463
.1168
.0910
.0687
.0497
.1555
.1249
.0979
.0746
.0547
.1649
.1331
.1051
.0807
.0598
.1745
.1415
.1124
.0870
.0652
.1841
.1500
.1198
.0935
.0707
.1940
.1587
.1275
.1001
.0763
5.2
5.4
5.6
.0191
.0090
.0010
.0225
.0116
.0030
.0261
.0145
.0052
.0298
.0175
.0076
.0338
.0206
.0100
.0379
.0240
.0127
.0422
.0275
.0155
.0466
.0311
.0184
.0512
.0349
.0215
.0560
.0389
.0247
1.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
3.6
3.6
3.7
3.8
[ 33 ]
Table XV
b=.8
Values of - du/dx
a=.1
a=.2
.5
.0000
.0092
.0170
.0235
.0287
.0329
.0000
.0184
.0340
.0468
.0573
.0657
.6
.7
.8
.9
1.0
.0362
.0387
.0404
.0414
.0420
1.1
1.2
1.3
b=.8
a=.4
a=.5
a=.6
a=.7
a=.8
a=.9
a=1.0
.0000
.0277
.0509
.0702
.0858
.0982
.0000
.0369
.0679
.0935
.1142
.1306
.0000
.0461
.0848
.1167
.1425
.1629
.0000
.0553
.1017
.1398
.1707
.1949
.0000
.0645
.1185
.1630
.1987
.2268
.0000
.0737
.1354
.1860
.2267
.2585
.0000
.0829
.1522
.2090
.2545
.2900
.0000
.0921
.1691
.2320
.2823
.3214
.0721
.0770
.0803
.0824
.0834
.1078
.1149
.1198
.1229
.1243
.1432
.1526
.1589
.1628
.1646
.1784
.1899
.1977
.2024
.2045
.2134
.2268
.2360
.2415
.2438
.2481
.2635
.2740
.2801
.2826
.2825
.2999
.3115
.3183
.3210
.3168
.3360
.3487
.3561
.3589
.3508
.3717
.3866
.3934
.3963
1.4
1.5
.0421
.0417
.0411
.0401
.0390
.0835
.0828
.0815
.0796
.0773
.1243
.1233
.1212
.1184
.1149
.1646
.1631
.1603
.1565
.1519
.2044
.2024
.1989
.1941
.1884
.2435
.2411
.2368
.2311
.2243
.2822
.2792
.2742
.2675
.2596
.3203
.3168
.3111
.3035
.2944
.3579
.3539
.3474
.3388
.3287
.3951
.3906
.3832
.3737
.3626
1.6
1.7
1.8
1.9
2.0
.0376
.0361
.0345
.0329
.0311
.0746
.0716
.0685
.0652
.0618
.1109
.1066
.1019
.0970
.0920
.1467
.1409
.1348
.1284
.1218
.1819
.1748
.1672
.1593
.1513
.2165
.2081
.1991
.1898
.1803
.2506
.2409
.2306
.2199
.2090
.2843
.2733
.2617
.2496
.2374
.3174
.3052
.2923
.2790
.2654
.3501
.3367
.3226
.3080
.2931
2.1
2.5
.0294
.0276
.0259
.0241
.0224
.0583
.0549
.0515
.0481
.0448
.0869
.0819
.0768
.0718
.0670
.1152
.1085
.1019
.0954
.0891
.1431
.1349
.1268
.1189
.1111
.1707
.1611
.1515
.1422
.1329
.1980
.1870
.1760
.1653
.1547
.2250
.2126
.2003
.1883
.1766
.2517
.2380
.2245
.2112
.1981
.2782
.2632
.2484
.2339
.2196
2.6
2.7
2.8
2.9
3.0
.0208
.0192
.0177
.0162
.0148
.0415
.0384
.0354
.0325
.0298
.0622
.0576
.0532
.0490
.0449
.0829
.0769
.0711
.0655
.0602
.1035
.0961
.0890
.0822
.0757
.1240
.1153
.1070
.0989
.0913
.1445
.1346
.1250
.1158
.1069
.1650
.1538
.1430
.1327
.1227
.1854
.1730
.1611
.1496
.1386
.2057
.1923
.1792
.1667
.1546
3.2
3.3
3.4
3.5
.0135
.0122
.0111
.0100
.0089
.0272
.0247
.0224
.0202
.0182
.0410
.0375
.0340
.0308
.0278
.0552
.0504
.0459
.0416
.0376
.0695
.0636
.0580
.0528
.0478
.0839
.0770
.0704
.0641
.0583
.0985
.0905
.0829
.0757
.0689
.1133
.1042
.0956
.0875
.0799
.1281
.1181
.1086
.0995
.0910
.1431
.1321
.1217
.1118
.1024
3.6
3.7
3.8
3.9
4.0
.0080
.0071
.0063
.0065
.0048
.0163
.0145
.0129
.0114
.0100
.0249
.0223
.0199
.0176
.0155
.0339
.0304
.0272
.0242
.0214
.0432
.0388
.0348
.0311
.0276
.0527
.0476
.0428
.0383
.0342
.0626
0566
.0510
.0458
.0410
.0727
.0659
.0596
.0536
.0481
.0830
.0754
.0683
.0617
.0555
.0935
.0852
.0774
.0700
.0632
4.2
4.4
4.6
4.8
.0036
.0026
.0017
.0011
.0006
.0075
.0055
.0038
.0025
.0014
.0118
.0088
.0062
.0042
.0026
.0165
.0124
.0090
.0062
.0040
.0215
.0163
.0120
.0085
.0056
.0268
.0205
.0153
.0110
.0075
.0324
.0251
.0189
.0138
.0097
.0383
.0299
.0228
.0169
.0120
.0444
.0349
.0269
.0202
.0147
.0509
.0403
.0313
.0238
.0175
.0022
.0009
.0034
.0016
.0003
.0047
.0026
.0009
.0063
.0037
.0017
.0001
.0081
.0050
.0026
.0007
.0102
.0065
.0037
.0015
.0124
.0082
.0050
.0024
.0003
x
.1
.2
.3
.4
2.2
2.3
2.4
5.0
5.2
5.4
5.6
5.8
6.0
.0002
-.0001
.0006
.0001
-.0004
.0013
.0003
-.0004
-.0001
-.0007
[ 34 ]
-.0004
Table XV
Values of - du/dx
b=.8
b =.8
a=1.9 a=2.0
a=1.6 a=1.7
a=-_-1.8
.0000
.1473
.2696
.3687
.4467
.5063
.0000
.1564
.2863
.3913
.4738
.5366
.0000
.1656
.3029
.4138
.5007
.5667
.0000
.1748
.3196
.4363
.6276
.5966
.0000
.1839
.3362
.4587
.5543
.6265
.5173
.5463
.5647
.5744
.5771
.5500
.5804
.5995
.6095
.6121
.5825
.6142
.6341
.6443
.6467
.6147
.6477
.6683
.6787
.6809
.6468
.6810
.7022
.7128
.7148
.6786
.7141
.7358
.7465
.7483
.5392
.5322
.5218
.5087
.4935
.5741
.5666
.5554
.5413
.5252
.6087
.6005
.5885
.5736
.5565
.6429
.6340
.6213
.6055
.5874
.6766
.6672
.6537
.6370
.6180
.7101
.6999
.6857
.6681
.6482
.7431
.7323
.7173
.6989
.6781
.4456
.4288
.4112
.3930
.3746
.4767
.4587
.4400
.4207
.4012
.5073
.4884
.4685
.4482
.4275
.5376
.5176
.4967
.4754
.4537
.5676
.5466
.6247
.5023
.4795
.5972
.5752
.5523
.5289
.5052
.6265
.6035
.5795
.5553
.5306
.6555
.6316
.6068
.5814
.5559
.3303
.3130
.2958
.2790
.2624
.3561
.3376
.3193
.3013
.2837
.3816
.3620
.3426
.3236
.3049
.4068
.3862
.3658
.3457
.3260
.4319
.4102
.3888
.3677
.3471
.4568
.4341
.4117
.3896
.3680
.4814
.4578
.4344
.4116
.3889
.5059
.4814
.4571
.4332
.4097
.5302
.5048
.4796
.4548
.4305
3.0
.2261
.2115
.1974
.1838
.1707
.2463
.2307
.2165
.2009
.1869
.2665
.2499
.2337
.2181
.2032
.2867
.2690
.2519
.2354
.2195
.3068
.2882
.2701
.2527
.2359
.3269
.3073
.2883
.2700
.2523
.3470
.3264
.3066
.2874
.2688
.3669
.3455
.3248
.3047
.2854
.3869
.3646
.3430
.3221
.3020
.4068
.3837
.3613
.3396
.3186
3.1
3.2
3.3
3.4
3.5
.1582
.1463
.1350
.1242
.1140
.1735
.1606
.1484
.1367
.1257
.1889
.1751
.1620
.1495
.1376
.2042
.1896
.1757
.1624
.1497
.2197
.2043
.1895
.1754
.1620
.2353
.2190
.2035
.1886
.1744
.2510
.2339
.2175
.2019
.1870
.2667
.2488
.2317
.2153
.1997
.2826
.2639
.2460
.2288
.2125
.2984
.2790
.2603
.2425
.2254
3.6
3.7
3.8
3.9
4.0
.1043
.0952
.0866
.0786
.0711
.1153
.1054
.0961
.0874
.0792
.1264
.1158
.1058
.0964
.0876
.1378
.1264
.1157
.1057
.0962
.1493
.1372
.1259
.1151
.1050
.1610
.1482
.1361
.1247
.1140
.1728
.1594
.1466
.1346
.1232
.1848
.1706
.1572
,1446
.1326
.1969
.1821
.1680
.1547
.1421
.2092
.1937
.1790
.1650
.1518
4.2
4.4
4.6
4.8
5.0
.0576
.0459
.0360
.0276
.0206
.0645
.0517
.0408
.0316
.0238
.0717
.0578
.0459
.0358
.0273
.0791
.0641
.0612
.0402
.0309
.0867
.0706
.0567
.0449
.0348
.0945
.0773
.0625
.0497
.0388
.1025
.0842
.0684
.0547
.0430
.1107
.0914
.0745
.0599
.0474
.1190
.0986
.0808
.0653
.0520
.1276
.1061
.0873
.0709
.0561
5.2
5.4
5.6
5.8
6.0
.0148
.0101
.0064
.0035
.0012
.0174
.0122
.0080
.0046
.0021
.0202
.0144
.0097
.0060
.0031
.0232
.0168
.0116
.0074
.0042
.0263
.0193
.0136
.0090
.0054
.0297
.0220
.0158
.0107
.0067
.0332
.0249
.0181
.0126
.0081
.0368
.0279
.0206
.0145
.0097
.0407
.0311
.0232
.0166
.0113
.0446
.0344
.0259
.0189
.0131
x
a=1.3 a=-1.4
a=1.5
.0000
.1197
.2194
.3006
.3649
.4146
.0000
.1289
.2361
.3233
.3923
.4453
.0000
.1381
.2529
.3460
.4195
.4759
.4181
.4424
.4582
.4670
.4699
.4514
.4773
.4940
.5032
.5060
.4845
.5119
.5295
.5390
.5418
.4318
.4266
.4185
.4082
.3960
.4680
.4623
.4534
.4421
.4289
.5038
.4975
.4878
.4756
.4614
.3824
.3678
.3525
.3366
.3206
.4142
.3985
.3820
.3650
.3477
.3044
.2882
.2722
.2565
.2411
a=1.1 a=1.2
.3
.4
.5
.0000
.1013
.1859
.2549
.3100
.3526
.0000
.1105
.2026
.2778
.3375
.3837
.6
.7
.8
.9
1.0
.3845
.4072
.4221
.4304
.4333
1.1
1.2
1.3
.1
.2
1.4
1.5
1.6
1.7
1.8
1.9
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
[ 35 ]
Table XV
Values of - du/dx
b=--.8
x
a=2.1 a=2.2
a=2.3
a=2.5 a=2.6
b=.8
a=2.7 a=2.8 a=2.9
a=3.0
.1
.2
.8
.4
.5
.0000
.1931
.8529
.4811
.5810
.6561
.0000
.2023
.3695
.5035
.6076
.6857
.0000
.2114
.3860
.5257
.6341
.7150
.0000
.2206
.4026
.5480
.6604
.7443
.0000
.2298
.4191
.5702
.6867
.7734
.0000
.2389
.4356
.6923
.7129
.8023
.0000
.2481
.4522
.6144
.7390
.8311
.0000
.2572
.4686
.6365
.7651
.8598
.0000
.2664
.4851
.6585
.7910
.8884
.0000
.2755
.5016
.6804
.8168
.9168
.6
.7
.8
.9
1.0
.7103
.7469
.7692
.7800
.7815
.7417
.7794
.8023
.8131
.8144
.7729
.8117
.8350
.8459
.8469
.8040
.8438
.8676
.8784
.8791
.8349
.8757
.8998
.9106
.9110
.8655
.9073
.9318
.9426
.9426
.8960
.9387
.9635
.9742
.9739
.9263
.9699
.9950
1.0056
1.0050
.9566
1.0008
1.0262
1.0367
1.0357
.9864
1.0315
1.0572
1.0675
1.0661
1.1
1.2
1.3
1.4
1.5
.7758
.7644
.7486
.7294
.7077
.8082
.7961
.7795
.7595
.7370
.8402
.8275
.8102
.7893
.7659
.8719
.8585
.8405
.8189
.7946
.9033
.8892
.8704
.8481
.8230
.9343
.9197
.9001
.8770
.8511
.9651
.9498
.9295
.9056
.8790
.9956
.9796
.9586
.9340
.9065
1.0258
1.0092
.9875
.9621
.9339
1.0557
1.0384
1.0160
.9899
.9609
1.6
1.7
1.8
1.9
2.0
.6842
.6593
.6336
.6074
.5809
.7125
.6868
.6602
.6331
.6057
.7406
.7140
.6866
.6586
.6303
.7685
.7410
.7127
.6838
.6548
.7960
.7677
.7386
.7089
.6790
.8233
.7942
.7643
.7338
.7031
.8504
.8205
.7897
.7585
.7270
.8772
.8465
.8150
.7830
.7508
.9038
.8723
.8400
.8073
.7744
.9301
.8979
.8649
.8315
.7979
2.1
2.2
2.3
2.4
2.5
.5543
.6280
.6019
.4763
.4511
.5783
.5511
.5242
.4977
.4717
.6021
.5740
.5463
.5190
.4923
.6257
.5969
.5683
.5403
.5127
.6492
.6195
.5902
.5614
.5331
.6725
.6421
.6120
.5825
.5535
.6967
.6645
.6337
.6034
.5737
.7187
.6868
.6553
.6243
.5939
.7816
.7090
.6768
.6451
.6140
.7643
.7310
.6982
.6658
.6341
2.6
2.7
2.8
2.9
3.0
.4266
.4027
.3795
.3570
.3353
.4464
.4217
.3977
.3745
.3520
.4661
.4407
.4159
.3919
.3687
.4858
.4596
.4341
.4094
.3855
.5055
.4785
.4523
.4269
.4023
.5251
.4974
.4705
.4444
.4191
.5446
.5163
.4887
.4619
.4359
.5641
.5351
.5068
.4794
.4527
.5836
.5539
.5260
.4969
.4696
.6030
.5727
.5431
.5144
.4866
3.1
3.2
3.3
3.4
3.5
.3143
.2942
.27 8
.2562
.2385
.3308
.3094
.2893
.2701
.2517
.3463
.3247
.3040
.2840
.2649
.3624
.3401
.3186
.2980
.2783
.3785
.3555
.3334
.3121
.2917
.3946
.3710
.3482
.3263
.3053
.4108
.3865
.3631
.3406
.3189
.4270
.4021
.3780
.3549
.3326
.4432
.4176
.3980
.8692
.3464
.4594
.4333
.4081
.3837
.3602
3.6
3.7
.2216
.2064
.2341
.2173
.2981
.2658
.2474
.2298
.2783
.2592
.2411
.3244
.3377
.2126
.1968
.1818
.2535
.2357
.2187
.2025
.2851
.2013
.1861
.1717
.2413
.2241
.2077
.1921
.2722
.1901
.1755
.1617
.2467
.2293
.2594
3.8
8.9
4.0
.2131
.2238
4.2
4.4
4.6
4.8
5.0
.1363
.1137
.0939
.0766
.0616
.1451
.1215
.1007
.0825
.0667
.1542
.1295
.1077
.0885
.0719
.1633
.1376
.1148
.0948
.0772
.1726
.1821
.1543
.1295
.1076
.0884
.2014
.1715
.2112
.1221
.1011
.0828
.1917
.1628
.1371
.1143
.0942
.1448
.1210
.1002
.1526
.1280
.1062
.1606
.1350
.1124
5.2
5.4
6.6
5.8
6.0
.0488
.0379
.0288
.0212
.0150
.0531
.0415
.0318
.0237
.0170
.0575
.0453
.0349
.0263
.0191
.0621
.0492
.0382
.0290
.0213
.0669
.0532
.0416
.0318
.0237
.0717
.0574
.0451
.0347
.0261
.0767
.0617
.0487
.0378
.0286
.0819
.0661
.0525
.0410
.0313
.0872
.0706
.0564
.0442
.0340
.0926
.0763
.0604
.0476
.0368
.1459
[ 36 ]
.3112
.2908
.2711
.2524
.2346
.3034
.2832
.2639
.2455
.1803
.3160
.2953
.2754
.2565
.2212
.1892
Table XV
Values of - du/dx
b=-.9
x
a=.1
a=.2
a=.3
a=.4
b=.9
a=.5
a=.6
a=.7
a=.8
a=.9
a=1.0
.0000
.0639
.1164
.1586
.1919
.2175
.0000
.0730
.1329
.1810
.2189
.2479
.0000
.0821
.1494
.2034
.2458
.2782
.0000
.0912
.1659
.2258
.2727
.3083
.5
.0000
.0091
.0167
.0228
.0277
.0316
.0000
.0183
.0333
.0456
.0553
.0629
.0000
.0274
.0499
.0683
.0828
.0941
.0000
.0365
.0666
.0909
.1102
.1252
.0000
.0456
.0832
.1135
.1376
.1561
.0000
.0548
.0998
.1361
.1648
.1869
.6
.7
.8
.9
1.0
.0344
.0365
.0380
.0388
.0391
.0687
.0728
.0756
.0772
.0778
.1027
.1088
.1128
.1151
.1160
.1364
.1444
.1497
.1527
.1537
.1699
.1798
.1862
.1898
.1910
.2033
.2149
.2224
.2265
.2279
.2364
.2497
.2583
.2629
.2643
.2693
.2842
.2938
.2989
.3003
.3019
.3185
.3290
.3345
.3360
.3344
.3525
.3639
.3698
.3712
1.1
1.2
1.3
1.4
1.5
.0391
.0386
.0379
.0370
.0359
.0776
.0767
.0753
.0735
.0712
.1156
.1143
.1122
.1094
.1061
.1532
.1514
.1485
.1448
.1404
.1903
.1880
.1844
.1797
.1743
.2269
.2241
.2197
.2142
.2077
.2631
.2597
.2546
.2482
.2407
.2988
.2949
.2890
.2817
.2732
.3341
.3296
.3230
.3148
.3053
.3691
.3640
.3566
.3475
.3370
1.6
1.7
1.8
1.9
2.0
.0346
.0333
.0318
.0304
.0288
.0688
.0661
.0632
.0603
.0573
.1024
.0984
.0942
.0899
.0855
.1356
.1303
.1248
.1191
.1133
.1683
.1618
.1550
.1480
.1409
.2005
.1928
.1848
.1765
.1681
.2324
.2235
.2142
.2047
.1951
.2638
.2538
.2433
.2326
.2218
.2948
.2837
.2721
.2602
.2482
.3255
.3133
.3006
.2875
.2744
2.1
2.2
2.3
2.4
2.5
.0273
.0258
.0243
.0228
.0214
.0543
.0484
.0455
.0427
.0811
.0767
.0723
.0681
.0639
.1075
.1018
.0961
.0905
.0850
.1338
.1267
.1197
.1128
.1061
.1597
.1513
.1431
.1360
.1271
.1854
.1758
.1663
.1570
.1480
.2109
.2001
.1894
.1790
.1688
.2361
.2242
.2124
.2008
.1895
.2612
.2481
.2352
.2226
.2102
2.6
2.7
2.8
2.9
3.0
.0200
.0187
.0174
.0161
.0150
.0399
.0373
.0348
.0323
.0300
.0599
.0560
.0522
.0486
.0452
.0797
.0746
.0697
.0650
.0605
.0996
.0933
.0873
.0815
.0759
.1194
.1120
.1048
.0980
.0914
.1392
.1307
.1225
.1146
.1070
.1589
.1493
.1401
.1312
.1227
.1786
.1680
.1577
.1479
.1384
.1982
.1866
.1754
.1646
.1542
3.1
3.2
3.3
3.4
3.5
.0138
.0128
.0118
.0108
.0100
.0278
.0257
.0238
.0219
.0201
.0420
.0389
.0359
.0332
.0306
.0562
.0522
.0483
.0446
.0412
.0706
.0656
.0608
.0563
.0521
.0852
.0792
.0735
.0682
.0631
.0998
.0929
.0864
.0802
.0743
.1145
.1068
.0994
.0924
.0857
.1294
.1207
.1125
.1047
.0973
.1443
.1348
.1258
.1172
.1090
3.6
3.7
3.8
3.9
4.0
.0091
.0083
.0076
.0070
.0063
.0185
.0170
.0155
.0142
.0129
.0281
.0258
.0237
.0217
.0198
.0380
.0349
.0321
.0294
.0269
.0480
.0443
.0407
.0374
.0343
.0583
.0538
.0496
.0456
.0419
.0688
.0636
.0586
.0540
.0497
.0794
.0735
.0679
.0627
.0577
.0903
.0836
.0774
.0715
.0660
.1013
.0939
.0870
.0805
.0744
4.2
4.4
4.6
4.8
5.0
.0052
.0043
.0034
.0028
.0022
.0107
.0088
.0071
.0058
.0046
.0164
.0136
.0111
.0090
.0073
.0224
.0186
.0163
.0125
.0101
.0287
.0238
.0197
.0161
.0131
.0352
.0293
.0243
.0200
.0164
.0419
.0350
.0292
.0241
.0198
.0488
.0410
.0342
.0284
.0234
.0559
.0471
.0395
.0328
.0272
.0632
.0534
.0449
.0375
.0311
5.2
5.4
5.6
5.8
6.0
.0017
.0013
.0010
.0008
.0006
.0037
.0029
.0022
.0017
.0013
.0058
.0046
.0036
.0028
.0021
.0081
.0064
.0051
.0040
.0030
.0106
.0085
.0067
.0053
.0041
.0133
.0107
.0085
.0067
.0052
.0161
.0130
.0104
.0083
.0065
.0191
.0155
.0125
.0100
.0079
.0223
.0182
.0147
.0118
.0094
.0257
.0210
.0171
.0138
.0110
7.0
8.0
.0001
.0002
.0004
.0006
.0009
.0012
.0000
.0016
.0001
.0020
.0002
.0025
.0003
.0030
.0004
.1
.2
.3
.4
.C513
[ 37 ]
Table XV
Values of - du/dx
b=.9
a=1.2 a=1.3
=1.4
.0000
.1094
.1989
.2704
.3260
.3681
.0000
.1185
.2153
.2926
.3526
.3978
.0000
.1276
.2318
.3147
.3790
.4273
.0000
.1367
.2482
.3368
.4056
.4567
.3666
.3862
.3984
.4047
.4061
.3987
.4197
.4327
.4393
.4406
.4306
.4529
.4667
.4735
.4747
.4622
.4859
.5004
.5074
.5085
.4036
.3979
.3898
.3798
.3684
.4377
.4315
.4226
.4117
.3993
.4715
.4646
.4551
.4433
.4300
2.0
.3559
.3425
.3287
.3146
.3003
.3858
.3714
.3566
.3414
.3260
2.1
2.2
2.3
2.4
2.5
.2861
.2719
.2579
.2442
.2308
2.6
2.7
2.8
2.9
3.0
a=1.5 a=1.6
b=.9
a=1.9 a=2.0
a=1.7
a=1.8
.0000
.1458
.2646
.3589
.4316
.4860
.0000
.1549
.2810
.3809
.4578
.5151
.0000
.1640
.2973
.4028
.4839
.5441
.0000
.1781
.8137
.4248
.5099
.5729
.0000
.1822
.3300
.4466
.5358
.6016
.4937
.5186
.5338
.5410
.5420
.5249
.5611
.5669
.5743
.5751
.5560
.5834
.5998
.6073
.6079
.5869
.6154
.6323
.6400
.6403
.6176
.6472
.6646
.6724
.6725
.6481
.6747
.6967
.7045
.7043
.5049
.4974
.4871
.4745
.4602
.5379
.5298
.5187
.5053
.4902
.5706
.5619
.5501
.5358
.5198
.6029
.5936
.5810
.5660
.5491
.6349
.6250
.6117
.5958
.5780
.6666
.6561
.6421
.6254
.6067
.6980
.6869
.6721
.6546
.6352
.4154
.4001
.3841
.3679
.3515
.4447
.4284
.4114
.3942
.3768
.4737
.4564
.4385
.4202
.4018
.5024
.4841
.4652
.4460
.4267
.5308
.5116
.4918
.4716
.4514
.5589
.5388
.5180
.4970
.4758
.5867
.5657
.5441
.5222
.5001
.6143
.5924
.6699
.5471
.5242
.3107
.2955
.2805
.2667
.2514
.3351
.3189
.3029
.2872
.2719
.3594
.3422
.3252
.3085
.2922
.3835
.3653
.3474
.3298
.3126
.4074
.3883
.3694
.3509
.3328
.4311
.4111
.3913
.3720
.3530
.4547
.4338
.4132
.3929
.3732
.4781
.4563
.4349
.4138
.3932
.5014
.4788
.4565
.4346
.4132
.2178
.2052
.1931
.1814
.1701
.2374
.2239
.2108
.1982
.1860
.2569
.2425
.2285
.2150
.2020
.2764
.2611
.2462
.2319
.2181
.2958
.2796
.2639
.2487
.2341
.3152
.2982
.2816
.2657
.2503
.3346
.3167
.2993
.2826
.2664
.3539
.3352
.3171
.2995
.2826
.3732
.3537
.3348
.3165
.2989
.3924
.3722
.3525
.3335
.3151
3.1
3.2
3.3
3.4
3.5
.1593
.1490
.1392
.1298
.1209
.1744
.1633
.1527
.1425
.1329
.1896
.1777
.1663
.1554
.1451
.2048
.1921
.1800
.1684
.1573
.2201
.2067
.1938
.1815
.1697
.2355
.2213
.2077
.1947
.1823
.2509
.2360
.2217
.2080
.1949
.2664
.2507
.2357
.2213
.2076
.2819
.2655
.2498
.2348
.2204
.2974
.2804
.2640
.2483
.2333
8.6
3.7
3.8
3.9
4.0
.1124
.1044
.0969
.0897
.0830
.1238
.1151
.1069
.0991
.0918
.1352
.1259
.1170
.1087
.1008
.1468
.1368
.1274
.1184
.1099
.1585
.1479
.1378
.1283
.1192
.1704
.1591
.1484
.1383
.1287
.1824
.1705
.1592
.1484
.1382
.1945
.1819
.1700
.1587
.1480
.2066
.1935
.1810
.1691
.1578
.2189
.2052
.1921
.1797
.1678
4.2
4.4
4.6
4.8
5.0
.0707
.0599
.0505
.0423
.0352
.0784
.0667
.0563
.0473
.0395
.0863
.0735
.0623
.0525
.0440
.0944
.0806
.0685
.0578
.0486
.1026
.0878
.0748
.0633
.0533
.1110
.0952
.0812
.0690
.0582
.1195
.1027
.0879
.0747
.0633
.1281
.1104
.0946
.0807
.0685
.1369
.1182
.1016
.0868
.0738
.1459
.1262
.1086
.0930
.0793
5.2
5.4
5.6
6.8
6.0
.0292
.0240
.0195
.0158
.0127
.0328
.0271
.0222
.0180
.0145
.0366
.0303
.0249
.0203
.0165
.0406
.0337
.0278
.0227
.0185
.0447
.0372
.0307
.0253
.0206
.0489
.0408
.0338
.0279
.0228
.0533
.0446
.0371
,0306
.0252
.0578
.0485
.0404
.0335
.0276
.0624
.0525
.0439
.0365
.0301
.0672
.0566
.0474
.0395
.0327
7.0
8.0
9.0
.0036
.0006
.0043
.0008
.0050
.0010
.0058
.0012
.0066
.0015
.0000
.0074
.0018
.0001
.0084
.0021
.0001
.0093
.0024
.0002
.0104
.0027
.0003
.0115
.0031
.0004
x
a=1.1
.1
.2
.3
.4
.5
.0000
.1003
.1824
.2481
.2994
.3383
.6
.7
1.1
1.2
.8
.9
1.0
1.3
1.4
1.5
1.6
1.7
1.8
1.9
[ 38 ]
Table XV
b=.9
Values of
x
a=2.1
a=2.2
a=2.3
a=2.4
o
.1
.2
.3
.4
.6
.0000
.1912
.3463
.4684
.5616
.6302
.0000
.2003
.3626
.4902
.5873
.6586
.0000
.2094
.3789
.5119
.6130
.6869
.0000
.2185
.3952
.5336
.6385
.7151
.6
.7
.8
.9
1.0
.6784
.7101
.7284
.7363
.7359
.7086
.7412
.7600
.7678
.7671
.7386
.7721
.7913
.7991
.7981
1.1
1.2
1.3
1.4
1.5
.7290
.7173
.7018
.6836
.6633
.7598
.7476
.7313
.7123
.6912
1.6
1.7
1.8
1.9
2.0
.6416
.6189
.5955
.5719
.5482
2.1
2.2
2.3
2.4
2.5
b=.9
du/dx
a=2.5 a=2.6
a=2.7
.0000
.2276
.4114
.5552
.6640
.7431
.0000
.2366
.4276
.5768
.6893
.7711
.0000
.2457
.4438
.5983
.7146
.7988
.0000
.2547
.4600
.6198
.7398
.8265
.0000
.2638
.4762
.6413
.7650
.8540
.0000
.2728
.4923
.6626
.7900
.8814
.7684
.8028
.8223
.8301
.8288
.7980
.8333
.8531
.8608
.8592
.8275
.8636
.8837
.8913
.8894
.8567
.8936
.9140
.9216
.9192
.8859
.9235
.9441
.9516
.9489
.9149
.9532
.9740
.9813
.9783
.9437
.9827
1.0037
1.0108
1.0074
.7903
.7773
.7604
.7407
.7188
.8205
.8069
.7893
.7688
.7462
.8504
.8362
.8179
.7967
.7733
.8801
.8652
.8463
.8243
.8001
.9094
.8940
.8743
.8517
.8268
.9385
.9225
.9022
.8788
.8532
.9674
.9508
.9298
.9057
.8794
.9961
.9788
.9572
.9324
.9053
.6686
.6451
.6209
.5965
.5719
.6954
.6711
.6461
.6209
.5955
.7220
.6969
.6711
.6451
.6190
.7484
.7224
.6959
.6691
.6423
.7745
.7478
.7206
.6930
.6654
.8004
.7730
.7450
.7167
.6884
.8261
.7980
.7692
.7403
.7112
.8516
.8227
.7933
.7636
.7339
.8768
.8473
.8172
.7869
.7565
.5245
.5011
.4780
.4553
.4332
.5475
.5232
.4994
.4760
.4531
.5703
.5453
.5207
.4965
.4729
.5930
.5672
.5419
.5170
.4926
.6155
.5891
.5630
.5374
.5123
.6379
.6108
.5840
.5577
.5319
.6602
.6323
.6049
.5779
.5515
.6824
.6538
.6257
.5981
.5710
.7044
.6752
.6463
.6182
.5905
.7263
.6965
.6670
.6381
.6098
2.6
2.7
2.8
2.9
3.0
.4116
.3906
.3702
.3505
.3314
.4307
.4090
.3879
.3675
.3477
.4498
.4274
.4056
.3845
.3641
.4689
.4457
.4233
.4015
.3804
.4879
.4640
.4409
.4185
.3968
.5068
.4823
.4586
.4355
.4131
.5257
.5006
.4762
.4525
.4295
.5446
.5188
.4938
.4695
.4459
.5634
.5370
.5114
.4865
.4623
.5822
.5552
.5289
.5034
.4787
3.1
3.2
3.3
3.4
3.5
.3130
.2953
.2783
.2620
.2463
.3287
.3103
.2926
.2757
.2594
.3443
.3253
.3070
.2894
.2726
.3600
.3404
.3215
.3033
.2858
.3758
.3555
.3360
.3172
.2991
.3915
.3706
.3505
.3311
.3125
.4073
.3858
.3651
.3451
.3259
.4231
.4010
.3797
.3591
.3393
.4389
.4162
.3943
.3732
.3529
.4547
.4315
.4090
.3874
.3665
3.6
3.8
3.9
4.0
.2313
.2170
.2033
.1903
.1779
.2438
.2289
.2147
.2011
.1881
.2564
.2409
.2261
.2119
.1985
.2690
.2530
.2376
.2229
.2089
.2817
.2651
.2492
.2340
.2195
.2945
.2774
.2609
.2452
.2301
.3074
.2897
.2727
.2564
.2408
.3203
.3021
.2845
.2677
.2516
.3333
.3145
.2965
.2791
.2626
.3464
.3270
.3085
.2906
.2736
4.2
4.4
4.6
4.8
5.0
.1549
.1343
.1158
.0994
.0848
.1641
.1425
.1231
.1059
.0906
.1734
.1508
.1306
.1125
.0964
.1828
.1593
.1381
.1192
.1024
.1924
.1679
.1458
.1261
.1084
.2020
.1766
.1536
.1330
.1146
.2118
.1854
.1615
.1401
.1209
.2216
.1943
.1696
.1473
.1274
.2315
.2033
.1777
.1546
.1339
.2416
.2124
.1859
.1620
.1405
5.2
5.4
5.6
5.8
6.0
.0721
.0609
.0511
.0427
.0354
.0771
.0652
.0549
.0460
.0383
.0822
.0697
.0588
.0494
.0412
.0875
.0743
.0628
.0528
.0442
.0928
.0791
.0670
.0564
.0473
.0983
.0839
.0712
.0601
.0504
.1039
.0888
.0755
.0638
.0537
.1096
.0938
.0799
.0677
.0570
.1154
.0989
.0844
.0716
.0605
.1213
.1042
.0890
.0757
.0640
7.0
8.0
9.0
10.0
.0126
.0035
.0005
.0138
.0039
.0006
.0150
.0044
.0007
.0164
.0049
.0009
.0177
.0054
.0010
.0191
.0059
.0012
.0206
.0065
.0014
.0000
.0222
.0071
.0016
.0000
.0237
.0078
.0018
.0001
.0254
.0084
.0020
.0001
3.7
[ 39 ]
a=2.8 a=2.9
a=3.0
Table XVI
Values of - dis/dx
x
b=.1
b=.2
b =.3
b=.3
b=.4
b =.4
b =.4
b = .4
b =.5
b =.5
uo
.0200
.0800
.1000
-1800
.1000
.2000
.3000
.3200
.1000
.2000
0
.1
.2
.3
.4
.5
.2000
.1990
.1960
.1911
.1842
.1756
.4000
.3980
.3920
.3821
.3685
.3512
.2000
.1990
.1961
.1914
.1850
.1771
.6000
.5970
.5880
.5732
.5527
.5268
.1367
.1360
.1341
.1310
.1268
.1217
.3101
.3086
.3041
.2969
.2872
.2751
.6000
.5970
.5882
.5738
.5540
.5292
.8000
.7960
.7841
.7643
.7370
.7026
.1056
.1051
.1036
.1013
.0982
.0944
.2254
.2243
.2211
.2161
.2093
.2011
.6
.7
.8
.9
1.0
.1661
.1531
.1396
.1248,
.1089
.3304
.3066
.2799
.2507
.2195
.1678
.1572
.1455
.1329
.1196
.4958
.4603
.4206
.3776
.3316
.1157
.1090
.1017
.0939
.0858
.2610
.2451
.2276
.2090
.1893
.4999
.4665
.4296
.3899
.3479
.6614
.6143
.5620
.5054
.4455
.0900
.0852
.0800
.0745
.0687
.1915
.1809
.1693
.1571
.1443
1.1
1.2
1.3
.0921
.0745
.0563
.0379
.0194
.1867
.1527
.1181
.0832
.0486
.1058
.0916
.0772
.0628
.0486
.2838
.2344
.1848
.1351
.0866
.0773
.0688
.0601
.0515
.0430
.1690
.1483
.1275
.1068
.0865
.3044
.2601
.2155
.1714
.1283
.3833
.3198
.2562
.1935
.1326
.0628
.0569
.0509
.0450
.0392
.1312
.1180
.1047
.0916
.0787
.0148
.0346
.0211
.0081
-.0052
.0395
.0348
.0268
.0191
.0118
.0050
.0667
.0477
.0296
.0125
.0868
.0473
.0102
-.0315
.0744
.0195
.0335
.0281
.0229
.0179
.0133
.0662
.0542
.0428
.0320
.0219
.0090
.0049
.0013
-.0040
b=.6
1.4
1.5
1.6
1.7
1.8
1.9
2.0
.0011
-.0169
-.0180
-.0042
-.0034
-.0242
-.0014
2.1
2.2
2.3
2.4
-.0021
.0125
.0039
Table XVI
Values of - du/dx
b=.5
b=.5
b=.5
b=.6
b=.6
b=.6
b=.6
b=.6
b=.6
uo
.3000
.4000
.5000
.1000
.2000
.3000
.4000
.5000
.6000
.7000
0
.1
.2
.3
.4
.5
.3675
.3657
.3605
.3521
.3409
.3270
.5528
.5501
.5421
.5292
.5117
.4901
1.0000
.9950
.9801
.9554
.9213
.8783
.0865
.0860
.0849
.0830
.0805
.0776
.1802
.1793
.1768
.1729
.1677
.1615
.2835
.2821
.2782
.2719
.2636
.2535
.4000
.3981
.3924
.3835
.3716
.3570
.5367
.5341
.5264
.5142
.4979
.4778
.7101
.7066
.6964
.6799
.6576
.6308
1.0000
.9950
.9803
.9563
.9234
.8822
.6
.7
.8
.9
1.0
.3109
.2929
.2732
.2524
.2305
.4647
.4362
.4050
.3717
.3368
.8271
.7686
.7039
.6341
.5607
.0742
.0705
.0666
.0623
.0580
.1543
.1464
.1378
.1288
.1196
.2420
.2291
.2153
.2008
.1857
.3403
.3216
.3014
.2801
.2580
.4546
.4286
.4004
.3706
.3396
5981
.5621
5229
.4812
.4377
.8338
.7788
.7186
.6541
.5868
1.1
1.2
1.3
1.4
1.5
.2081
.1854
.1627
.1403
.1184
.3009
.2645
.2282
.1923
.1574
.4849
.4082
.3320
.2574
.1858
.0536
.0492
.0447
.0404
.0361
.1101
.1006
.0911
.0817
.0726
.1703
.1548
.1393
.1241
.1093
.2354
.2127
.1900
.1678
.1461
.3079
.2760
.2442
.2130
.1827
.3933
.3485
.3040
.2604
.2183
.5177
.4483
.3795
.3125
.2482
1.6
1.7
1.8
1.9
2.0
.0971
.0768
.0576
.0395
.0227
.1238
.0918
.0618
.0338
.0081
.1179
.0547
.0320
.0280
.0242
.0206
.0172
.0638
.0562
.0471
.0395
.0323
.0950
.0812
.0682
.0559
.0444
.1252
.1053
.0864
.0687
.0522
.1536
.1260
.0999
.0757
.0533
.1780
.1400
.1044
.0715
.0415
.1874
.1306
.0784
.0310
2.1
2.2
2.3
2.4
2.5
.0073
-.0153
.0140
.0111
.0083
.0058
.0036
.0256
.0194
.0137
.0085
.0038
.0337
.0240
.0150
.0069
.0371
.0232
.0106
.0328
.0143
.0144
.0099
.0015
-.0004
2.6
2.7
-.0068
-.0034
-.0003
[ 40 ]
-.0003
-.0006
-.0023
-.0115
Table XVI
Values of - du/dx
b=.7
b=.6
b=.7
b=.7
b=.7
b=.7
b=.7
b=.7
b=.7
b=.7
.7200
.1000
.2000
.3000
.4000
.5000
.6000
.7000
.8000
.9000
.0734
.0730
.2338
.2327
.2295
.2245
.2179
.2100
.3230
.3214
.3170
.3100
.3008
.2897
.4202
.4182
.4124
.4032
.3910
.3763
.5282
.5257
.6183
.5066
.4910
.4721
.6517
.6485
.6393
.6247
.6051
.5812
.8000
.7961
.7847
.7664
.7417
.7115
1.0000
.0705
.0685
.0661
.1510
.1503
.1482
.1450
.1408
.1358
.9930
.9233
.8464
.7639
.6775
.0634
.0604
.0572
.0539
.0505
.1301
.1239
.1173
.1103
.1032
.2011
.1912
.1807
.1696
.1583
.2770
.2631
.2482
.2325
.2164
.3594
.3409
.3209
.3000
.2783
.4504
.4263
.4005
.3733
.3452
.5536
.5230
.4899
.4550
.4190
.6765
.6373
.5949
.6501
.5037
.8400
.7883
.7321
.6725
.6105
.5888
.4997
.4119
.3267
.2455
.0471
.0437
.0403
.0369
.0336
.0959
.0887
.0815
.0744
.0675
.1468
.1353
.1238
.1126
.1017
.2001
.1837
.1675
.1515
.1361
.2564
.2344
.2126
.1913
.1706
.3166
.2880
.2597
.2319
.2051
.3823
.3455
.3092
.2736
.2393
.4564
.4090
.3623
.3167
.2727
.5474
.4842
.4220
.3615
.3035
.1694
.0991
.0352
-.0221
2.0
.0305
.0274
.0245
.0217
.0191
.0609
.0545
.0484
.0426
.0372
.0911
.0810
.0714
.0623
.0538
.1211
.1069
.0933
.0806
.0687
.1506
.1316
.1137
.0968
.0811
.1793
.1548
.1317
.1102
.0902
.2065
.1754
.1463
.1192
.0942
.2309
.1915
.1548
.1210
.0900
.2487
.1975
.1502
.1071
.0682
2.1
2.2
2.3
2.4
2.5
.0167
.0144
.0123
.0103
.0085
.0321
.0273
.0229
.0189
.0152
.0458
.0385
.0317
.0254
.0198
.0576
.0473
.0380
.0294
.0217
.0666
.0532
.0411
.0301
.0202
.0718
.0550
.0398
.0262
.0141
.0715
.0509
.0324
.0159
.0015
.0621
.0370
.0148
-.0237
2.6
2.7
2.8
.0118
.0088
.0061
.0037
.0015
.0146
.0101
.0060
.0024
-.0008
.0148
.0086
.0031
-.0016
.0114
.0037
-.0031
.0034
-.0111
-.0059
2.9
3.0
.0068
.0053
.0040
.0028
.0017
3.1
3.2
.0007
-.0004
-.0002
x
tto
0
.1
.2
.3
.4
.5
.6
.7
.8
.9
1.0
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.2000
1.1940
1.1761
1.1465
1.1056
1,0542
0720
[41]
-.0047
.9951
.9806
.9571
.9253
.8860
.0335
.0030
Table XVI
Values of du/dx
b=.7
b=.8
b=.8
b=.8
b=.8
b=.8
b=.8
b=.8
b=.8
b=.8
uo
.9800
.1000
.2000
.3000
.4000
.5000
.6000
.7000
.8000
.9000
0
.1
.2
.3
.4
.5
1.4000
1.3930
1,3721
1.3376
1.2900
1.2301
.0638
.0635
.0626
.0613
.0596
.0576
.1303
.1297
.1280
.1253
.1218
.1176
.2000
.1991
.1964
.1922
.1868
.1804
.2734
.2720
.2684
.2626
.2551
.2462
.3610
.3493
.3446
.3371
.3274
.3158
.4338
.4317
.4258
.4165
.4043
.3897
.5230
.5205
.5133
.5020
.4871
.4692
.6202
.6172
.6086
.5950
.5771
.5555
.7145
.6984
.6770
.6510
.6
.7
.8
.9
1.0
1.1591
1.0782
.9895
.8945
.7957
.0554
.0529
.0504
.0477
.0460
.1130
.1079
.1026
.0970
.0913
.1731
.1652
.1568
.1481
.1392
.2361
.2251
.2134
.2012
.1888
.3026
.2882
.2728
.2568
.2405
.3731
.3549
.3355
.3153
.2945
.4487
.4263
.4023
.3772
.3515
.5307
.6034
.4742
.4436
.4121
.6211
.5882
.5528
.5157
.4775
1.1
1.2
1.3
1.4
1.5
.6947
.5942
.4955
.4009
.3111
.0422
.0395
.0367
.0341
.0315
.0855
.0798
.0742
.0686
.0632
.1302
.1212
.1123
.1036
.0952
.1762
.1637
.1513
.1391
.1274
.2239
.2074
.1911
.1752
.1597
.2735
.2526
.2320
.2118
.1922
.3255
.2996
.2740
.2490
.2249
.3803
.3486
.3174
.2869
.2575
.4389
.4004
.3624
.3255
.2899
1.6
.2280
.1515
.0830
.0217
.0290
.0266
.0243
.0221
.0200
.0580
.0530
.0482
.0437
.0394
.0870
.0792
.0718
.0647
.0581
.1160
.1052
.0948
.0851
.0759
.1449
.1307
.1172
.1045
.0926
.1735
.1556
.1387
.1228
.1079
.2017
.1797
.1589
.1395
.1214
.2294
.2027
.1776
.1542
.1325
.2560
.2240
.1940
.1661
.1404
2.1
2.2
2.3
2.4
2.5
.0180
.0162
.0145
.0129
.0114
.0354
.0316
.0281
.0248
.0218
.0518
.0460
.0406
.0356
.0310
.0673
.0593
.0519
.0451
.0389
.0815
.0712
.0617
.0530
.0460
.0941
.0813
.0696
.0589
.0492
.1046
.0892
.0752
.0624
.0509
.1126
.0943
.0777
.0628
.0494
.1168
.0955
.0762
.0589
.0436
2.6
2.7
2.8
2.9
3.0
.0100
.0088
.0076
.0065
.0056
.0190
.0165
.0141
.0120
.0101
.0268
.0229
.0194
.0163
.0134
.0331
.0280
.0233
.0190
.0153
.0378
.0313
.0254
.0201
.0155
.0405
.0326
.0255
.0193
.0138
.0405
.0313
.0231
.0169
.0097
.0375
.0269
.0176
.0095
.0026
.0301
.0183
.0080
3.1
.0047
.0039
.0032
.0026
.0020
.0084
.0068
.0054
.0042
.0031
.0108
.0085
.0065
.0047
.0031
.0119
.0089
.0063
.0040
.0020
.0114
.0077
.0046
.0018
.0089
.0047
.0011
-.0005
.0042
-.0034
-.0084
.0016
.0011
.0007
.0004
.0001
.0022
.0013
.0006
.0001
-.0004
1.7
1.8
1.9
2.0
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
-.0314
.0018
.0006
.0003
-.0012
-.0024
-.0006
[ 42 ]
-.0005
-.0020
.7282
.7247
-.0009
-.0084
Table XVI
Values of du/dx
x
ik
b =.8
1.0000
b=.8
b=.8
1.1000
1.2000
b=.8
b=.9
b=.9
b=.9
b=.9
b=.9
b=.9
1.2800
.1000
.2000
.3000
.4000
.5000
.6000
.1752
.1744
.1721
.1685
.1639
.1585
.2380
.2368
.2337
.2288
.2225
.2151
.3033
.3019
.2979
.2916
.2835
.2740
.3717
.3699
.3650
.3573
.3473
.3354
.8517
.8475
.8365
.8164
.7908
.7597
1.0000
.9951
.9809
.9580
.9272
.8895
1.2000
1.1941
1.1767
1.1486
1.1104
1.0633
1.6000
1.5920
1.5681
1.5287
1.4743
1.4060
.0564
.0562
.0554
.0543
.0529
.0512
.1148
.1142
.1127
.1104
.1074
.1040
1.0
.7237
.6839
.6411
.5960
.5496
.8458
.7972
.7447
.6893
.6322
1.0083
.9467
.8799
.8092
.7361
1.3252
1.2335
1.1330
1.0260
.9151
.0493
.0472
.0451
.0429
.0406
.1001
.0958
.0914
.0868
.0822
.1525
.1460
.1391
.1320
.1248
.2068
.1978
.1883
.1785
.1685
.2632
.2516
.2393
.2265
.2135
.3220
.3075
.2921
.2762
.2600
1.1
1.2
1.3
1.4
1.5
.5026
.4557
.4097
.3649
.3219
.5743
.5167
.4600
.4051
.3526
.6620
.5883
.5161
.4464
.3800
.8027
.6912
.5828
.4793
.3823
.0384
.0361
.0339
.0317
.0296
.0775
.0729
.0683
.0638
.0595
.1175
.1103
.1032
.0963
.0895
.1585
.1485
.1387
.1291
.1198
.2005
.1875
.1747
.1623
.1502
.2436
.2274
.2114
.1959
.1808
1.6
1.7
2.0
.2810
.2426
.2067
.1736
.1433
.3029
.2565
.2134
.1740
.1382
.3177
.2600
.2070
.1590
.1160
.2927
.2115
.1387
.0746
.0188
.0276
.0256
.0238
.0220
.0203
.0553
.0512
.0474
.0437
.0402
.0830
.0768
.0708
.0652
.0598
.1108
.1022
.0940
.0862
.0789
.1386
.1274
.1169
.1068
.0974
.1663
.1525
.1393
.1269
.1153
2.1
2.2
2.3
2.4
2.5
.1158
.0909
.0688
.0491
.0318
.1059
.0772
.0518
.0295
.0102
.0778
.0443
.0152
-.0290
.0187
.0172
.0157
.0144
.0132
.0369
.0338
.0309
.0282
.0257
.0547
.0500
.0456
.0114
.0376
.0720
.0655
.0595
.0539
.0486
.0886
.0803
.0726
.0654
.0588
.1043
.0942
.0847
.0760
.0679
.0167
.0037
-.0063
.0120
.0109
.0099
.0090
.0081
.0233
.0211
.0191
.0173
.0155
.0340
.0307
.0276
.0248
.0222
.0438
.0394
.0353
.0315
.0281
.0527
.0471
.0420
.0373
.0330
.0606
.0538
.0476
.0420
.0369
3.1
3.2
3.3
3.4
3.5
.0073
.0066
.0059
.0053
.0047
.0140
.0125
.0112
.0100
.0089
.0199
.0177
.0157
.0140
.0123
.0250
.0221
.0195
.0172
.0151
.0291
.0256
.0225
.0196
.0170
.0323
.0281
.0244
.0210
.0180
3.6
3.7
3.8
3.9
4.0
.0042
.0038
.0033
.0029
.0026
.0079
.0070
.0061
.0054
.0047
.0109
.0095
.0083
.0073
.0063
.0132
.0114
.0099
.0085
.0073
.0147
.0126
.0108
.0091
.0077
.0153
.0129
.0108
.0090
.0074
4.2
4.4
4.6
4.8
5.0
.0020
.0015
.0011
.0008
.0006
.0036
.0026
.0019
.0013
.0009
.0046
.0033
.0023
.0015
.0009
.0052
.0036
.0023
.0014
.0007
.0052
.0034
.0020
.0009
.0001
.0047
.0026
.0011
.0000
0
.1
.2
.3
.4
.5
.6
.7
.8
9
1.8
1.9
2.6
2.7
2.8
2.9
3.0
-.0075
-.0098
[ 43 ]
-.0008
Table XVI
Values of - du/dx
b=.9
b=.9
b=.9
b=.9
b=.9
b=.9
b=.9
b=.9
b=.9
1.1000
1.2000
1.3000
1.4000
1.5000 1.6000
1.6200
.6865
.6832
.6737
.6590
.6396
.6163
.7802
.7764
.7657
.7487
.7263
.6993
.8835
.8792
.8669
.8474
.8217
.7905
1.3101
1.3037
1.2848
1.2544
1.2135
1.1631
1.6000
1.5921
1.5686
1.5301
1.4776
1.4122
1.8000
1.7910
1.7641
1.7198
1.6587
1.5821
.5167
.4919
.4655
.4380
.4100
.5897
.5607
.5298
.4976
.4647
.6685
.6347
.5987
.5610
.5225
.7548
.7155
.6734
.6295
.5845
.8514
.8055
.7563
.7049
.6520
.9637
.9096
.8515
.7905
.7279
1.1045
1.0393
.9688
.8947
.8185
1.3356
1.2494
1.1556
1.0666
.9545
1.4914
1.3890
1.2770
1.1584
1.0359
.3341
.3104
.2871
.2644
.2425
.3818
.3537
.3262
.2993
.2735
.4315
.3986
.3662
.3348
.3046
.4838
.4452
.4074
.3707
.3355
.5392
.4941
.4500
.4072
.3662
.5988
.5460
.4942
.4442
.3964
.6647
.6021
.5408
.4817
.4254
.7416
.6654
.5911
.5197
.4520
.8515
.7498
.6511
.5569
.4685
.9125
.7908
.6734
.5620
.4584
.1940
.1772
.1613
.1463
.1323
.2215
.2015
.1826
.1658
.1483
.2488
.2253
.2031
.1824
.1631
.2756
.2483
.2225
.1985
.1762
.3019
.2702
.2405
.2128
.1873
.3272
.2906
.2564
.2247
.1955
.3511
.3086
.2692
.2329
.1997
.3724
.3229
.2773
.2355
.1976
.3886
.3299
.2762
.2275
.1838
.3868
.3122
.2450
.1852
.1326
.3634
.2778
.2016
.1349
.0771
2.1
2.2
2.3
2.4
2.5
.1192
.1070
.0957
.0854
.0759
.1329
.1186
.1054
.0933
.0823
.1452
.1286
.1134
.0996
.0870
.1556
.1367
.1195
.1038
.0896
.1638
.1423
.1229
.1053
.0894
.1689
.1447
.1229
.1033
.0858
.1696
.1424
.1180
.0964
.0772
.1635
.1330
.1059
.0822
.0614
.1449 .0869
.1106 .0476
.0706 .0141
.0546 -.0140
.0323
2.6
2.7
2.8
2.9
3.0
.0671
.0592
.0520
.0454
.0395
.0722
.0631
.0549
.0474
.0408
.0756
.0653
.0560
.0477
.0403
.0768
.0653
.0551
.0460
.0379
.0753
.0627
.0515
.0417
.0330
.0703
.0566
.0446
.0341
.0249
.0605
.0458
.0330
.0220
.0126
.0434
.0133
.0278 -.0027
.0145
.0033
3.1
3.2
3.3
3.4
3.5
.0342
.0295
.0252
.0214
.0180
.0348
.0295
.0248
.0206
.0169
.0338
.0280
.0228
.0183
.0144
.0308
.0246
.0191
.0144
.0103
.0255
.0171
.0046
.0104 -.0021
.0189
.0047
.0133
.0084 -.0002
.0043
3.6
3.7
3.8
3.9
4.0
.0150
.0124
.0101
.0080
.0063
.0136
.0108
.0083
.0062
.0043
.0110
.0068
.0008
.0080
.0038 -.0022
.0012 -.0046
.0054
.0033 -.0009
.0014 -.0027
4.2
4.4
4.6
-.0002
x
uo
.7000
.8000
.9000 1.0000
0
.1
.2
.3
.4
.5
.4435
.4414
.4354
.4262
.4141
.3997
.5194
.5169
.5099
.4989
.4846
.4675
.6000
.5971
.5889
.5762
.5594
.5394
.6
.7
.8
.9
1.0
.3835
.3659
.3472
.3278
.3080
.4482
.4272
.4048
.3816
.3579
1.1
1.2
1.3
1.4
1.5
.2881
.2683
.2489
.2299
.2116
1.6
1.7
1.8
1.9
2.0
.0034
.0014
.0013 -.0007
-.0015
[44]
1.0000 1.1367
.9951 1.1311
.9811 1.1150
.9588 1.0892
.9290 1.0546
.8929 1.0124
-.0061
b=.9
b=.9
.0278
-.0138
Table XVII
Values of u. Downward Flow.
b=.1
x
0
.2
.4
.6
.8
1.0
1.2
b=.1
a=.5
a=1.0
a=2.5
a--=3.0
.2809
.2765
.2595
.2337
.1992
.4515
.4427
.4168
.3750
.3189
.5602
.5492
.5169
.4648
.3952
.6309
.6185
.5821
.5234
.4448
.6775
.6642
.6251
.5620
.4777
.7082
.6942
.6532
.6872
.4994
.1577
.1105
.0600
.0077
.2514
.1749
.0933
.0094
.3114
.2170
.1167
.0141
.3507
.2449
.1329
.0187
.3768
.2637
.1441
.0225
.3939
.2763
.1515
.0255
a=1.5
a=2.0
1.4
1.6
1.8
-.0438
-.0728
-.0860
-.0925
-.0956
-.0975
2.0
2.2
2.4
2.6
2.8
-.0929
-.1373
-.1756
-.2060
-.2275
-.1509
-.2211
-.2819
-.3300
-.3646
-.1806
-.2655
-.3388
-.3968
-.4385
-.1971
-.2911
-.3717
-.4360
-.4821
-.2067
-.3062
-.3917
-.4597
-.5087
-.2123
-.3158
-.4040
-.4748
-.5254
3.0
3.2
-.2395
-.2417
-.3873
.3836
-.4618
-.4665
-.5082
-.5134
-5363
-.5424
-.5543
-.5607
Table XVII
Values of u. Downward Flow.
b =.2
x
o
.2
.4
.6
.8
1.0
1.2
1.4
1.6
1.8
b=.2
a=.5
a=1.0
.2066
.2027
.1913
.1733
.1497
.3349
.3284
.3098
.2804
.2419
.4169
.4088
.3855
.3488
.3006
.4699
.4608
.4345
.3930
.3386
.5044
.4946
.4664
.4214
.3632
.5267
.5164
.4869
.4403
.3790
.1220
.0912
.0589
.0262
.1963
.1455
.0918
.0372
.24133
.2736
.2008
.1232
.0439
.2932
.2146
.1307
.0448
.3057
.2234
.1354
.0452
a=1.5
.1793
.1112
.0418
a=2.0
a=2.5
a=3.0
-.0053
-.0166
-.0261
-.0340
-.0397
-.0435
2.8
-.0348
-.0609
-.0831
-.1005
-.1129
-.0658
-.1103
-.1480
-.1776
-.1985
-.0899
-.1471
-.1956
-.2338
-.2606
-.1073
-.1733
-.2293
-.2734
-.3044
-.1195
-.1912
-.2524
-.3004
-.3343
-.1274
-.2028
-.2673
-.3179
-.3536
3.0
3.2
-.1200
-.1220
-.2103
-.2132
-.2756
-.2789
-.3216
-.3252
-.3529
-.3568
-.3731
-.3772
2.0
2.2
2.4
2.6
[ 45 ]
Table XVII
Values of u. Downward Flow.
b=--.3
a=.5
a=1.0
.8
.1475
.1448
.1369
.1247
.1090
.2398
.2353
.2223
.2024
.1769
1.0
1.2
1.4
1.6
1.8
.0909
.0712
.0509
.0308
.0116
2.0
2.2
2.4
2.6
2.8
3.0
3.2
3.4
x
0
.2
.4
.6
a=1.5
b=.3
a=2.0
a=2.5
a=3.0
.2985
.2928
.2767
.2518
.2198
.3359
.3295
.3113
.2832
.2472
.3597
.3528
.3338
.3032
.2646
.3747
.3675
.3472
.3158
.2755
.1473
.1150
.0814
.0482
.0162
.1827
.1421
.1001
.0581
.0180
.2052
.1594
.1117
.0642
.0185
.2195
.1703
.1189
.0678
.0186
.2285
.1771
.1233
.0700
.0184
-.0060
-.0216
-.0346
-.0450
-.0525
-.0130
-.0390
-.0606
-.0779
-.0902
-.0191
-.0517
-.0792
-.1008
-.1165
-.0234
-.0607
-.0918
-.1166
-.1341
-.0267
-.0668
-.1004
-.1270
-.1459
-.0288
-.0709
-.1059
-.1338
-.1535
-.0573
-.0593
-.0589
-.0980
-.1010
-.1001
-.1259
-.1298
-.1281
-.1450
-.1489
-.1471
-.1575
-.1617
-.1595
-.1656
-.1698
-.1675
Table XVII
Values of u. Downward Flow.
b=.4
b=.4
x
a=.5
a=1.0
a=1.5
a=2.0
a=2.5
a=3.0
0
.2
.4
.6
.8
.1007
.0988
.0935
.0856
.0756
.1634
.1603
.1517
.1388
.1225
.2024
.1986
.1880
.1719
.1518
.2266
.2223
.2104
.1924
.1698
.2416
.2370
.2243
.2051
.1810
.2507
.2460
.2329
.2130
.1879
1.2
1.4
1.6
1.8
1.0
.0643
.0522
.0400
.0279
.0167
.1041
.0842
.0642
.0444
.0259
.1288
.1042
.0791
.0545
.0313
.1440
.1164
.0882
.0606
.0345
.1534
.1240
.0938
.0644
.0364
.1593
.1286
.0973
.0666
.0375
2.0
2.2
2.4
2.6
2.8
-.0027
-.0104
-.0165
-.0212
-.0059
-.0165
-.0285
-.0361
-.0084
-.0241
-.0366
-.0460
-.0102
-.0278
-.0419
-.0524
-.0114
-.0302
-.0453
-.0565
-.0122
-.0318
-.0474
-.0592
3.0
3.2
3.4
-.0244
-.0262
-.0268
-.0412
-.0441
-.0449
-.0523
-.0558
-.0567
-.0595
-.0633
-.0643
-.0641
-.0681
-.0692
-.0669
-.0713
-.0721
.0063
.0089
.0101
[ 46 ]
.0107
.0110
.0111
Table XVII
Values of u. Downward Flow.
b=.5
b=.5
a=.5
a=1.0
.0642
.0630
.0597
.0549
.0489
.1031
.1012
.0960
.0882
.0786
1.4
1.6
1.8
.0423
.0352
.0283
.0214
.0152
.0679
.0566
.0452
.0343
.0239
2.0
2.2
2.4
2.6
2.8
-.0001
-.0035
-.0064
-.0005
-.0063
-.0107
-.0010
-.0081
-.0135
-.0013
-.0092
-.0152
-.0015
-.0098
-.0162
-.0017
-.0103
-.0168
3.0
3.2
3.4
3.6
-.0083
-.0098
-.0105
-.0108
-.0140
-.0161
-.0174
-.0176
-.0176
-.0201
-.0217
-.0219
-.0197
-.0226
-.0242
-.0246
-.0209
-.0240
-.0257
-.0261
-.0218
-.0248
-.0267
-.0270
x
o
.2
.4
.6
.8
1.0
1.2
.0093
.0044
.0147
.0064
x
a=.5
a=1.0
0
.0366
.0359
.0341
.0315
.0283
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
a=2.5
a=3.0
.1266
.1242
.1178
.1082
.0964
.1405
.1379
.1308
.1202
.1071
.1487
.1459
.1384
.1272
.1133
.1535
.1506
.1428
.1312
.1169
.0832
.0693
.0553
.0418
.0291
.0924
.0770
.0614
.0464
.0322
.0978
.0814
.0650
.0490
.0341
.1008
.0839
.0669
.0504
.0350
.0195
.0082
.0205
.0086
.0177
.0075
Table XVII
Values of u. Downward Flow.
b=.6
.2
.4
.6
.8
a=2.0
a=1.5
.0210
.0087
b=.6
a=1.5
a=2.0
a=2.5
a=3.0
.0577
.0566
.0538
.0497
.0446
.0697
.0684
.0650
.0600
.0539
.0764
.0750
.0712
.0657
.0590
.0800
.0785
.0746
.0688
.0618
.0820
.0805
.0764
.0705
.0633
.0248
.0211
.0175
.0140
.0108
.0391
.0333
.0276
.0221
.0170
.0472
.0402
.0333
.0266
.0204
.0517
.0440
.0364
.0291
.0223
.0541
.0461
.0381
.0305
.0234
.0553
.0472
.0391
.0312
.0240
.0078
.0053
.0030
.0012
.0124
.0083
.0048
.0018
.0148
.0098
.0056
.0020
.0161
.0107
.0060
.0022
.0169
.0113
.0063
.0023
.0173
.0115
.0064
.0023
2.8
-.0004
-.0005
-.0008
-.0010
-.0010
-.0011
3.0
3.2
3.4
3.6
3.8
-.0015
-.0025
-.0029
-.0033
-.0035
-.0024
-.0037
-.0047
-.0052
-.0056
-.0031
-.0047
-.0059
-.0065
-.0068
-.0034
-.0052
-.0064
-.0072
-.0075
-.0035
-.0054
-.0067
-.0075
-.0078
-.0037
-.0057
-.0070
-.0078
-.0081
4.0
-.0037
-.0055
-.0068
-.0075
-.0078
-.0081
[47]
Table XVII
Values of u. Downward Flow.
b=.7
b=.7
a=.5
n=1.0
a=1.5
a=2.0
a=2.5
a=3.0
.2
.4
.6
.8
.0171
.0168
.0160
.0148
.0134
.0261
.0256
.0244
.0226
.0205
.0307
.0301
.0286
.0265
.0240
.0330
.0324
.0308
.0285
.0258
.0342
.0336
.0320
.0296
.0268
.0348
.0342
.0325
.0301
.0273
1.0
1.2
1.4
1.6
1.8
.0119
.0103
.0088
.0073
.0060
.0181
.0157
.0133
.0111
.0090
.0212
.0184
.0156
.0130
.0105
.0228
.0198
.0168
.0140
.0113
.0237
.0206
.0175
.0146
.0119
.0242
.0210
.0179
.0149
.0121
2.0
2.2
2.4
2.6
2.8
.0047
.0036
.0026
.0018
.0011
.0071
.0054
.0040
.0027
.0017
.0083
.0063
.0046
.0031
.0019
.0090
.0068
.0050
.0034
.0021
.0094
.0071
.0052
.0036
.0023
.0096
.0073
.0054
.0037
.0023
3.0
3.2
3.4
3.6
3.8
.0006
.0001
.0008
.0002
.0009
.0002
.0010
.0002
-.0002
-.0005
-.0006
-.0003
-.0007
-.0009
-.0004
-.0008
-.0011
-.0005
-.0010
-.0013
-.0005
-.0009
-.0011
-.0004
-.0009
-.0013
4.0
4.2
4.4
-.0007
-.0008
-.0008
- .0011.
-.0013
-.0014
-.0014
-.0015
-.0016
-.0016
-.0013
-.0015
-.0015
-.0015
-.0016
-.0016
x
0
-.0012
-.0012
.0012
.0003
Table XVII
Values of u. Downward Flow.
b=.8
.0012
.0003
b=.8
a=.5
a=1.0
a=1.5
a=2.0
a=2.5
a=3.0
.8
.0052
.0051
.0049
.0046
.0042
.0076
.0075
.0071
.0066
.0060
.0086
.0085
.0081
.0075
.0068
.0090
.0088
.0084
.0078
.0071
.0092
.0090
.0086
.0080
.0073
.0093
.0091
.0087
.0081
.0074
1.0
1.2
1.4
1.6
1.8
.0038
.0033
.0029
.0025
.0021
.0054
.0047
.0041
.0035
.0029
.0061
.0053
.0046
.0039
.0033
.0064
.0056
.0049
.0042
.0035
.0065
.0057
.0049
.0042
.0035
.0066
.0058
.0050
.0043
.0036
2.0
2.2
2.4
2.6
2.8
.0017
.0014
.0011
.0009
.0007
.0024
.0019
.0016
.0012
.0009
.0027
.0022
.0017
.0013
.0010
.0029
.0024
.0019
.0015
.0011
.0029
.0023
.0019
.0014
.0011
.0030
.0025
.0020
.0015
.0012
3.0
3.2
3.4
3.6
3.8
.0005
.0004
.0003
.0002
.0001
.0007
.0005
.0003
.0002
.0001
.0007
.0005
.0003
.0002
.0001
.0008
.0006
.0004
.0002
.0001
.0008
.0006
.0004
.0003
.0002
.0009
.0007
.0005
.0003
.0002
4.0
4.2
4.4
4.6
4.8
.0001
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
.0000
-.0001
-.0001
-.0001
-.0001
-.0001
-.0001
.0001
.0000
.0000
.0000
.0001
.0000
.0000
.0000
.0000
0
.2
.4
.6
[ 48 ]
-.0001
-.0001
-.0002
-.0002
-.0001