*
This research was supported in part by the ~1athematics Division of
the United States Air Force Office of Scientific Research under Contract No.
AFOSR-68-1415 and the National Institutes of Health, Institute of General
Hedica1 Sciences under Grant No. 5R01GM-12868.
TABLES TO FACILITATE FITTING SB
CURVES
by N.L. Johnson*
With an Appendix on Calculation of the Tables
by J .0. Kitchen
Department of Statistics
University of North Carolina at Chapel Hill
Institute of Statistics MimeD Series No. 683
l4AY 1970
•
TABLES TO FACILITATE FITTING SB ·CURVES
by
N.
L~
Johnson. *
llnivePsity of NOl'th CaroUna at Chapet Bitt
With an Appendix on Calculation of the Tables
by
J. O. Kitchen,
llnivereity of North Caf'OUna at Chapel, Bitt
1.
DESCRIPTIOO OF TABLES.
The SB
family of frequency curves is defined (Johnson, 1949) by the
distribution of x,
where
z = y+
(~ <
x
< ;+A)
(l)
6 log[(x-~)/(~+A-X)]
is a unit normal variable.
The parameters
;
and
A af-
feet only the location and seale, respectively, of the distribution.
shape of the distribution of
x
is the same as that of a variable y
The
for
which
z
•
•
y
+ o£og[y/(l-y)]
(2)
This research was supported in part by the Mathematics Division of
the United States Air Foree Office of Scientific Research under Contract No.
AFOSR-68-l4l5 and the National Institutes of Health, Institute of General
Medical Sciences under Grant No. 5ROlGM-12868 •
2
(0 < y < 1)
is a unit normal variable.
by the values of
and
y
6.
This shape is therefore determined
By convention, we make
The r-th moment about zero of
y
6
positive.
is
(3)
An
explicit expression «56) of Johnson (1949» can be given for
lJi
and
by repeated use of the relationship
(4)
formulae for
can be obtained.
lJ t
r
These are, however, too complicated,
except for occasional use as checks on computation.
Formulae for
can also be obtained using the relationship
lJ t
r
(5)
This is even less convenient than (4).
In order to fit an
SB
curve by making the first four moments agree
with those of a given distribution, the first, and most difficult, step is
to determine
1(31
and
y
and
6
to give the required values of the moment-ratio .
6 •
2
Table 3 gives values of
6
2
at intervals of
lated
6
2
0.1.
y
and
6
161
For a given value of
value is the smallest quantity
6 > <ra;:-)2+1.
2
for
A few values of
6 ,
2
1~
x
= 0.00(0.05)2.00
113 1 ,
the smallest tabu-
(integer)
a lognormal fit would usually be very nearly-the same as an
would also be simpler.
such that
close to the lognormal line, are
omitted at the upper end of the range of possible values.
•
with
For such values,
SB
fit, and
3
Once the appropriate values of y
and
and
6 have been determined,
~
A can be obtained from the formulae
standard deviation of x =mean of
; + AlJi
x ...
where a •
standard deviation of y.
and a to aid these calculations.
Table 3 includes values of lJi
the sign of
lJi by
.ra;:-
(l-lJi>.
be changed (to -),
The values of
The present tables of y
tables relating to the
x'
has an
Su
~',
A'(>O)
such that
AO;
Su
6 and
and
should be replaced by -y,
y
a
If
and
remain unaltered.
0 are similar in form to earlier
class of frequency curves (Johnson, 1965).
distribution, there are parameter values
y',
If
6' (>O),
(6)
has a unit normal distribution.
that if x has an
has an
Su
SB
Comparison of
cS' == a,
The present Table 3 also includes values of
Su
and
(6)
shows
distribution then
distribution with y' =- y,
.ponding values for
(1)
~'
== 0,
lJ'1 and
A' ... 1.
6.
The corre-
distributions are much easier to calculate, and
so it was not necessary to give these values in the earlier Tables.
The methods used in calculating Table 3 are described in the Appendix.
4
2.
INTERPOLATIOO.
~
It will usually be necessary to interpolate with respect to both
and
13 2
62 •
Except near the lognormal line (i.e. for the highest values of
for given
113 ), repeated univariate interpolation using Everett's
1
second central difference formula should give results correct to -within
about 2 units in the last decimal place shown..
than
1,
~ greater
linear interpolation is adequate over about the lowest
13 " When y and/or
2
advantageous to 1Ilterpolate for
values of
6
For values of
directly_
0
are large (greater than
yo
-1 and
15
Interpolation with respect to
-2 ,
1,
say) it is
rather than for
(rather than
<Sa
15-20
and
y
a) can also
be useful"
The very slow initial variation of
e
(1+131 )
~
with
fixed, is notable.
lli,
13
as
2
increases from
There is a remarkably flat maximum
which gradually moves away from the lowest
13
2
value as
~
increases.
,
We will now use the present tables to fit an
data of Example 2 (p.169) of Johnson (1969).
sample values:
mean
=
29.5291
standard deviation
~ = 0.3112
8
2
=
2.4303.
=
6.1663
SB
distribution to the
For these data we have the
5
Using Everett's central second difference interpolation formula, we
obtain
=
y
0.5369;
o ==
y ==
0.5362; 0
1.1904.)
~i ==
give
~
= 1.193.
and
Direct calculation and second difference interpolation both
0.4034,
ai == 0.1774).
1.191
==
13 • (Linear interpolation
2
Direct calculation gives y == 0.5371;
corresponding to the sample values of
gives
o
Then
a
==
~
0.1773
and
t + 0.4034A
=
(linear interpolation gives
~i ==
0.4034,
A are estimated from the formulae
29.5291;
0.1773A ==
6.1663
whence
A ==
34.78.
"
(By comparison, Johnson (1949), using a method based on equating certain observed and fitted percentile points, obtained the values
y
=
0.5918;
IS
==
1.2536;
~
==
15.0;
A
=
36.5.)
Table 1 shows the original data together with fitted frequencies from
a Type I curve (Pretorius, 1930), the
the
SB
curve fitted by moments.
SB
curve from Johnson (1949), and
The fit of the latter is relatively poor
in the tails, but quite good in the major part of the distribution.
•
7
4.
Blt'ODALITY.
SB curves are bimodal i f
otherwise they are unimodal.
Some values of
2 on the boundary
between unomodal and bimodal curves are shown in Table 2 (which is an extension of DrapeT (1952, F1g. 2».
13
For a given
1
and
61'
13
curves are bimodal if
82 is less than the value shown in this table.
TABLE 2
Boundar'y of Bimodal SB
e
6
(12)-1
0.70
0.68
0.66
0.64
0.62
0.60
0.58
0.56
0.54
0.52
•
61
0.000
0.003
0.020
0.048
0.086
0.132
0.188
0.254
0.331
0.421
0.524
82
1.866
1.859
1.838
1.818
1.802
1. 792
1.792
1.805
1.837
1.895
1.989
0
0.50
0.49
0.48
0.47
0.46
0.45
0.44
0.43
0.42
0.41
CU1'tXl8
81
0.644
0.711
0.784
0.862
0.947
1.040
1.141
1.251
1.373
1.508
82
2.133
2.230
2.346
2.487
2.657
2.861
3.107
3.406
3.768
4.210
8
5.
FITTING AN
Sa
DISTRIBtI11~, USING 00 KM:NTS.
It often happens that the range of variation of x is known -- that
is,
15.
u
A are known -- and it is only necessary to estimate y
and
~
and
A simple way to do this is to use the first and second moments of
= log[(x-~)/(~+A-x)],
taking
(standard deviation of
15
=
y
= - 15 (mean u) •
However, the moments of
u)-l
u may be difficult to evaluate (for a
theoretical distribution) or tedious to compute (for an empirical distribution).
In the latter case, there may be additional problems associated
with unequal group-widths for
x.
consequent upon equal group-widths for
In such cases the first two moments of y =
values of
of
u,
y
1.1'
1
and 15.
(x-F~)/A
may be used.
The
and a in Table 3 give some indication of appropriate values
Further tables, giVing
y
and
15 with arguments
1.Ii,
a
at regular intervals are now being computed.
REFERENCES
DRAPER, J. (1952) Properties of distributions resulting from certain
simple transformations of the normal distribution,
Biometpi.ka~ 39, 290-301.
JOHNSON, N. L. (1949) Systems of frequency curves generated by methods of
translation, Biomemka, 36, 149-176.
JOHNSON, N. L. (1965)
Tables to facilitate fitting
52, 547-558.
Biometrika~
P"RETORIUS,
•
s.
Su
frequency curves,
J. (1930) Skew bivariate frequency surfaces, examined in the
light of numerical illustrations, Biometrika, 22, 109-223 •
9
TABLE 3
~
rs;:- .. 0.05
= 0.00
a
e
a
y
lJt1
y
a
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
0.0883 0.4639
0.1692 0.4306
0.2465 0.3999
0.3227 0.3714
0.3994 0.3451
0.4780 0.3204
0.5599 0.2913
0.6465 0.2754
0.73930.2547
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
0.0316
0.0327
0.0342
0.0361
0.0385
0.0413
0.0447
0.0487
0.0536
0.0862
0.1671
0.2444
0.3205
0.3972
0.4757
0.5573
0.6436
0.7360
0.4875
0.4875
0.4875
0.4874
0.4874
0.4873
0.4871
0.4868
0.4865
0.4646
0.4313
0.4006
0.3721
0.3457
0.3210
0.2919
0.2760
0.2552
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
0.0633
0.0652
0.0681
0.0719
0.0765
0.0820
0.0887
0.0966
0.1061
0.0796
0.1608
0.2382
0.3141
0.3904
0.4685
0.5496
0.6351
0.7266
0.4750
0.4750
0.4750
0.4149
0.4748
0.4746
0.4142
0.4737
0.4732
0.4669
0.4334
0.4025
0.3740
0.3475
0.3227
0.2995
0.2776
0.2568
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
0.8403
0.9522
1.079
1.225
1.398
1.613
1.892
2.286
2.918
4.241
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
0.0594
0.0666
0.0757
0.0873
0.1029
0.1244
0.1571
0.2111
0.3180
0.6295
0.83670.4862
0.9480 0.4857
1.074 0.4851
1.219 0.4844
1.391 0.4834
1.603 0.4822
1.879 0.4804
2.266 0.4777
2.885 0.4733
4.195 0.4632
0.2353
0.2162
0.1976
0.1794
0.1613
0.1433
0.1247
0.1053
0.0840
0.0584
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
0.1174
0.1315
0.1491
0.1717
0.2014
0.2429
0.3043
0.4044
0.5957
1.109
0.8259
0.9357
1.059
1.201
1.369
1.574
1.839
2.205
2.776
3.916
0.4725
0.4716
0.4704
0.4691
0.4672
0.4648
0.4614
0.4564
0.4482
0.4308
0.2370
0.2177
0.1992
0.1810
0.1631
0.1450
0.1266
0.1074
0.0864
0.0617
(y
0.2347
0.2157
0.1971
0.1788
0.1608
0.1421
0.1241
0.1046
0.0833
0.0582
= OJ lJi· 0.5
in all cases)
•
~ .. 0.10
10
Ii\ = 0.15
82
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
e
•
y
0
~ ... 0.20
ll'
0.0946 0.0688
0.0972 0.1504
0.1014 0.2277
0.1068 0.3034
0.1136 0.3793
0.1216 0.4567
0.1314 0.5368
0.1428 0.6211
0.1565 0.7111
I
1
0.4626
0;4626
0.4626
0.4624
0.4622
0.4620
0.4614
0.4608
0.4600
2.0 0.1729 0.8084 0.4590
2.1 0.1932 0.9154 0.4577
2.2 0.2182 1.035 0.4561
2.3 0.2502 1.173 0.4541
2.4 0.2922 1.334 0.4515
2.5 0.3499 1.529 0.4481
2.6 0.4338 1.777 0.4434
2.7 0.5671 2.113 0.4366
2.8 0.8098 2.619 0.4259
2.9 1.391 3.546 0.4050
a
82
y
0
ll'
1
0.4503
0.4503
0.4502
0.4501
0.4499
0.4495
0.4488
0.4480
0.4471
a
0.4758
0.4706
0.4369
0.4059
0.3772
0.3505
0.3256
0.3024
0.2804
0.2595
0.0536
0.1358
0.2133
0.2887
0.1493 0.3640
1.6 0.1597 0.4404
1.7 0.1720 0.5193
1.8 0.1866 0.6020
1.9 0.2040 0.6899
0.2396
0.2204
0.2019
2.0 0.2248 0.7846 0.4458 0.2433
2.1 0.2502 0.8883 0.4442 0.2241
2.2 0.2816 1.004 0.4422 0.2056
2.3 0.3211 1.135 0.4397 0.1875
2.4 0.3726 1.287 0.4365 0.1696
2.5 0.4423 1.410 0.4323 0.1519
2.6 0.5411 1.698 0.4267 0.1339
0.4188 0.1154
2.7 0.6932 1.999
2.8 0.9548 2.433 0.4068 0.0956
2.9 1.513 3.163 0.3853 0.0733
0.1837
0.1658
0.1479
0.1297
0.1108
0.0903
0.0668
1.1
1.2
1.3
1.4
1.5
0.1256
0.1287
0.1338
0.1407
0.4419
0.4106
0.3817
0.3548
0.3298
0.3064
0.2843
0.2633
11
re;:62
e
•
y
6
ca
lei' = 0.30
0.25
\.I'
1
a
62
y
6
\.1'
1
a
1.1 0.1564 0.0337 0.4380 0.4826
1.2 0.1595 0.1171 0.4380 0.4483
1.3 0.1653 0.1948 0.4380 0.4167
1.4 0.1733 0.2700 0.4378 0.3874
1.5 0.1834 0.3446 0.4376 0.3603
1.6 0.1957 0.4199 0.4371 0.3351
1.7 0.2102 0.4977 0.4365 0.3114
1.8 0.2276 0.5783 0.4356 0.2892
1.9 0.2480 0.6636 0.4344 0.2682
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
0.1870
0.1896
0.1957
0.2045
0.2158
0.2295
0.2458
0.2652
0.2880
0.0091
0.0942
0.1725
0.2475
0.3213
0.3956
0.4716
0.5502
0.6329
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
0.2724
0.3019
0.3380
0.3833
0.4414
0.5184
0.6253
0.7834
1.041
2.9
0.3151
0.3477
0.3873
0.4362
0.4980
0.5786
0.6875
0.8431
1.082
1.496
0.7212 0.4205 0.2539
0.8164 0.4185 0.2346
0.9209 0.4161 0.2160
1.037 0.4130 0.1980
1.170 0.4091 0.1803
1.324 0.4042 0.1629
1.510 0.3978 0.1455
1.741 0.3893 0.1277
2.046 0.3773 0.1094
2.486 0.3590 0.0896
1.532
1.872
2.238
2.804
0.4330
0.4311
0.4288
0.4260
0.4224
0.4177
0.4116
0.4031
0.3908
0.3705
0.2481
0.2289
0.2103
0.1922
0.1745
0.1569
0.1392
0.1211
0.1020
0.0810
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3.0 2.839
3.911
0.3285
0.0557
3.0 2.388
3.1 5.711
0.7552
0.8550
0.9650
1.089
1.232
1.401
1.607
3.225
5.004
0.4258
0.4258
0.4258
0.4257
0.4255
0.4250
0.4243
0.4234
0.4221
0.4909
0.4562
0.4241
0.3945
0.3671
0.3415
0.3177
0.2952
0.2740
0.3265 0.0670
0.2439 0.0367
12
~. 0.35
82
e
y
0
~ .. 0.40
]..I'
1
a
82
y
6
0.2488
0.2536
0.2625
0.2147
0.0349
0.1159
0.1914
0.2643
0.3364
0.4091
0.4835
0.5605
]..I'
1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
0.2193 0.0668 0.4138
0.2251 0.1462 0.4138
0.2342 0.2213 0.4138
0.2461 0.2945 0.4135
0.2609 0.3671 0.4131
0.2785 0.4419 0.4124
0.2993 0.5185 0.4114
0.3239 0.5983 0.4101
0.4655
0.4330
0.4029
0.3751
0.3492
0.3251
0.3024
0.2810
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
0.3301
0.3558
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
0.3529
0.3875
0.4291
0.4799
0.5433
0.6242
0.7309
0.8780
1.093
1.435
0.2607
0.2413
0.2226
0.2046
0.1871
0.1698
0.1526
0.1353
0.1176
0.0989
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
0.3859 0.6415 0.3969
0.4215 0.7277 0.3948
0.4640 0.8206 0.3923
0.5153 0.9221 0.3891
0.5784 1.035 0.3852
0.6574 1.162 0.3803
0.7591 1.309 0.3742
0.8946 1.483 0.3664
1.084 1.697 0.3562
1.366 1.973 0.3421
3.0 2.068
3.1 3.626
0.6830
0.7738
0.8737
0.9814
1.104
1.244
1.409
1.610
1.865
2.211
0.4085
0.4065
0.4039
0.4007
0.3968
0.3918
0.3855
0.3772
0.3660
0.3498
2.730
3.676
0.3241 0.0783
0.2749 0.0536
0.2900
0.3084
3.0 1.831
3.1 2.739
3.2 5.312
2.354
2.945
4.107
a
0.4764
0.4433
0.4127
0.3844
0.3581
0.3336
0.3998 0.3106
0.3985 0.2890
0.4019
0.4020
0.4019
0.4018
0.4014
0.4007
0.2685
0.2490
0.2303
0.2122
0.1947
0.1776
0.1606
0.1437
0.1265
0.1087
0.3215 0.0897
0.2878 0.0683
0.2180 0.0413
13
Iel
JB;" • 0.45
62
y
0
1.3 0.2816 0.0813
1.4 0.2896 0.1578
1.5 0.3015 0.2308
1.6 0.3168 0.3019
11'
1
0'
0.3903
0.3903
0.3902
0.3899
0.3893
0.3885
0.3950
0.3683
0.3433
0.3200
0.5974
0.6792
0.7665
0.8611
0.9645
1.080
1.211
1.362
1.543
1.766
0.3857
0.3837
0.3812
0.3781
0.3743
0.3697
0.3639
0.3567
0.3475
0.3353
0.2773
0.2576
0.2388
0.2207
0.2032
0.1862
0.1694
0.1528
0.1361
0.1191
2.056
2.461
3.104
4.425
0.3186 0.1013
0.2938 0.0821
0.2524 0.0600
0.1632 0.0308
0.4551
0.4239
1.7 0.3355 0.3732
1.8 0.3576 0.4455
1.9 0.3837 0.5198 0.3872 0.2981
e
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
0.4142
0.4501
0.4925
0.5432
0.6045
0.6801
0.7752
0.8982
1.063
1.296
3.0 1.647
3.1 2.239
3.2 3.447
3.3 7.310
. 62
y
CIl
0.50
0
1.3 0.3097 0.0423
1.4 0.3159 0.1206
1.5 0.3269 0.1939
1.6 0.3417 0.2647
1.7 0.3601 0.3346
1.8 0.3822 0.4051
1.9 0.4081 0.4769
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
0.4385
0.4739
0.5154
0.5646
0.6234
0.6945
0.7823
0.8929
1.036
1.229
3.0 1.502
3.1 1.917
3.2 2.622
3.3 4.088
11'
1
0.3788
0.3788
0.3787
0.3785
0.3781
0.3713
0.3762
0'
0.4683
0.4364
0.4070
0.3796
0.3542
0.3304
0.3082
0.5511 0.3748 0.2812
0.6288 0.3729 0.2673
0.7109 0.3706 0.2483
0.7989 0.3677 0.2301
0.8943 0.3641 0.2125
0.9988 0.3598 0.1956
1.116 0.3545 0.1790
1.248 0.3479 0.1626
1.402 0.3397 0.1463
1.586 0.3292 0.1298
1.814
2.109
2.524
3.181
0.3154 0.1130
0.2964 0.0953
0.2682 0.0760
0.2213 0.0536
14
;a;. 0.60
~. 0.55
62
e
y
t5
11'
1
a
62
y
0
11'
a
1
1.4
1.5
1.6
1.7
1.8
1.9
0.3421
0.3513
0.3649
0.3826
0.4040
0.4294
0.0792
0.1536
0.2245
0.2935
0.3624
0.4319
0.3674
0.3674
0.3674
0.3670
0.3664
0.3655
0.4505
0.4203
0.3923
0.3663
0.3421
0.3194
1.4 0.3689
1.5 0.3753
1.6 0.3872
1.7 0.4034
1.8 0.4237
1.9 0.4480
0.0330
0.1097
0.1812
0.2499
0.3177
0.3852
0.3563
0.3564
0.3563
0.3561
0.3557
0.3550
0.4659
0.4350
0.4063
0.3797
0.3548
0.3317
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
0.4590
0.4934
0.5336
0.5807
0.6362
0.7025
0.7829
0.8818
1.006
0.5032
0.5710
0.6545
0.7365
0.8244
0.9198
1.025
0.3642
0.3626
0.3603
0.3577
0.3544
0.3504
0.3456
0.2980
0.2779
0.2586
0.2403
0.2227
0.2057
0.1892
2.0
2.1
2.2
2.3
2.4
2.5
2.6
0.3099
0.2894
0.2699
0.2514
0.2337
0.2165
0.2001
1.141
1.274
1.428
0.3397 0.1730
0.3324 0.1570
0.3233 0.1410
1.611
1.837
0.3119 0.1249
0.2968 0.1083
0.2762 0.0909
0.2462 0.0718
0.1969 0.0495
1.168
1.384
1.690
2.152
2.935
3.4 4.551
3.0
3.1
3.2
3.3
2.130
2.536
3.170
0.4764
0.5095
0 .. 5478
0.5923
0.6444
0.7058
0.7788
2.7 0.8670
2.8 0.9754
2.9 1.111
0.4539
0.5243
0.5975
0.6744
0.7557
0.8432
0.9375
0.3539
0.3524
0.3505
0.3481
0.3451
0.3415
0.3372
1.041
1.157
1.288
0.3319 0.1840
0.3255 0.1682
0.3178 0.1526
3.0
3.1
3.2
3.3
3.4
3.5
1.439
1.618
1.837
2.118
2.501
3.079
0.3081
0.2959
0.2801
0.2588
0.2284
0.1800
1.287
1.520
1.847
2.337
3.151
4.770
0.1370
0.1212
0.1050
0.0879
0.0693
0.0477
15
~
132
e
y
1.5
1..6
1..7
1..8
1.9
0.4000
0.4090
0 .. 4232
0.4418
0.4645
2.0
2.1
2.2
2.3
2.4
0.4914
0.5227
0.5588
0.6006
0.6490
a
~ .. 0.70
0.65
IS
lit
1
0.0616
0.1348
0.2037
0.2707
0.3370
0.3455
0.3455
0.3454
0.3452
0.3446
0.4034
0.4711
0.5404
0.6127
0.6885
2.5 0.7054 0.7687
2.6 0.7717 0.8545
2.7 0.8504 0.9476
2.8 0.9449 1.049
0.3438
0.3425
0.3409
0.3387
0.3361
0.3329
0.3292
0.3245
0.3189
CJ
13 2
y
tS
lJt
1
CJ
0.4688
0.4512
0 .. 4217
0.3944
0.3689
0.3451
1.5
1.6
1.7
1.8
1.9
0.4627
0.4315
0.4427
0.4590
0.4796
0.0081
0.0845
0.1548
0.2217
0.2870
0.3348
0.3349
0.3348
0.3348
0.3344
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
0.5045
0.5336
0.5674
0.6063
0.6510
0.7027
0.7626
0.8328
0.9156
1.015
0.3518
0.4170
0.4834
0.5516
0.6225
0.6967
0.7754
0.8594
0.9500
1.049
0.3338
0.3328
0.3314
0.3296
0.3274
0.3246
0.3213
0.3172
0.3124
0.3061
0.3369
0.3155
0.2953
0.2761
0.2580
0.2406
0.2239
0.2018
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
1.135
1.284
1.412
1.718
2.052
2.529
3.271
4.580
7.610
1.158
1.280
1.420
1..582
1.776
2.015
2.325
2.754
3.419
0.2998
0.2915
0.2814
0.2690
0.2533
0.2329
0.2052
0.1652
0.1005
0.1620
0.1412
0.1324
2.9
1.060
1.162
0.3228
0.3019
0.2821
0.2633
0.2454
0.2282
0.2117
0.1956
0.1800
0.3122 0.1645
3.0
3.1
3.2
3.3
3.4
3.5
3.6
1.204
1..389
1.632
1.969·
2.463
3.258
4.759
1..289
1.436
1.608
1.816
2.078
2.426
2.932
0.3040
0.2941
0.2816
0.2656
0.2444
0.2149
0.1100
0.1493
0.1342
0.1188
0.1030
0.0864
0.0684
0.0479
0.4385
0.4104
0.3842
0.3598
0.1924
0.1170
0.~175
0.1022
0.0862
0.0691
0.0498
0.0266
16
•
re;:- • 0.75
62
1.6
1.7
1.8
1.9
e
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.. 8
2.9
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
•
y
0.4560
0.4629
0.4760
0.4940
0.5165
0.5432
0.5743
0.6101
0.6512
0.6983
0.7525
0.8150
0.8879
0.9735
1.075
1.198
1.348
1.535
1. 777
2.098
2.545
3.210
4.309
6.510
6
0.0294
0.1024
0.1703
0.2354
0.2991
0.3624
0.4262
0.4911
0.5579
0.6272
0.6997
0.7762
0.8577
0.9453
1.040
1.145
1.261
1.393
1.544
1.722
1.937
2.207
2.565
3.080
~ .. 0.80
ll'
1
0.3244
0.3245
0.3245
0.3243
0.3239
0.3232
0.3222
0.3207
0.3189
0.3164
0.3136
0.3101
0.3060
0.3011
0.2953
0.2883
0.2801
0.2702
0.2580
0.2430
0.2238
0.1987
0.1639
0.1118
a
0.4568
0.4277
0.4008
0.3756
0.3521
0.3301
0.3094
0.2899
0.2714
0.2537
0.2369
0.2207
0.2050
0.1898
0.1751
0.1604
0.1461
0.1318
0.1173
0.1025
0.0872
0.0710
0.0531
0.0324
82
y
15
ll'
1
a
1.7 0.4853 . 0.0455 0.3143 0.4466
1.8 0.4939 0.1156 0.3144 0.4188
1.9 0.5085 0.1815 0.3143 0.3928
0.3141
0.3137
0.3129
0.3118
0.3103
0.3084
0.3060
0.3030
0.2995
2.9 0.9367 0.8503 0.2954
0.3685
0.3459
0.324';
0.3046
0.2856
0.2677
0.2506
0.2342
0.2184
0.2032
0.9344
1.025
1.125
1.234
1.357
1.496
1.657
1.849
2.082
2.379
0.2904
0.2847
0.2778
0.2698
0.2602
0.2486
0.2345
0.2169
0.1944
0.1646
0.1884
0.1740
0.1599
0.1459
0.1320
0.1180
0.1039
0.0893
0.0739
0.0574
2.780
0.1227 0.0389
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
0..5280
0.5519
0.5802
0.6129
0.6503
0.6931
0.7420
0.7979
0.8622
0.2448
0.3070
0.3689
0.4312
0.4946
0.5597
0.6271
0.6975
0.7717
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
1.024
L126
1.248
1.397
1.580
1.811
2.112
2.520
3.102
4.003
4.0
5.602
17
•
rat· 0.85
62
3.0
3.1
3.2
3.3
3.4
3.5
3.6
0.5401
0.5607
0.5858
0.6152
0.6490
0.6877
0.7316
0.7816
0.8385
0.9036
0.9786
1.065
1.167
1.287
1.431
1.606
1.824
3.7 2.102
2.8 2.467
3.9 2.968
4.0 3.701
4.1 4.876
4.2 7.146
•
<5
lJ'
1
a
1.8 0.5144 0.0570 0.3045 0.4381
1.9 0.5243 0.1249 0.3045 0.4112
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
e
y
I6i" • 0.90
0.1889
0.2506
0.3112
0.3715
0.4322
0.4941
0.5572
0.6226
0.6907
0.7621
0.8377
0.9180
1.004
1.098
1.201
1.314
1.442
1.587
1.756
1.957
2.204
2,519
2.948
0.3045
0.3042
0.3037
0.3030
0.3018
0.3003
0.2983
0.2959
0.2930
0.2896
0.2854
0.2806
0.2749
0.2683
0.2605
0.2514
0.2405
0.2274
0.2115
0.1916
0.3862
0.3628
0.3409
0.3203
0.3008
0.2824
0.2650
0.2483
0.2324
0.2171
0.2022
0.1879
0.1738
0.1601
0.1465
0.1331
0.1196
0.1060
0.0921
0.0777
0.1663· 0.0624
0.1325 0.0457
0.0850 0.0266
62
y
<5
lJ'
1
a
1.9 0.5429
0.0644 0.2948 0.4311
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
0.5538
0.5704
0..5918
0.6177
0.6479
0.6825
0.7219
0.7665
0.8170
0.8742
0.1303
0.1925
0.2527
0.3118
0.3706
0.4296
0.4896
0.5510
0.6142
0.6799
0.2949
0.2948
0.2946
0.2941
0.2933
0.2922
0.2907
0.2887
0.2863
0.2835
0.4052
0.3810
0.3583
0.3370
0.3170
0.2981
0.2802
0.2632
0.2470
0.2315
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.. 8
3.9
0.9393
1.014
1.099
1.199
1.315
1.413
1.619
1.821
2.. 073
2.396
0.7483
0.8205
0.8969
0.9785
1.067
1.162
1.267
1.383
1.513
1.662
0.2801
0.2760
0.2714
0.2659
0.2596
0.2522
0.2436
0.2335
0 .. 2216
0.2073
0.2165
0.2021
0.1881
0.1745
0.1611
0.1480
0.1349
0.1220
0.1089
0.0957
1.835
2.041
2.293
2.615
0.1899
0.1684
0.1409
0.1046
0.0821
0.0679
0.0527
0 ..0361
4.0 2.. 824
4.1 3.417
4.2 4.298
4.3 5.764
18
~
62
2.0
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
e
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
4.4
4.5
•
y
0.5708
0.5823
0.5993
0.6213
0.6476
0.6782
0.7132
0.7529
0.7977
0.8480
D
~ = 1.00
0.95
c5
0.0680
0.1322
0.1928
0.2515
0.3091
0.3662
0.4237
0.4818
0.5412
0.6020
jJ'
1
0.2855
0.2855
0.2855
0.2853
0.2848
0.2840
0.2829
0.2815
0.2795
0.2772
C1
0.4255
0.4004
0.3769
0.3549
0.3342
0.3148
0.2964
0.2790
0.2623
0.2465
0.9049 0.6651 0.2744 0.2313
0.9693 0.7308 0.2712 0.2168
1.042 0.7995 0.2673 0.2027
1.126 0.8719 0.2628 0.1890
1.222 0.9488 0.2577 0.1758
1.333 1.031 0.2517 0.1628
1.464 1.120 0.2448 0.1500
1.619 1.216 0.2368 0.1374
1.804 1.321 0.2275 0.1249
2.032
1.438 0.2167 0.1124
2.315
2.679
3.162
3.835
4.846
6.574
1.570
1.719
1.892
2.097
2.346
2.661
0.2039
0.1887
0.1704
0.1479
0.1194
0.0823
0.0998
0.0870
0.0737
0.0599
0.0450
0.0287
13 2
y
15
l1'
1
0.2764
0.2765
0.2764
0.2762
0.2758
0.2750
0.2740
0.2726
0.2708
C1
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
0.5980
0.6095
0.6268
0.6489
0.6754
0.7061
0.7411
0.7807
0.8251
0.0681
0.1307
0.1900
0.2472
0.3032
0.3589
0.4146
0.4710
0.5283
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
0.8749
0.9309
0.9939
1..065
1.146
1.238
1.343
1.466
1.609
1.779
0.5869 0.2685 0.2468
0.6473 0.2659 0.2322
0.7100 0.2628 0.2177
0.7753 0.2591 0.2040
0.8438 0.2549 0.1907
0.9160 0.2500 0.1777
0.9929 0.2444 0.1651
1.075 0.2380 0.1527
1.163 0.2306 0.1405
1.259 0.2222 0.1285
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
1.982
2.230
2.540
2.935
3.459
4.192
5.292
7.201
1.364
1.480
1.609
1.756
1.925
2.124
2.363
2.659
0.2124
0.2011
0.1819
0.1723
0.1536
0.1308
0.1025
0.0662
0.4212
0.3968
0.3740
0.3526
0.3324
0.3135
0.2955
0.2785
0.2622
0.1164
0.1044
0.0922
0.0798
0.0670
0.0536
0.0393
0.0236
19
e
e
13 2
y
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
0.6243
0.6357
0.6528
0.6748
0.7011
0.7315
0.7662
0.8053
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
4..4
4.5
4.6
4.7
4.8
4.9
•
ra;:-. 1.10
~ = 1.05
0.8490
0.8979
0.9525
1.013
1.082
1.160
1.247
1.346
1.461
1.592
1.746
1.928
2.145
2.409
2.736
3.153
3.702
4.462
5.597
7.559
<5
0.0649
0.1263
0.1842
0.2401
0.2947
0.3488
0.4029
0.4574
0.5125
0.5690
0.6268
0.6865
0.7485
0.8133
0.8811
0.9526
1.029
1.110
1.197
1.291
1.394
1.501
1.633
1.775
1.936
2.124
2.346
2.616
u'1
a
13 2
0.2676
0.2677
0.2677
0.2675
0.2670
0.2664
0.2654
0.2641
0.4181
0.3943
0.3721
0.3512
0.3315
0.3130
0.2954
0.2787
2.3
2.4
2.5
2.6
2.7
2.8
2.9
0.6498
0.6607
0.6774
0.6988
0.7247
0.7546
0.7887
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
0.2623
0.2603
0.2577
0.2548
0.2514
0.2414
0.2429
0.2377
0.2318
0.2251
0.2174
0.2087
0.1981
0.1811
0.1737
0.1581
0.1396
0.1173
0.0901
0.0560
0.2629
0.2471
0.2332
0.2193
0.2059
0.1929
0.1802
0.1680
0.1559
0.1441
0.1324
0.1209
0.1093
0.0978
0~0861
0.0741
0.0619
0.0490
0.0354
0.0206
u'1
a
0.0587
0.1189
0.1757
0.2303
0.2838
0.3361
0.3887
0.2591
0.2591
0.2591
0.2590
0.2586
0.2580
0.2571
0.4160
0.3929
0.3711
0.3507
0.3314
0.3133
0.2961
0.8269
0.8696
0.9171
0.9701
1.029
1.• 095
1.168
1.251
1.344
1.450
0.4414
0.4947
0.5487
0.6041
0.6609
0.7198
0.7809
0.8445
0.9113
0.9816
0.2559
0.2543
0.2523
0.2500
0.2473
0.2441
0.2404
0.2362
0.2315
0.2261
0.2797
0.2642
0.2494
0.2352
0.2215
0.2084
0.1956
0.1833
0.1713
0.1596
1.571
1.710
1.812
2.062
2.288
2.562
2.898
3.. 323
3.817
4.634
1.056
1.135
1.220
1.312
1.411
1.521
1.641
1.175
1.921
2.100
0.2200
0.2131
0.2053
0.1964
0.1863
0.1748
0.1616
0.1463
0.1284
0.1072
0.1481
0.1368
0.1257
0.1146
0.1036
0.0924
0.0813
0.0699
0.0582
0.0460
5.0 5.745
5.1 7.622
2.303
2.543
0.0819 0.0332
0.0508 0.0194
y
<5
20
e
~ .. 1.15
62
2.4
2.5
2.6
2.7
2.8
2.9
e
•
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
Y
0.6747
0.6846
0.7004
0.7213
0.7464
0.1144
0.8086
0.8458
0.8871
0.9331
0.9840
1.040
1.103
1.173
1.250
1.337
4.0
4.1
4.2
4.3
1.435
1.545
1.671
1.816
4.4
4.5
4.6
4.7
4.8
4.9
1.982
2.177
2.408
2.684
3.021
3.442
5.0
5.1
5.2
5.3
3.983
4.709
5.148
7.431
IS
0.0496
0.1088
0.1647
0.2181
0.2102
0.3214
0.3723
0.4232
0.4745
0.5265
0.5795
0.6337
0•.6894
0.7471
0.8066
0.8692
0.9344
1.003
1.075
1.152
1.234
1.323
1.418
1.521
1.635
1.760
1.900
2.058
2.240
2.450
~ .. 1.20
lJI
1
0.2508
0.2509
0.2509
0.2507
0.2504
0.2499
0.2491
0.2480
0.2466
0.2448
0.2427
0.2402
0.2373
0.2339
0.2301
0.2257
0.2208
0.2153
0.2091
0.2021
0.1943
0.1855
0.1755
0.1643
0.1514
0.1368
0.1199
0.1002
0.0170
0.0493
a
0.4150
0.3924
0.3110
0.3510
0.3321
0.3143
0.2974
0.2814
0.2662
0.2516
0.2376
0.2243
0.2114
0.1989
0.1868
0.1751
0.1637
0.1526
0.1416
0.1308
0.1201
0.1095
0.0990
0.0884
0.0717
0.0669
0.0558
0.0443
0.0323
0.0196
82
y
c5
lJI
1
a
2.5
2.6
2.1
2.8
2.9
0.6990
0.7077
0.1224
0.7422
0.7662
0.0378
0.0963
0.1513
0.2038
0.2546
0.2428
0.2428
0.2429
0.2428
0.2426
0.4148
0.3927
0.3718
0.3521
0.3336
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
0.7943
0.8263
0.8621
0.9020
0.9461
0.9948
1.049
1.108
1.173
1.246
0.3046
0.3540
0.4032
0.4528
0.5027
0.5534
0.6051
0.6580
0.7124
0.7686
0.2421
0.2414
0.2405
0.2392
0.2376
0.2357
0.2334
0.2307
0.2277
0.2242
0.3161
0.2995
0.2837
0.2687
0.2544
0.2407
0.2275
0.2148
0.2026
0.1908
4.0
4.1
4.2
4.3
4.4
0.8269
0.8874
0.9507
1.017
1.087
4.5
4.6
4.7
4.8
4.9
1.327
1.417
1.518
1.632
1.761
1.908
2.076
2.272
2.502
2.775
1.161
1.240
1.324
1.414
1.511
0.2202
0.2158
0.2108
0.2053
0.1991
0.1921
0.1844
0.1758
0.1661
0.1553
0.1793
0.1682
0.1573
0.1466
0.1362
0.1259
0.1157
0.1056
0.0955
0.0854
5.0
5.1
5.2
5.3
5.4
5.5
3.105
3.511
4.025
4.699
5.636
7.073
1.617
1. 733
1.861
2.003
2.164
2.346
0.1431
0.1293
0.1136
0.0956
0.0748
0.0505
0.0753
0.0650
0.0545
0.0438
0.0326
0.0210
21
~
e
e
/3 2
y
III
0
~ .. 1.30
1.25
~'
a
2.6
2.7
2.8
2.9
0.7231
0.7300
0.7433
0.7617
0.0232
0.0814
0.1357
0.1873
1
0.. 2350
0.2351
0.2351
0.2351
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
0.7846
0.8114
0.8419
0.8763
0.9144
0.9566
1.003
1.054
1.110
1.172
0.2372
0.2859
0.3338
0.3817
0.4294
0.4775
0.5260
0.5753
0.6256
0.6771
0.2349
0.2346
0.2340
0.2332
0.2321
0.2306
0.2289
0.2269
0.2245
0.2217
0.3357
0.3184
0.3021
0.2865
0.2718
0.2576
0.2442
0.2312
0.2187
0.2067
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
1.239
1.314
1.397
1.490
1.592
1.708
1.837
1.985
2.153
2.346
0.7301
0.7846
0.8410
0.8996
0.9607
1.025
1.092
1.162
1.237
1.. 316
0.2186
0.2150
0.2110
0.2066
0.2016
0.1961
0.1900
0.1832
0.1757
0.1674
0.1952
0.1839
0.1729
0.1623
0.1519
0.1418
0.1318
0.1219
0.1122
0.1026
5.0
5.1
5.2
5.3
5.4
5.5
5.6
5.1
2.571
2.836
3.152
3.536
4.013
4.624
5.445
6.638
1.401
1.492
1.590
1.696
1.812
1.940
2.080
2.238
0.1582
0.1479
0.1365
0.1236
0.1092
0.0929
0.0744
0.0533
0..4155
0.3938
0.3733
0.3539
0.0930
0.0834
0.0738
0.0641
0.0542
0.0442
0.0339
0.0232
a
1
2.7 0.7471 0.0061 0.2275 0.4170
2.8 0.7519 0.0643 0.2276 0.3956
2.9 0.7634 0.1180 0.2276 0.3755
y
15
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
0.7802
0.8015
0.8268
0.8559
0.8885
0.9248
0.9648
1.009
1.057
1.110
0.1689
0.2179
0.2655
0.3123
0.3587
0.4049
0.4511
0.4977
0.5448
0.5927
0.2276
0.2275
0.2273
0.2268
0.2261
0.2252
0.2239
0.2224
0.2206
0.2185
0.3564
0.. 3384
0.3214
0.3053
0.2899
0.2753
0.2614
0.2481
0.2353
0.2230
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
1.167
1.231
1.300
1.376
1.460
1.554
1.657
1.172
1.901
2.047
0.6415
0.6914
0.7426
0.7954
0.8498
0.9062
0.9648
1.026
1.090
1.157
0.2160
0.2132
0.2100
0.2064
0.2025
0.1981
0.1932
0.1878
0.1818
0.1753
0.2112
0.1998
0.1888
0.1780
0.1676
0.1575
0.1478
0.1379
0.1283
0.1189
5.0
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
2.212
2.400
2.617
2.870
3.168
3.523
3.959
4.503
5.210
6.184
1.228
1.302
1.381
1.466
1.556
1.652
1.757
1.870
1.994
2.130
0.1681
0.1602
0.1515
0.1418
0.1312
0.1194
0.1063
0.0917
0.0753
0.0570
0.1096
0.1004
0.0913
0.0822
0.0731
0.0639
0.0547
0.0453
0.0358
0.0260
6.0
7.687
2.280
0.0362 0.0158
/3 2
lJ'
~
13 2
III
lit
es
1
2.9 0.7737 0.0446 0.2203 0.3982
e
•
y
15
22
~ .. 1.40
1.35
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
0.1830
0.7919
0.8175
0.8411
0.8684
0.8992
0.9335
0.9713
1.013
1.058
0.0982
0.1481
0.1969
0.2436
0.2894
0.3344
0.3791
0.4238
0.4686
0.5131
0.2204
0.2204
0.2204
0.2202
0.2198
0.2193
0.2185
0.2175
0.2161
0.2145
0.3783
0.3595
0.3418
0.3249
0.3090
0.2938
0.2794
0.2656
0.2525
0.2399
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
1.108
1.162
1.220
1.285
1.355
1.432
1.516
1.609
1.712
1.826
0.5594
0.6058
0.6529
0.7011
0.7506
0.8014
0.8537
0.9078
0.9637
1.022
0.2126
0.2104
0.2080
0.2051
0.2020
0.1985
0.1946
0.1902
0.1855
0.1803
0.2277
0.2160
0.2048
0.1939
0.1834
0.1732
0.1632
0.1535
0.1441
0.1348
5.0
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
1.952
2.094
2.254
2.435
2.642
2.880
3.157
3.484
3.875
4.353
1.083
1.146
1.212
1.282
1.356
1.434
1.516
1.604
1.698
1.799
0.1746
0.1683
0.1615
0.1540
0.1458
0.1368
0.1270
0.1163
0.1044
0.0914
0.1257
0.1167
0.1078
0.0991
0.0904
0.0817
0.0731
0.0645
0.0558
0.0470
6.0 4.955
6.1 5.748
6.2 6.873
1.908
2.025
2.153
0.0770 0.0382
0..0611 0.0291
0.0433 0.0199
132
y
3.0
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
0.7958
0.8025
0.8151
0.8326
0.8543
0.8797
0.9085
0.9408
0.9763
1.015
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
6
lit
es
0.0228
0.0765
0.1266
0.1744
0.2204
0.2650
0.3090
0.3524
0.3956
.0.4389
1
0.2133
0.2133
0.2134
0.2134
0.2133
·0.2131
0.2126
0.2120
0.2112
0.2100
0.4015
0.3818
0.3633
0.3456
0.3290
0.3132
0.2982
0.2839
0.2703
0.2572
1.058
1.104
1.154
1.209
1.268
1.333
1.403
1.480
1.564
1.656
0.4821
0.5258
0.5700
0.6147
0.6602
0.7067
0.7543
0.8030
0.8532
0.9048
0.2087
0.2070
0.2051
0.2029
0.2005
0.1977
0.1945
0.1911
0.1873
0.1831
0.2448
0.2328
0.2213
0.2101
0.1994
0.1890
0.1790
0.1692
0.1597
0.1504
5.0
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
1.757
1.868
1.992
2.129
2.282
2.454
2.649
2.871
3.126
3.422
0.9581
1.013
1.071
1.130
1.193
1.258
1.326
1.397
1.473
1.553
0.1785
0.1735
0.1681
0.1621
0.1557
0.1487
0.1411
0.1328
0.1238
0.1140
0.1413
0.1325
0.1237
0.1152
0.1067
0.0984
0.0901
0.0819
0.0737
0.0656
6.0
6.1
6.2
6.3
6.4
6.5
3.771
4.188
4.700
5.346
6.206
7.458
1.637
1.727
1.823
1.925
2.035
2.153
0.1034
0.0918
0.0791
0.0653
0.0501
0.0334
0.0574
0.0492
0.0410
0.0326
0.0242
0.0156
24
~
e
•
13 2
3.5
3.6
3.7
3.8
3.9
0.8677
0.8772
0.8916
0.9102
0.9323
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5.0
5.1
5.2
5.3
5.4
y
= 1.55
~
lJ'
~ = 1.60
(1
0.0498
0.0974
0.1423
0.1854
0.2270
1
0.1938
0.1938
0.1939
0.1939
0.1938
0.3776
0.3604
0.3441
0.3286
0.3139
0.9577
0.9863
1.018
1.052
1.089
1.130
1.173
1.220
1.270
1.324
0.2674
0.3074
0.3467
0.3856
0.4245
0.4633
0.5023
0.5415
0.5810
0.6209
0.1935
0.1931
0.1926
0.1919
0.1909
0.1897
0.1883
0.1867
0.1849
0.1829
0.2999
0.2865
0.2738
0.2616
0.2498
0.2386
0.2277
0.2173
0.2073
0.1976
5.6
5.7
5.8
5.9
1.382
1.445
1.512
1.585
1.663
1.748
1.839
1.938
2.047
2.165
0.6614
0.7024
0.7442
0.7867
0.8301
0.8745
0.9199
0.9665
1.014
1.064
0.1805
0.1780
0.1752
0.1721
0.1688
0.1652
0.1613
0.1571
0.1525
0.1477
0.1882
0.1791
0.1702
0.1618
0.1532
0.1450
0.1370
0.1292
0.1215
0.1140
6.0
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
7.0
7.1
7.2
7.3
2.294
2.435
2.591
2.764
2.956
3.171
3.413
3.689
4.005
4.372
4.806
5.331
5.989
6.857
1.114
1.166
1.220
1.276
1.334
1.394
1.456
1.521
1.588
1.658
1.731
1.807
1.886
1.969
0.1425
0.1369
0.1310
0.1246
0.1173
0.1106
0.1029
0.0947
0.0860
0.0767
0.0669
0.0564
0.0453
0.0335
0.1066
0.0994
0.0922
0.0851
0.0781
0.0712
0.0644
0.0576
0.0508
0.0440
0.0373
0.0306
0.0239
0.0172
5.S
13 2
3.6
3.7
3.8
3.9
0.8872
0.8931
0.9045
0.9203
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.• 9
y
<5
lJ'
(1
0.0203
0.0685
0.1136
0.1565
1
0.1877
0.1877
0.1878
0.1878
0.3834
0.3663
0.3501
0.3346
0.9399
0.9629
0.9889
1.018
1.050
1.084
1.121
1.161
1.205
1.251
0.1977
0.2377
0.2767
0.3151
0.3531
0.3906
0.4281
0.4656
0.5032
0.5408
0.1878
0.1877
0.1874
0.1870
0.1864
0.1857
0.1848
0.1836
0.1822
0.1807
0.3200
0.3060
0.2927
0.2799
0.2677
0.2561
0.2449
0.2341
0.2238
0.2138
5.0
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
1.301
1.354
1.411
1.472
1.537
1.607
1.683
1. 764
1.852
1.946
0.5789
0.6172
0.6560
0.6953
0.7351
0.7756
0.8169
0.8590
0.9019
0.9459
0.1789
0.1770
0.1748
0.1723
0.1696
0.1668
0.1636
0.1602
0.1565
0.1526
0.2041
0.1948
0.1858
0.1770
0.1685
0.1602
0.1522
0.1443
0.1366
0.1291
6.0
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
2.049
2.159
2.280
2.412
2.555
2.713
2.887
3.080
3.295
3.807
0.9908
1.037
1.084
1.133
1.183
1.234
1.287
1.342
1.398
1.517
0.1484
0.1439
0.1391
0.1340
0.1285
0.1227
0.1166
'0.1101
0.1032
0.0959
0.1218
0.1146
0.1075
0.1005
0.0937
0.0869
0.0803
0.0737
0.0672
0.0607
7.0
7.1
7.2
7.3
7.4
7.5
7.6
7.7
3.807
4.117
4.474
4.893
5.396
6.017
6.826
7.982
1.517
1.579
1.644
1.711
1.780
1.852
1.927
2.004
0.0881
0.0800
0.0713
0.0622
0.0526
0.0424
0.0318
0.0205
0.0543
0.0480
0.0417
0.0354
0.0292
0.0229
0.0167
0.0106
25
e
e
e
1Bi". 1.70
~. 1.65
}.I'
y
6
<1
(32
1
3.8 0.9105 0.0377 0.1818 0.3726
3.9 0.9184 0.0834 0.1819 0.3564
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
0.9312
0.9480
0.9683
0.9917
1.018
1.047
1.079
1.113
1.150
1.190
0.1264
0.1674
0.2071
0.2456
0.2831
0.3202
0.3569
0.3932
0.4293
0.4654
0.1820
0.1820
0.1820
0.1818
0.1815
0.1811
0.1805
0.1798
0.1789
0.1778
0.3410
0.3264
0.3124
0.2991
0.2865
0.2743
0.2626
0.2515
0.2408
0.2304
5.0
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
1.233
1.278
1.327
1.379
1.434
0.5016
0.5318
0.5743
0.6111
0.6483
0.6858
0.7239
0.7625
0.8017
0.8415
0.1765
0.1750
0.1733
0.1714
0.1693
0.1670
0.1645
0.1618
0.1588
0.1556
0.2205
0.2190
0.2016
0.1927
0.1840
0.1755
0.1674
0.1594
0.1516
0.1441
1.i194
1.557
1.625
1.697
1.775
}.I'
y
<1
6
62
1
3.9 0.9301 0.0049 0.1762 0.3793
4.0
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
0.9339
0.9432
0.9569
0.9744
0.9951
1.019
1.045
1.074
1.106
1.140
0.0514
0.0949
0.1361
0.1756
0.2138
0.2509
0.2872
0.3229
0.3582
0.3933
0.1762
0.1763
0.1764
0.1764
0.1763
0.1762
0.1759
0.1755
0.1749
0.1742
0.3632
0.3478
0.3332
0.3193
0.3059
0.2932
0.2811
0.2695
0.2583
0.2476
5.0
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
1.176
1.215
1.257
1.302
1.349
1.400
1.454
1.511
1·.572
1.637
0.4281
0.4629
0.4977
0.5326
0.5677
0.6030
0.6385
0.6744
0.7107
0.7475
0.1733
0.1723
0.1110
0.1696
0.1680
0.1662
0.1642
0.1621
0.1597
0.1572
0.2373
0.2274
0.2179
0.2086
0.1997
0.1911
0.1827
0.1746
0.1666
0.1590
6.0
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
1.707
1.781
1.860
1.944
2.035
2.132
2.237
2.350
2.471
2.603
0.1841
0.8226
0.8609
0.9000
0.9398
0.9803
1.022
1.064
1.107
1.151
0.1544
0.1514
0.1483
0.1449
0.1412
0.1374
0.1333
0.1290
0.1245
0.1197
0.lS15
0.1442
0.1371
0.1301
0.1233
0.1166
0.1101
0.1037
0.0974
0.0912
6.0
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
1.858
1.948
2.044
2.148
2.261
2.383
2.515
2.659
2.816
2.990
0.8822
0.9235
0.9658
1.009
1.053
1.098
1.145
1.192
1.241
1.291
0.1522
0.1485
0.1447
0.1405
0.1361
0.1314
0.1264
0.1211
0.1156
0.1096
0.1367
0.1294
0.1224
0.1154
0.1086
0.1020
0.0954
0.0889
0.0826
0.0763
7.0
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
3.180
3.391
3.626
3.890
4.188
4.530
4.927
5.391
5.970
6.699
1.343
1.396
1.450
1.506
1.564
1.623
1.685
1.748
1.813
1.880
0.1035
0.0969
0.0900
0.0828
0.0752
0.0672
0.0588
0.0501
0.0409
0.0313
0.0101
0.0640
0.0579
0.0519
0.0460
0.0401
0.0343
0.0284
0.0227
0.0170
1.0
7.1
1.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
2.745
2.900
3.070
3.256
3.461
3.687
3.940
4.223
4.545
4.915
1.196
1.242
1.289
1.337
1.387
1.437
1.489
1.543
1.597
1.653
0.1147
0.1094
0.1038
0.0919
0.0918
0.0853
0.0786
0.0716
0.0642
0.0566
0.0851
0.0791
0.0732
0.0673
0.0616
0.0559
0.0503
0.0447
0.0392
0.0331
8.0
7.702
1.948
0.0213 0.0113
8.0
8.1
8.2
8.3
5.347
5.865
6.508
1.355
1.710
1.769
1.829
1.891
0.0486
0.0403
0.0317
0.0228
0.0283
0.0230
0.0177
0.0125
26
2
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
0.9517
0.9569
0.9672
0.9815
0.9994
1.020
1.044
1.070
1.099
5.0
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
1.130
1.164
1.200
1.238
1.279
1.322
1.369
1.418
1.470
1.525
0.3575
0.3913
0.4249
0.4585
0.4920
0.5255
0.5592
0.5931
0.6271
0.6614
0.1696
0.1689
0.1680
0.1670
0.1658
0.1645
0.1630
0.1613
0.1595
0.1574
0.2547
0.2444
0.2345
0.2250
0.2158
0.2069
0.1982
0.1899
0.1818
0.1740
6.0
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
1.583
1.646
1.712
1.782
1.857
1.937
2.022
2.113
2.210
2.314
0.6960
0.7310
0.7665
0.8023
0.8386
0.8755
0.9129
0.9510
0.9897
1.029
0.1552
0.1528
0.1503
0.1415
0.1446
0.1414
0.1381
0.1346
0.1309
0.1269
0.1664
0.1589
0.1517
0.1447
0.1378
0.1311
0.1245
0.1181
0.1118
0.1056
7.0
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
8.0
8.1
8.2
8.3
8.4
8.5
8.6
8.7
2.426
2.546
2.675
2.815
2.966
3.130
3.310
3.506
3.722
3.961
4.227
4.527
4.867
5.259
5.720
6.278
6.986
7.957
1.069
1.110
1.152
1.194
1.237
1.282
1.327
1.373
1.420
1.468
1.517
1.567
1.618
1.670
1.723
1.777
1.832
1.888
0.1228
0.1184
0.1138
0.1090
0.1040
0.0988
0.0933
0.0875
0.0815
0.0153
0.0688
0.0621
0.0551
0.0479
0.0404
0.0327
0.0248
0.0166
0.0995
0.0936
0.0878
0.0820
0.0763
0.0108
0.0653
0.0598
0.0545
0.0492
0.0440
0.0388
0.0337
0.0287
0.0237
0.0188
0.0140
0.0092
13
e
•
lSi" • 1.80
~. 1.75
6
lJ'
1
0.0175 0.1708
0.0619 0.1708
0.1035 0.1709
0.1431 0.1709
0.1811 0.1710
0.2178 0.1709
0.2537 0.1708
0.2888 0.1705
0.3233 0.1701
y
a
0.3703
0.3550
0.3404
0.3264
0.3131
0.3004
0.2882
0.2766
0.2654
B2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
y
6
0.9131
0.9194
0.9903
1.005
1.023
1.044
1.067
5.0
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
lJ'
a
0.0270
0.0694
0.1095
0.1416
0.1843
0.2198
0.2545
1
0.1655
0.1656
0.1651
0.1657
0.1657
0.1657
0.1656
0.3625
0.3419
0.3339
0.3206
0.3078
0.2956
0.2840
1.093
1.122
1.152
1.185
1.220
1.258
1.298
1.340
1.384
1.432
0.2884
0.3218
0.3547
0.3874
0.4199
0.4522
0.4846
0.5168
0.5491
0.5815
0.1653
0.1649
0.1644
0.1637
0.1629
0.1620
0.1609
0.1597
0.1582
0.1567
0.2728
0.2620
0.2518
0.2418
0.2323
0.2231
0.2142
0.2056
0.1973
0.1892
6.0
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
1.482
1.535
1.591
1.650
1.713
1.780
1.851
1.926
2.005
2.090
0.6142
0.6470
0.6801
0.1134
0.7471
0.1811
0.8155
0.8503
0.8855
0.9214
0.1550
0.1531
0.1510
0.1488
0.1464
0.1439
0.1412
0.1383
0.1352
0.1320
0.1814
0.1738
0.1664
0.1593
0.1523
0.1455
0.1388
0.1323
0.1260
0.1198
7.0
7.1
7.2
7.3
7.4
2.180
2.277
2.379
2.489
2.606
0.9577
0.9945
1.032
1.070
1.109
0.1286
0.1250
0.1212
O.U72
0.1131
0.1137
0.1077
0.1019
0.0962
0.0905
1.5
7.6
7.7
7.8
7.9
8.0
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
9.0
9.1
2.732
2.868
3.014
3.172
3.344
3.530
3.734
3.958
4.205
4.481
4.791
5.143
5.550
6.032
6.620
7.380
8.466
1.148
1.188
1.228
1.270
1.312
1.355
1.398
1.443
1.488
1.534
1.581
1.628
1.676
1.. 725
1.774
1.824
1.875
0.1088
0.1042
0.0995
0.0945
0.0894
0.0841
0.0785
0.0728
0.0669
0.0607
0.0544
0.0478
0.0411
0.0342
0.0271
0.0198
0.0124
0.0850
0.0796
0.0742
0.0690
0.0638
0.0587
0.0536
0.0487
0.0438
0.0389
0.0342
0.0295
0.0248
0.0203
0.0158
0.0113
0.0070
27
~
82
e
-
y
III
15
~ .. 1.90
1.85
).1'
1
a
4.5 0.9943 0.0336
4.6 1.001 0.0144
4.7 1.012 0.1130
4.8 1.027 0.1499
4.9 1.045 0.1854
0.3558
0.3411
0.3284
0.3156
0.1607 0.3033
5.0 1.066
5.1 1.089
5.2 1.114
5.3 1.142
5.4 1.172
5.5 1.204
5.6 1.238
5.7 1.275
5.8 1.313
5.9 1.354
0.2197
0.2533
0.2861
0.3184
0.3503
0.3819
0.4132
0.4444
0.4755
0.5066
0.1607
0.1605
0.1603
0.1599
0.1595
0.1589
0.1581
0.1572
0.1562
0.1551
6.0
6.1
6.2
6.3
6.4
6.5
6.6
6.7
1.397
1.443
1.491
1.542
1.595
1.652
1.112
1.715
6.8 1.842
6.9 1.912
0.5376 0.1538 0.1967
0.5688 0.1523 0.1889
0.6000 0.1507 0.1813
0.6314 0.1490 0.1740
0.6630 0.1471 0.1668
0.6948 0.1450 0.1599
0.7268 0.1428 0.1531
0.7592 0.1405 0.1465
0.7918 0.1380 0.1401
0.8248 0.1353 0.1338
7.0
7.1
7.2
7.3
7.4
0.8581
0.8918
0.9259
0.9604
0.9955
1.987
2.066
2.149
2.238
2.333
0.1605
0.1606
0.1606
0.1607
0.1325
0.1296
0.1264
0.1232
0.1197
0.2916
0.2804
0.2696
0.2593
0.2493
0.2398
0.2305
0.2216
0.2131
0.2047
0.1277
0.1216
0.1158
0.1100
0.1044
82
y
4.1 1.015
4.8 1.022
4.9 1.033
5.0
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
1.048
1.066
1.086
1.109
1.134
1.161
1.190
1.221
1.254
1.290
6.0
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
7.0
7.1
7.2
7.3
7.4
7.5
7.6
15
lJ'
1
a
0.0378 0.1556 0.3499
0.0712 0.1557 0.3364
0.1145 0.1558 0.3236
0.1502
0.1845
0.2178
0.2504
0.2822
0.3135
0.3443
0.3748
0.4051
0.4352
0.1559
0.1559
0.1558
0.1557
0.1555
0.1552
0.1547
0.1542
0.1535
0.1527
0.3113
0.2995
0.2883
0.2714
0.2671
0.2571
0.2475
0.2382
0.2293
0.2207
1.327
1.366
1.408
1.451
1.498
1.546
1.597
1.651
1.708
1.767
0.4652
0.4951
0.5249
0.5549
0.5849
0.6149
0.6451
0.6754
0.7060
0.7367
0.1518
0.1507
0.1495
0.1482
0.1467
0.1451
0.1433
0.1415
0.1394
0.1373
0.2123
0.2043
0.1965
0.1889
0.1816
0.1745
0.1675
0.1608
0.1542
0.1478
1.830
1.896
1.966
2.040
2.117
2.199
2.286
0.7676
0.7989
0.8304
0.8623
0.8944
0.9268
0.9597
0.1350
0.1325
0.1300
0.1273
0.1244
0.1214
0.1183
0.1416
0.1355
0.1295
0.1237
0.1180
0.1125
0.1070
28
~
e
1..031
1.067
1.103
1.140
1.178
0.1161
0.1123
0.1084
0.1043
0.1000
8.0 3.046
8.1 3.197
8.2 3.360
8.3 3.535
8.4 3.726
1.216
1.255
1.294
0.0956 0.0727
0.0910 0.0677
0.0862 0.0628
8.5 3.935
8.6 4.163
8.7 4.415
8.8 4.695
1.334
1.374
1.415
1.457
1.499
1.542
0.0813
0.0762
0.0709
0.0655
0.0598
0.0541
8.9
1.585
0.0481 0.0305
1.629
1.673
1. 718
1.763
1.808
0.0421 0.0262
0.0358 0.0219
5.010
IBi" • 1.90
(cont.)
7.5 2.433
7.6 2.540
7.7 2.654
7.8 2.776
7.9 2.906
0.0989
0.0934
0.0881
9.5
6.273
6.877
7.666
8.830
1.853
0.0295
0.0230
0.0164
0.0097
(cont.)
7.7 2.379
1.8 2.476
7.9 2.580
0.9929
1.027
1.060
0.1150 0.1016
0.1116 0.0964
0.1080 0.0912
8.0
1.095
1.130
1.165
0.1043
0.1005
0.0965
0.0923
0.0880
0.0836
0.0790
0.0743
0.0695
0.0645
0.0829
0.0777
0.0580
0.0532
0.0485
0.0439
0.0394
0.0349
2.690
8.1 2.807
8.2 2.932
8.3 3.065
8.4 3.208
8.5 3.361
8.6 3.526
8.7 3.703
8.8 3.896
8.9 4.106
1.201
1.237
1.273
1.311
1.348
1.386
1.424
I
9.0 5.368
9.1 5.782
9.2
9.3
9.4
•
= 1.85
0.0177
0.0136
0.0095
0.0056
9.0
4.335
9.1 4.587
9.2 4.868
9.3 5.182
9.4 5.538
9.5 5.950
9.6 6.438
9.7 7.039
9.8 7.828
9.9 9.006
1.463
1.502
1.542
1.582
1.622
1.662
1.703
1.744
1.785
1.826
0.0862
0.0812
0.0764
0.0716
0.0669
0.0622
0.0577
0.0532
0.0488
0.0444
0.0402
0.0360
0.0318
0.0277
0.0376 0.0237
0.0319 0.0198
0.0261 0.0159
0.0202 0.0122
0.0143 0.0084
0.0083 0.0048
0.0594
0.0541
0.0487
0.0432
29
~. 2.00
~:II 1.95
82
e
-
y
6
lJ'
1
0
4.9
1.035
0.0396 0.1510 0.3449
5.0
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
1.042
1.053
1..068
1.085
1.105
1.127
1.151
1.178
1.206
1.236
0.0778
0.1140
0.1486
0.1820
0.2143
0.2458
0.2767
0.3070
0.3369
0.3664
0.1510
0.1511
0.1512
0.1512
0.1512
0.1511
0.1509
0.1506
0.1502
0.1497
0.3319
0.3196
0.3077
0.2964
0.2855
0.2751
0.2650
0.2553
0.2460
0.2371
6.0
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
1.268
1.302
1.338
1.376
1.416
1.458
1.502
1.548
1.597
1.648
0.3957
0.42'48
0.4537
0.4825
0.5113
0.5400
0.5688
0.5976
0.6264
0.6554
0.1491
0.1484
0.1475
0.1465
0.1454
0.1442
0.1428
0.1414
0.1398
0.1381
0.2284
0.2201
0.2120
0.2042
0.1966
0.1893
0.1821
0.1752
0.1684
0.1619
7.0
7.1
7.2
7.3
7.4
7.5
7.6
7.7
1.8
7.9
1.701
1.758
1.817
1.879
1.944
2.013
2.085
2.161
2.241
2.326
0.6846
0.7138
0.7433
0.7130
0.8028
0.8329
0.8633
0.8939
0.9248
0.9559
0.1362
0.1342
0.1321
0.1299
0.1276
0.1251
0.1225
0.1197
0.1169
0.1139
0.1555
0.1493
0.1433
0.1373
0.1316
0.1259
0.1204
0.1150
0.1097
0.1045
8.0
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
2.415
2.509
2.609
2.714
2.826
2.945
3.071
3.206
3.349
3.503
0.9874
1.019
1.051
1.084
1.116
1.150
1.183
1.217
1.251
1.285
0.1108
0.1076
0.1042
0.1007
0.0971
0.0934
0.0895
0.0856
0.0815
0.0772
0.0994
0.0945
0.0896
0.0848
0.0801
0.0754
0.0709
0.0664
0.0620
0.0577
9.0
9.1
3.668
3.846
1.320
1.355
0.0729
0.0685
0.0535
0.0493
62
y
6
ll'
0
0.1465
0.1465
0.1466
0.1467
0.1467
0.1467
0.1466
0.1465
0.1462
0.3406
0.3281
0.3162
0.3048
0.2938
0.2833
0.2732
0.2634
0.2541
0.1459
0.1454
0.1449
0.1442
0.1434
0.1425
0.1415
0~5243 0.1404
0.5520 0.1392
0.5196 0.1379
0.2451
0.2364
0.2280
0.2198
0.2120
0.2044
0.1970
0.1899
0.1830
0.1762
1
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
1.054
1.061
1.012
1.086
1.103
1.122
1.144
1.167
1.193
0.0394
0.0765
0.lll7
0.1453
0.1778
0.2092
0.2399
0.2698
0.2993
6.0
6 .. 1
6.2
6.3
6.4
6.5
6.6
6.1
6.8
6.9
1.220
1.249
1.280
1.313
1.347
1.384
1.422
1.462
1.504
1.548
0.3283
0.3569
0.3852
0.4133
0.4412
0.4690
0.4967
7.0
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
1.594
1.643
1.693
1.746
1.802
1.860
1.921
1.985
2.053
2.123
0.6013
0.6350
0.6628
0.6908
0.7188
0.7471
0.7754
0.8039
0.8326
0.8615
0.1364
0.1348
0.1331
0.1313
0.1294
0.1274
0.1252
0.1230
0.1206
0.1181
0.1697
0.1633
0.1571
0.1510
0.1451
0.1394
0.1337
0.1283
0.1229
0.1176
8.0
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
2.197
2.275
2.357
2.443
2.533
2.629
2.729
2.836
2.948
3.067
0.8906
0.9200
0.9495
0.9794
1.009
1.040
1.070
1.101
1.132
1.164
0.1155
0.1128
0.1100
0.1071
0.1040
0.1009
0.0976
0.0942
0.0908
0.0872
0.1125
0.1075
0.1025
0.0971
0.0930
0.0883
0.0838
0.0793
0.0749
0.0706
1.195
1.227
1.259
1.291
1.324
0.0835
0.0797
0.0158
0.0718
0.0677
0.0663
0.0622
0.0581
0.0540
0.0501
9.0 3.194
9.1 3.328
9.2 3.471
9.3 3.623
9.4 3.787
y:
30
e
•
~. 1.95
ra;. 2.00
(cont.)
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
4.037
4.246
4.472
4.721
4.997
5.305
5.653
6.054
1.391
1.427
1.463
1.499
1.535
1.572
1.609
1.646
0.0639
0.0592
0.0544
0.0496
0.0446
0.0395
0.0344
0.0292
0.0452
0.0412
0.0372
0.0333
0.0295
0.0257
0.0220
0.0184
10.0
10.1
10.2
10.3
6.527
7.107
7.863
8.978
1.683
1.720
1.757
1.794
0.0239
0.0185
0.0132
0.0078
0.0148
0.0113
0.0079
0.0046
9.S 3.962
9.6 4.150
9.7 4.354
9.8 4.576
9.9 4.818
10.0
10.1
10.2
10.3
10.4
10.5
10.6
10.7
5.085
5.381
5.715
6.098
6.546
7.090
7.787
8.780
(cOnt.)
1.357
1.390
1.423
1.457
1.490
0.0635
0.0593
0.0549
0.0505
0.0460
0.0462
0.0424
0.0386
0.0349
0.0313
1.524
1.558
1.591
1.625
1.659
1.693
1.726
1.760
0.0414
0.0367
0.0320
0.0273
0.0224
0.0176
0.0128
0.0079
0.0278
0.0243
0.0209
0.0175
0.0142
·0.0110
0.0019
0.0048
31
APPENDIX
CAlCULATION OF THE TABLES
In order to generate these tables, it was first necessary to find a
~ and
satisfactory method of computing
nations of values of
y
and
<5
6
2
for the various combi-
that were likely to occur.
To do this,
the parameter space was divided into four regions defined as follows
0
(1)
<5 ..
(2)
0
(3)
0.5 s
(4)
<5
For
<5 ...
< 0 <
~
0,
0.5
<5 <
2
2.
~ and
6 were obtained using the formulas
2
where
For the remaining three regions, the following method was used.
It was necessary to calculate
where
n ..
•
16<5
+Y
and
int{x)
..
integral part of
x.
32
We have
and so the first term in (A.l) 18 negligible.
For the third term in brackets in (A.l), we note that if
t·~
n
then
and so
(/2'iT)-l
f'
{l+e(y-t)/c5}-r e-%t
2
dt
n
with an error less than
e-
16
This latter integral was evaluated using a standard subroutine.
The
second term in (A. 1) was evaluated using a 32 point Legendre-Gauss quadrature,
0 < 6 < 0.5.
for
A 64 point Hermite-Gauss quadrature was used for the region
0.5
~
6 < 2
and a 24 point Hermite-Gauss quadrature for the region
~ and
The resulting values of
6
2
6
as functions of
~ and
(32'
•
desired range of
fixed
y
re;:-
and
6 •
2
when plotted in the
(y , 15)
(31
and
2
y
and
62 was
pairs which covered the
The curves corresponding to fixed
(31' 6
2.
All were foood to
As a first step in finding
a table of values of
computed for a pre-determined set of
~
were compared with the tables
in Johnson (1949) and closely examined for consistency.
be satisfactory to five decimal places.
15
0
and
space are app rox1mately linear over
33
small regions and this approximate linearity was satisfactory for most of
the iterations to be described in the next paragraph.
To find the values of
p
D
(~, 13 ),
2
and
y
0
that correspond to a desired point
the table generated above was searched until the 'quad-
rilateral' containing
.!
was found.
divided, first with respect to
0
and then with respect to
smaller 'quadrilateral' containing
the midpoint with respect to
0
then compared to the values for
This 'quadrilateral' was then sub-
X.
and
X.
~ and
The values of
y
to find a
6
for
2
of this new 'quadrilateral' were
If both coordinates were correct to
within a specified tolerance (0.00010), these values of
accepted as the desired values.
y
0
and
y
were
Otherwise, the process was continued un-
til this condition was achieved.
Occasionally, because of the non-linearity of the boundaries of the
'quadrilaterals' used,
X was found to be in a certain 'quadrilateral'
but not in any of the 'quadrilaterals' which resulted from the subdivision
process.
In these situations, the original 'quadrilateral' was enlarged
slightly and the subdividing process continued.
When required, one such
correction was usually sufficient.
The final print-out gave values of
rs;:-
and
6
2
at the vet'tices of
the last 'quadrilateral' as well as those at the center.
This enabled
final adjustments to be made, especially when deciding whether to 'roundoff' upwards or downwards.
All of the calculations were performed on an IBM 360/75 computer
using a lOOK partition of memory.
With the exception of the evaluation of
the cumulative normal function which was done using an IBM supplied routine,
•
all of the calculations were done using double precision arithmatic.
About
220 minutes of computer time were required to do all of the computing including the necessary preliminary investigations and debugging.