TABLES AND FORMULAS FOR CALCULATING
FOURIER COEFFICIENTS OF POWER-LAW DEVICES
PAUL PENFIELD, JR.
TECHNICAL REPORT 385
MARCH 20, 1962
I
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
RESEARCH LABORATORY OF ELECTRONICS
CAMBRIDGE, MASSACHUSETTS
The Research Laboratory of Electronics is an interdepartmental
laboratory in which faculty members and graduate students from
numerous academic departments conduct research.
The research reported in this document was made possible in part
by support extended the Massachusetts Institute of Technology,
Research Laboratory of Electronics, jointly by the U.S. Army
(Signal Corps), the U.S. Navy (Office of Naval Research), and the
U.S. Air Force (Office of Scientific Research) under Signal Corps
Contract DA 36-039-sc-78108, Department of the Army Task
3-99-20-001 and Project 3-99-00-000; and in part by Signal Corps
Contract DA-SIG-36-039-61-G14.
Reproduction in whole or in part is permitted for any purpose
of the United States Government.
MASSACHUSETTS
INSTITUTE
OF
TECHNOLOGY
RESEARCH LABORATORY OF ELECTRONICS
March 20, 1962
Technical Report 385
TABLES AND FORMULAS FOR CALCULATING FOURIER COEFFICIENTS
OF POWER-LAW DEVICES
Paul Penfield, Jr.
Abstract
Tables and formulas are given to facilitate the calculation of Fourier coefficients of
power-law devices that are driven from a sinusoidal source.
TABLE OF CONTENTS
Introduction
1
Formulas
2
Set I.
Integrals
2
Set II.
General Representation
5
Set III.
Recursion Relations
6
Set IV.
Differentiation Formulas
7
Set V.
Specific Values of n
8
Set VI.
Specific Values of
9
Set VII.
Specific Values of a
10
Set VIII.
Miscellaneous
13
14
Tables
I.
II.
Values of I(a, 0; p) for f3 between 0 and 1 in steps
of 0. 02, and a between 0 and 1 in steps of 0. 05
14
Values of I(a, 1; p) for
between 0 and 1 in steps
of 0. 02, and a between 0 and 1 in steps of 0. 05
16
I(a, 1; p)
III.
Values of
for
between 0 and 1 in steps
of 0. 02, and a between 0 and 1 in steps of 0. 05
18
20
Acknowledgment
iii
INT RODU CTION
Recently, the author described a method of calculating Fourier coefficients of powerlaw devices driven from sinusoidal sources.
I(a, n;Z)
(l+pcoswt)
2r
The integral
cos (nt)
d(t)
was expressed in terms of a hypergeometric series, and differentiation and recursion
formulas were given. In this report we give a summary of the pertinent formulas (many
of them already published ) and tables of values of I(a, n; P).
The first set of formulas includes
several examples of integrals that can be
expressed in terms of the function I(a, n; P).
Many of these arise naturally in computing
voltage and current coefficients and power for electrical power-law devices.
The sec-
ond set gives the general representation of I(a, n; P) in terms of a hypergeometric series,
with related definitions.
entiation formulas.
In the third and fourth sets are recursion relations and differ-
The fifth set includes formulas for specific values of n.
sixth set special formulas are stated for
mulas for special values of a.
= 0 and
In the
P = 1. The seventh set includes for-
The eighth set comprises some miscellaneous formulas.
The recursion relations given in the third set of formulas relate different I(a, n; P)
for values of n and a differing by integers.
By repeated use of these relations, any
I(a, n; P) can be expressed in terms of I(a', 0; P) and I(a', 1; P), where a' lies between 0
and 1 and differs from a by some integer.
We give in Tables I and II values of I(a, 0; P) and I(a, 1; P) for all values of a
between 0 and 1 in steps of 0. 05, and for all values of P between 0 and 1 in steps of
0. 02.
In Table III we give values of [I(a, 1; P)/P].
In principle these tables, when used
with the recursion relations given here, are sufficient for calculating any I(a, n; P).
In
practice, however, it is sometimes easier to sum the hypergeometric series directly,
rather than to use the cumbersome recursion relations.
Reference
1.
P. Penfield, Jr., "Fourier Coefficients of Power-Law Devices," J.
Inst. 273, 107-122 (February 1962).
1
Franklin
FORMULAS
Set I
Integrals
j2l
1s
27 )
Tr
S,1
(1
Pcost) a en
[(lip cos~
ct)a1
rw
ejn
I(a, n;
n
t d(t)
=
)
jnw I(a, n; P)
(l)n I(Za, n; B)
e - j nw t d(wt)
[d
[ tC(1 ± posat)a
O
j
2wt
n
d(w t) = (l)
cos t) a ] e - j n wt d(wt) = ()
dL (l
2
t
a2p2w
= (±1)n -
[2I(2a -2, n;p) -I(Za -2, n+ 2;P) -I(
2P 2
2
2 n2 - 2
n+
2a -[
+
n
2a
(n - 2a
n
-
-
4
a
2
n-
1+
n
2
a\
)
n 2 - Za n +- 1
n
n 2 -4a 2
1
2
a -2, n-2; p)]
I(2a-
2, n+1;[5)
2a) I(2 a -2, n- 1;5)]
2 22
a
= (i±l)n 2(2a-1) [(n+l)I(2a - 1, n+ 1; ) -(n-l)I(2a -1, n- 1;
)]
(±1 )n a
2
(2a-1)(2a-n) [(2a-n) I(2a - 1, n+ 1; P) -n(n-1) I(2a - 1, n; )]
(±l)n 2 2
=2a
- 1
= (l)n
2
2(2a-)
[ 1
[
2 )I(2a,
dI(2a, n;)
d
n;
)
) - I(2a - 1, n; 5)]
2
- nI(2a, n; )]
2
Z
[I(2a-2,;)-I(a-2,
Trrrd (i c Post)
d(w)
O; [P)=
Z
[I(2a - ,
I(Za - ,2; )]
ap w
[ (2a -
,
; )+I(za - 2, 1;
z2[I(2a1, 0; )-(p2)I(2a-,
= --
2
)]
; )]
FORMULAS
Set I (continued)
a 2c
2
= 2a - 1 I(2a - 1, 1; )
a
2 2
2=a -1 [I(za, 0; p) -I(2a -1, 0; P)]
apow
d I(2a, 0; )
d,
= 2(2a - 1)
n odd
I2Zwr
2
(1±ycos
"I f2
ae-jnwt
d(wt) =
e
t)
I (a,,
(+ 1)n/2
I...
(2
(~z2
)
2
n even
y
n odd
21
I ysin
(l±
0r
d(w t) =
e
t)
(:T 1)n/2 (2_
I(a,
)
22iY)
n even
n odd
s
-d
2
w
0.r
dt
(1 ± y cos2 ct)a ejnwt d(wt) =
jnw(± 1)n/
2
1)
2± y)
2;
I(a,;
n even
n odd
21 f2O dd (+Ysin
2
t)a e-jn t d(w t) =
jw+1n/2
l
I-~
1
[d(
S
1±ycos
t)a
( ±,P
I(a,;
22
e- jnwt d(wt)
n odd
0
(
n even
)n/2 (2
y
2 a) l
2
(2a-1)
2
\2+ 1) I(2a - 1,-2 + 1; 2
n - 1) I( Za - (2
3
1;
, -L'
2 -
IV
n even
FORMULAS
Set I (continued)
1
-r
Ldt
fZir
t) ai
(±Ysin
e - jn
t
d(wt)
n odd
n/
2y±
():1)n/2
(2a-1) a 2Y)
(22
a
1
(2
Y )
I (za -1 , '2-++ 1; 2-iY
Z
+
(-1
2
I
Zir
J[
r
d
di
2
(±y
azl
os wt)J
2ir
1
d(wt)
=
I2a-1,2
) (
2
(1 ±
sin
(2a-1) 2a
2
) (~~z ` 2)
4
1;
Kb]
t)aj
d(wt)
dr
L
2
2
22
-
-Ia (2
1 1; T±'
2i -
n even
FORMULAS
Set II
General Representation
(
;)
(-p)n
(1 +p)
r(n - a)
r(-a)r(n +
1) F(-a, n -a;
J' "
Pr(1+-)
2
(1 + p )a r(1 + a - n)r(n + 1) F(,
2
n + 1; p )
n-a; n
1;p2 )
where the hypergeometric series is
F(a,b;c;2) =1 + ab z +ab(a + )(b + 1) z2
2
c (c + 1)
ab(a + 1) (b + 1)(a + 2)(b + 2) z3
3! +
c(c + 1)(c + 2)
'
r(c) +-r(a + n)r(b + n) z
n!
r(c + n)
r(a)r(b) o
and the amplitude paremeters are related as
2
k
2p
1+ p2
P=
2-k
r +k2
-=
+
k _/
<11+,
2
-
2 p
1+P
5
_-I
FORMULAS
Set III
Recursion Relations
I(n
*(a
+ 2;
+ 2, n; )
(+
( +
I(a - 2, n;)-
t(a
21(a,n +1;8) +
n+l
)=[
+ I)I(a, n + 1;)
n;)]
n--a
n; )]
n; ) - (a + 1)(1 -fi)I(a,
2) --n' [(3 + 2a)I(a + 1,
2)n+ 2
-) (
+ (n-
(2a- l)I(a- , n;
) + n --
aI,n;
- a)I(a + 1, n; ) + (a + l)I(a, n; )
/5(n + 1 + a)I(a, n + 1; ) + 2n(a, n;/ ) + (n -
-a)I(a,n - 1; B)
/(n + 1 + a)I(a, n + 1;/ ) + (n -a)I(a, n; 8) + a(1 -2)I(a
(n + 1 + a)I(a + 1, n; i) - (a + )I(a, n; 8) -(a
)
0O
0O
- 1, n; ) = 0
+ )I(a, n - 1; P) = O
(n + 1 + a)(n - 1 - a)I(a + 1, n; ) + (a + 1)(1 + 2a)I(a, n; )
- a(a + 1)(1 -B'2)I(a - 1, n;)
(n + a)I(a, n; ) + P(n - 1 - a)I(a, n - 1; ) - a(l - #2)I(a - 1, n; ,) = 0
6
=0
FORMULAS
Set IV
Differentiation Formulas
dI(a,n;
a(a,
n; ) - I(a - 1,n;)]
d l (a, n; tt) =n
aP
- I (a, n; ) + n-
+ a I (a, n + 1; )
dI(a, n; )
da
1
dI(a,n;
dP 0)
n-a - ')1
+I(a,n;n;)- 1, n + 1;);2) _
00
n
dl(a, n; )
(-+= ,n;
dI(, n; )
d#
dI (a, n; )
dP
a+
0(1 -
=
-[I(a- n;,n
t
aEI(a 2
a,2(n;
')
+
(n ++
- (1
(a
n
+ 1;aI(a,n +
; )
a)(n-1
c)I(+
2)(1 + a)
)(
1;)
1,n) - 1;;
-I(a - ln
-1;
7
_____________II___·_
j)]
l)
1 n;)
FORMULAS
Set V
Specific Values of n
n=O
I(a,0;0)
F(-a, -a; ; p2 )
= (1 + p)a F(-a, -a; 1; P2 )
=
1 +(
I(a + 2, 0; ) =
1
(a(l - a)p')
Err(3
2
1
I(a-2,
I I(
-2, 0; ))== (a - 1) (1
dI(a, 0; )
d
'
= al(r(a - 1,
i2)
+
(a(
a)p3 )+
-- )(2
, 0; j3) - ( + 0
-
(a, 0; 0)
[(2 - 1)I(a - 1, 0; ) - aI(a, 0;,)]
; )
n= I
I(a, 1;)
= (1 + p2), F(-a, -a;
1 I(
0; ) - I(a, 0;)
+
I(ar, 1;)
2; P2)
at
, 0I(a
O;
)- ,
(1 - 0')I(,- 1, 0;)]
n-2
n
I(a, 2;)
=
I(a, 2;
=
p'a(a-
1)
2 +)
F(-a, 2 -a;3;p)
;
+
I(ao;
)
3
1
I(a, 3;0) =3
3 +
[( 8 1 +
-I)
(.,I;19- 'C I
(,
2 +-a +
0;
8
-
--
--
-----
)03
FORMULAS
Set VI
Specific Values of
I(a, 0O;O) = 1
I(a, n; 0) = 0
dI(a, 1;O)
dI;)
n = 1, 2, 3, ..-
a
2
dI(a, n; O)
d
0
n = 0, 2, 3, 4, -..
1
I(a, n; 1) =(-1)" oc
a-< -
I(an;1)
=1
r(+ 2)
2;)
1'(1 + a + n)r(l + a - n)
(- 1)"
r(n - a)r(l + 2a)
2
r(-a)r(l
+ a)r+(n
1+
+a)
2
r(2a)
I(a, O; 1) =a2[r(a)
a> -
2
I'(2a)
> -
(1+ a)2-[r(a)
2 a)211 -a
'(2a)
I(a 2; 1)
=- (2 +-)2
+ a [r(F)]
I(a, n + 1; 1) =
I(a + 1, n; 1) =
n +
(a, n; 1)
+
(1 + a)(l + 2a) I(
(1 + a)2 - n2(
n; 1)
a> - i
a > r >
-
-2
dI(a, n; )
d1(O, n; 1) =
dI(an;1)
=
dI(a, n; 1)
d(a, n; 1)
d#
( <
n2 + a
2a n2+a
2adI(a,O; 1)
dft
dI(a, 1; 1)
dOJ
a-
2
<
a I(a,n;)
r(l + 2a)
-a
1
1
2" r(l + a - n)r(n +
1 2
2a - 1 2
-a2-a+1
1 + a)
r(2 a)
r(o '-NU
2 r(2a)
2a + a -1 2
r(a)]2
9
--- ~ ~ ~ ~~~~~~~~~~~~~ ----~
-- ~~~~~~~~~~~~~~~~~~~~II
FORMULAS
Set VII
Specific Values of a
a =
I(O, ;)
= I
I(O,n; ) =0
n= 1, 2, 3,
-..
n = 2, 3, 4,
-
a =
I(1,0;0) =
I(1, 1; ) =
)=
I(1, n;
I(2,
/2
0
;) = I + 2
= 0
I(2,;1,)
I(2, 2; 0) = ,@/4
I(2, n; ) =0
a
n = 3, 4, 5,
--
- I
+
I(-1, n; ) = (p)n
l-p
2
(- )n
¥ -1 [ + t - a =-2
i-,
I(-2, n; ) =
i
-
[ +
fli
1 -
+
p'
(1 - 2)[1 +
"
T--3"i
[
1
+ n]
1
1(-2, 0; ) = (1 I)
(1
1(-2, 1;P) =
-
2) 312
2p(l + p2)'
(1 - p2 )
0
(1 - 6)31,
I(,
0; ) =- 2
[2E(p) - (1 - p2)K(p)]
T VI + p2
2
-=
2
-
1
[E(p) + p2B(p)]
V1 + p(k)
1+E(k)
T
10
---
----- -----------
FORMULAS
Set VII (continued)
I(2, 1;i)
2
=
[(1 + p)E(p) - (1
1
2
- p2)K(p)]
p
[E(p) + B(p)]
3 1 + p2
_2
7r
+l
[E(k) - ( -
30
2=
~+33
r
)K(k)]
[2B(k) - E(k)]
2v'2
I(-3, ; 1) 2=--
I(-I,
+ p'K(p)
I
1
2
+_ K(k)
2 41 + p2 [K(p) - E(p)]
1;0) =
p
= - 2pV
+ pD(p)
2
1
- #V
2
=- -
- ( + P)E(k)]
+ 8
20
(1 + 0)3I2 C(k)
I(-2, n; 1) = (-1)"
ai =
3
I(,
0; )
2
1
3(1 + p) 312 [3E(p) + 3p4 B(p) + 5pt E(p) + Sp'B(p)]
:
2
22 (1
(I +
+ #)3/" [(3 - 2k )E(k) + kB(k)]
r 3
I2 5(1 p
r 5(1 + p2)3J2
[B(p) + 7E(p) +
2
2
E(p) + 7p B(p)]
2 20V1 + 0 [2E(k) - (1 - k2)C(k)]
5
2
12v2
2
I(2, n; 1) =(-1)"
2
7r (4n -
2
1)(4n - 9)
11
_
I
_ II__
I
_
FORMULAS
Set VII (continued)
r =-
4n-1 I(, n;
-- 1 --
I(--}n;)e
I(-
1, 0; 0)
=
I(- , 1; ) =
2
2 (I + p')31
2 [E(p) + p B(p)]
p(1-2p)
2
1
r (1 -
) i +
E(k)
2 p(I + p2)31 2 [E(p) + B(p)]
,
2
= _ 2
2 2
(1 - p )
2_
_
0
[CD(k) - C(k) ]
1
- -2
F (
-
) 1
=_[2B(k) - E(k)]
+
I(-, n; ) = (-1)
where the elliptic integrals B, C, D, E, and K are related as
D (p) =
KD(p)
(p) - E(p)
1
P2
B(p) = K(p) - D(p) =
E(p) - (1 - p)K(p)
p2
C(p) = D(p) - B(p) = (2 - p2 )K(p) - 2E(p)
p'
P
12
FORMULAS
Set VIII
Miscellaneous
I(2a, 0; 3) = [I(a, 0; )I
I(2a, m; j3) = I
+ 2 n=l
ZI Cl(a,
I(a, n{; #)I(a,,
m-
13
__T
n;.8)]2
n;I,)
Table I.
- .00
.00
.05
.10
Values of I(a, 0; ) for
.15
.20
.25
.30
between
.35
0 and
.40
1 in
.45
.02
.o04
.06
.08
1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000
1.00000 1.00000 .99999 .99999 .99998 .99998 .99998 .99998 .99998 .99997
1.00000 .99998 .99996 .99995 .99994 .99992 .99992 .99991 .99990 .99990
1.00000 .99996 .99992 .99989 .99986 .99983 .99981 .99979 .99978 .99978
1.00000 .99992 .99986 .99980 .99974 .99970 .99966 .99964 .99962 .99960
.10
.12
.14
. 16
.18
1.oooo00000
.99988
1.00000 .99983
1.00000 .99977
1.00000 .99969
.99977
.99967
.99956
.99942
.99968
.99954
.99937
.99918
.99960 .99953
.99942 .99932
.99921 .99908
.99897 .99879
.99947
.99924
.99896
.99865
.99943
.99918
.99888
.99853
.99940
.99913
.99882
.99845
.99938
.99911
.99878
.99841
1.00000
.99961
.99926
.99896
.99869
.99847
.99828
.99814
.99804
.99798
.20
.22
1.00000
1.00000
.99952
.99941
.99909
.99889
.99871
.99843
.99838
.99803
.99810
.99770
.99787
.99742
.99770
.99721
.99757
.99706
.99750
.99697
1.00000
.99930
.99868
.99813
.99765
.99725
.99692
.99667
.99649
.99638
1.00000
1.00000
.99918
.99904
.99844
.99819
.99780
.99743
.99724
.99678
.99676
.99623
.99638
.99579
.99608
.99544
.99587
.99520
.99575
.99505
.30
.32
.34
.36
.38
1.00000
1.00000
1.00000
1.00000
1.00000
.99889
.99874
.99857
.99838
.99819
.99791
.99761
.99729
.99694
.99658
.99704
.99662
.99617
.99568
.99516
.99629
.99576
.99520
.99459
.99394
.99566
.99504
.99438
.99367
.99291
.99515
.99446
.99372
.99293
.99208
.99475
.99401
.99321
.99235
.99144
.99447
.99368
.99284
.99195
.99099
.99430
.99350
.99263
.99171
.99072
.40
.42
1.00000
1.00000
1.00000
1.00000
1.00000
.99798
.99776
.99752
.99727
.99700
.99618
.99576
.99532
.99485
.99434
.99461
.99402
.99339
.99273
.99202
.99325
.99251
.99173
.99090
.99002
.99211
.99125
.99034
.98937
.98835
.99118
.99023
.98921
.98814
.98700
.99047
.98944
.98835
.98719
.98597
.98997
.98889
.98774
.98653
.98525
.98968
.98857
.98739
.98615
.98484
.54
.56
.58
1.00000
1.00000
1.00000
1.00000
1.00000
.99672
.99642
.99610
.99577
.99541
.99381
.99325
.99265
.99202
.99135
.99127
.99048
.98965
.98876
.98782
.98909
.98811
.98707
.98597
.98481
.98727
.98613
.98492
.98365
.98230
.98580
.98453
.98319
.98178
.98028
.98467
.98331
.98187
.98036
.97876
.98389
.98247
.98096
.97938
.97771
.98345
.98199
.98045
.97883
.97713
.60
.62
.64
.66
.68
1.00000
1.00000
1.00000
1.00000
1.00000
.99503
.99463
.99420
.99375
.99326
.99065
.98989
.98910
.98825
.98735
.98683
.98578
.98467
.98349
.98224
.98358
.98228
.98091
.97945
.97791
.98088
.97938
.97779
.97611
.97433
.97871
.97705
.97530
.97345
.97149
.97707
.97530
.97343
.97145
.96937
.97595
.97410
.97215
.97010
.96794
.97534
.97345
.97147
.96938
.96719
.70
.72
.74
.76
.78
1.00000
1.00000
1.00000
1.00000
1.00000
.99275
.99220
.99161
.99098
.99030
.98640
.98538
.98429
.98312
.98187
.98091
.97950
.97799
.97638
.97465
.97628
.97454
.97269
.97071
.96860
.97245
.97046
.96834
.96608
.96367
.96942
.96716
.96723
.96483
.96491
.96244
.95981
.96237
.95975
.95697
.96565
.96324
.96069
.95799
.95513
.96487
.96243
.95985
.95713
.95424
.80
.82
.84
.86
.88
1.00000
1.00000
1.00000
1.00000
1.00000
.98956
.98876
.98789
.98692
.98585
.98051
.97905
.97745
.97570
.97375
.97279
.97078
.96860
.96621
.96358
.96634
.96389
.96125
.95837
.95519
.96109
.95831
.95532
.95206
.94850
.95699
.95398
.95073
.94722
.94338
.95401 .95208
.95084 .94883
.94743
.94535
.94376 .94161
.93976 .93755
.95118
.94792
.94444
.94071
.93668
.90
.92
.94
.96
.98
1.00000
1.00000
1.00000
1.00000
1.00000
.98464
.98324
.98159
.97954
.97673
.97157
.96907
.96614
.96254
.95770
.96063
.95728
.95339
.94866
.94240
.95167
.94769
.94309
.93757
.93039
.94455
.94013
.93505
.92902
.92132
.93916
.93445
.92909
.92278
.91485
.93539 .93313
.93053 .92824
.92504 .92276
.91864 .91643
.91072 .90870
.93230
.92749
.92212
.91598
.90860
1.00
1.00000
.96965
.94654
.92910
.91622
.90708
.90106
.89769
.89757
.24
.26
.28
.44
.46
.48
.50
.52
14
.89662
steps of 0. 02, and a
.50
.55
.60
between 0 and 1 in steps of 0. 05.
.85
.80
.75
.70
.65
.90
1.00oo
.95
1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000 1.00000
.99997
.99997
.99998
.99998
.99998
.99998
.99998
.99999
.99999 1.00000 1.00000
.99990
.99990
.99978
.99960
.99990
.99978
.99962
.99991
.99980
.99964
.99992
.99981
.99966
.99992
.99983
.99970
.99994
.99986
.99974
.99995
.99989
.99980
.99996
.99992
.99986
.99998 1.00000ooooo
.99996 1.0ooooo00
.99992 1.00000
.99938
.99911
.99878
.99841
.99798
.99940
.99913
.99882
.99846
.99804
.99943
.99918
.99888
.99854
.99814
.99947
.99924
.99897
.99865
.99829
.99953
.99932
.99908
.99879
.99847
.99960
.99942
.99921
.99897
.99870
.99968
.99954
.99937
.99918
.99896
.99977
.99967
.99956
.99942
.99927
.99988
.99983
.99977
.99969
.99961
1.00000
1.00000
1.00000
1.00000ooooo
1.ooo00000oo
.99750
.99697
.99639
.99575
.99506
.99758
.99707
.99650
.99588
.99522
.99771
.99722
.99669
.99610
.99547
.99788
.99744
.99694
.99640
.99582
.99811
.99771
.99727
.99679
.99627
.99839
.99805
.99767
.99726
.99682
.99872
.99845
.99815
.99782
.99747
.99909
.99890
.99869
.99846
.99822
.99952
.99942
.99931
.99919
.99906
1.00000
1.00000
1.00000
1.00000
1.00000ooooo
.99164
.99065
.99432
.99351
.99265
.99174
.99076
.99449
.99372
.99289
.99200
.99106
.99479
.99405
.99327
.99243
.99154
.99519
.99452
.99379
.99302
.99220
.99571
.99511
.99447
.99378
.99305
.99634
.99583
.99528
.99470
.99408
.99709
.99668
.99625
.99578
.99529
.99795
.99766
.99735
.99703
.99668
.99892
.99877
.99861
.99843
.99825
1.00000
1.00000ooooo
1.ooo00000oo
1.00000ooooo
1.00000
.98960
.98848
.98730
.98605
.98473
.98972
.98862
.98746
.98623
.98493
.99005
.98899
.98787
.98668
.98543
.99059
.98959
.98853
.98741
.98623
.99133
.99041
.98944
.98841
.98733
.99227
.99146
.99059
.98968
.98872
.99342
.99272
.99199
.99121
.99040
.99477
.99421
.99363
.99302
.99237
.99631
.99592
.99551
.99508
.99463
.99806
.99785
.99764
.99741
.99717
1.ooo00000oo
1.00000
1.ooo00000oo
1.ooo00000o
1.00000
.98334
.98188
.98034
.97872
.97701
.98357
.98213
.98062
.97902
.97735
.98412
.98273
.98128
.97975
.97814
.98499
.98369
.98232
.98088
.97938
.98619
.98500
.98374
.98243
.98105
.98771
.98665
.98554
.98437
.98315
.98954
.98864
.98770
.98672
.98568
.99169
.99098
.99023
.98945
.98864
.99415
.99365
.99313
.99258
.99201
.99692
.99666
.99639
.99610
.99580
1.00000
1.ooo00000oo
1.00000
1.00000ooooo
1.00000
.97522
.97334
.97136
.96929
.96710
.97560
.97376
.97182
.96979
.96766
.97646
.97469
.97284
.97090
.96886
.97780
.97614
.97441
.97259
.97068
.97960
.97809
.97651
.97485
.97311
.98187
.98054
.97914
.97768
.97615
.98460
.98347
.98229
.98106
.97977
.98778
.98689
.98596
.98499
.98397
.99141
.99079
.99014
.98946
.98875
.99549
.99516
.99482
.99447
.99410
1.00000ooooo
1.ooo00000oo
1.00000
1.00000
1.00000
.96480
.96238
.95982
.95713
.95428
.96542
.96306
.96058
.95797
.95521
.96672
.96447
.96211
.95963
.95701
.96856
.96659
.96439
.96208
.95965
.97130
.96939
.96740
.96530
.96310
.97455
.97287
.97112
.96928
.96736
.97842
.97701
.97554
.97400
.97239
.98291
.98181
.98065
.97945
.97819
.98801
.98724
.98644
.98560
.98473
.99371
.99331
.99289
.99246
.99200
1.00000
1.00000
1.00000
1.ooo00000oo
1.00000
.95126
.94806
.94465
.94099
.93707
.95230
.94920
.94592
.94242
.93867
.95425
.95133
.94823
.94494
.94142
.95709
.95439
.95153
.94850
.94527
.96079
.95836
.95579
.95307
.95018
.96534
.96321
.96098
.95862
.95611
.97070
.96893
.96707
.97687
.97548
.97403
.97251
.97090
.98381
.98286
.98186
.98081
.97970
.99153
.99104
.99052
.98998
.98941
1.00000
1.ooo00000oo
1.00000
1.ooo00000oo
1.00000ooooo
.93282
.92817
.92302
.91718
.91025
.93462
.93021
.92536
.91990
.91350
.93764
.93354
.92905
.92404
.91824
.94181
.93808
.93402
.92952
.92436
.94710
.94379
.94020
.93625
.93177
.95345
.95060
.94753
.94417
.94040
.96083
.95849
.95597
.95323
.95019
.96921
.96740
.96109
.97854
.97731
.97600
.97459
.97304
.98881
.98818
.98751
.98680
.98602
1.ooo00000oo
1.ooo00000oo
1.00000
1.00000
1.00000
.90032
.90467
.91050
.91703
.92614
.93578
.94654
.95838
.97126
.98515 1.00000
.99977
.99960
.99937
.99910
.99877
.99839
.99796
.99748
.99694
.99635
.99571
.99501
.99425
.99344
.99257
.96510
.96303
.96547
.96339
15
-
-
-
-
-
Table II.
a
= w00
.00
.o5
.10
Values
of I(a, 1;3)
.20
.15
.25
.00150 .00200 .00250
.OO00100
.00500
.00400
.00200 .00300
.00600 .00750
.00300 .oo00450
.010
.00401 .00601 .008
.o6
.08
. 10
.12
.14
.16
.18
.00000 .00251 .00501 .00751 .01002 .01252
.00000 .00301 .00602 .00oo903 .01203 .01504
.01756
.o14o5
0 .00352 .00703 .01054
.0000oooo
.o0oo00 .oo4o2 .o008o4 .01206 .01607 .02008
.01359 .01811 .02262
.00ooo,oo00453 .0090oogo6
.20
.22
.24
.01512
.00000 .00505 .01009
.00000 .00oo556 .01112 .01666
.00000 .00o608 .01215 .01821
.ooooo .0oo661 .01319 .01977
.00000 ,00713 ,01424 .02134
.26
.28
.30
.02015
.02220
.02426
.02633
.02841
.02517
.02772
.03029
.03287
.03547
.01752 .02002
.02104 .0244
.02456 .02807
.02810 .03210
.03164 .03614
.03018 .03519
.03324 .03875
.03632 .04233
.03940 .04592
.04251 .04953
.03102 ,04124 .05141
.03269 .o04345 .05415
.05691
.03438 .04568
.04794 .05970
.03609
.03783 .05023 .06253
.06153
.06479
.06807
.07139
.07474
.54
.56
.58
.00000 .01333 .02652
.00000 .01394 .02773
.01457 .02897
. 00000
.00000 .01521 ,03022
.01586 .03150
.00000
,.03959 .05254 .06539
.04138 ,o05489 .o06828
.05728 .07122
.04320
.o4505 .05970 .07420
.04693 .06217 .07722
.60
.62
.64
.66
.68
.01653
.00000ooooo
.00000 .01722
.01793
.00000
.01865
.00000
.01940
.00000ooooo
.70
.72
.78
.00000
.00000
.00000
,.00000
.00000
,02018 .03989 .05919
.02098 .04144 .06143
06376
.02182 .04305
.02269 ,o04472 .06616
.02360 .04647 .o06867
.07809
.080g99
.08397
.08706
.09026
.80
.82
.84
.86
.88
.00000
.00000
.00000
.00000
.00000
.02456
.02558
.02667
.02783
.02910
.04830
.05023
.05228
.05447
.05683
.07129
.07404
.07694
,08003
.09359
.09703
.10074
.10462
.10875
.90
.92
.00000
.00000
.00000
.00000
.00000
.03049
.03206
.03387
.03606
.03900
.05942
.06229
.04617
.00000ooooo
.42
.44
.46
.48
.50
.52
.74
.76
.94
.96
.98
1. 00
.45
.01502
.01804
.0216
.02409
.02713
.02074
.01040
.00000ooooo
.02187
.00000 ,01097
.02300
.00000 ,01155
.00000 .01213 .02416
.00000 .01272 .02533
.40
.4
in
.00350 .oo4oo .oo45o
oogo.00900
.00700.00o800
.01051 .01201 .01351
.001.01601 .01801
.01401
.04563
.04876
.05192
.05510
.05830
.36
.38
.35
0 and 1
300
.00oo
.006oo
.00900
.01201
.03051 .03808
.03262 .o04070
.03475 .04335
.03689 .04601
,03906 .04870
.30
.32
.34
between
.0000ooooo.0000ooooo.0000ooooo.0000ooooo.0000ooooo.0000ooooo.0000ooooo.0000ooooo.0000ooooo
.0000ooooo
.00ooo050
.00000ooo
.00000 .00100
.00000 .00150
.00000 .00200
.02
.o04
for
.00000 .00766
.00000 .00820
.00000 .00874
.00000 .00929
.o00984
.00000ooooo
.01530 .02292
.01637 .02451
.01744 ,02611
.01853 .02773
.01963 .02937
.05316
.0568
.06047
.o6416
.06737
.02252
.02704
.03157
.0361
.04064
.04020 .04519
.04426 .04976
.04834 .05434
.05243 .05893
.05654 .06354
.o06067
.06482
.06899
.07318
.07740
.06817
.07282
.07749
.08218
.08689
.07161 08164 .09164
.07538 .08591 .09641
.07917 .09022 .10121
.09455 .10604
.08300
.11091
.08687 .09893
.07813 .09078
.08156 .09472
.08503 .09871
.08854 .10275
.09211 .10684
.10334
.10779
.11229
.11684
.12144
.11582
.12077
.12577
.13081
.13591
.08030 .09573
.08343 .09941
.08663 .10316
.o08989 .10698
.09322 .11087
.11099
.11520
.11948
.12382
.12825
.14105
.12609og
.13081 .14626
.13560 .15154
.14045 .15689
.14539 .16231
.o09663
.11485
.100O14 .11892
,10374 .12310
,10745 .12740
.13182
,11129
.13277
.13738
.14210
.14693
.15190
.15041
.15553
.16076
.16dlO
.17157
.16782
.17342
.17912
.18494
.190G8
.11528
.11943
.12377
.12834
.13319
.13640
.14115
.14610
.15128
.15675
.15702
.16232
.16781
.17354
.17955
.17719
.18290
.18897
.19516
.20167
.19696
.20321
.20954
.21629
.22319
.07465
.08694 .11321
.09091 .11809
.09540 .12355
.10068 .12990
.10745 .13784
.13838
.14402
.15026
.15742
.16619
.16257
.16884
.17571
.18349
.19282
.18590
.19269
.20008
.20831
.21801
.20848
.21571
.22350
.23208
.24199
.23041
.23801
.24612
.25495
.26496
.08605
.12119
.15270
.18142
.20794
.23274
.25618
.27856
.06468
.03281 .04886
.03415 .05082 .06724
.03553 .0523 .06986
.03694 .05489 .07253
.03840 .05701 .07528
.06558
.06951
.08334
16
steps of 0. 02, and a between 0 and 1 in steps of 0. 05.
.50
.55
.60
.65
.70
.75
.80
.85
.90
.95
1.o00
.00000
.00000
.00000
.00000
.00000
.0000ooo0
.00000oo
.00000
.00000
.00ooo000
.00500
.01000l
.01500
.02001
.00550
.01100
.01650
.02201
.oo6oo
.01200
.01800
.02401
.oo650
.01300
.01950
.02601
.00700
.01400
.02100
.02801
.00750
.01500
.02250
.03001
.oo8
50
.01700
.02550
.03400
.00900
.01800
.02700
.03600
.00oo950o
.01000
.01900 .02000
.02850 .03000
.03800 .04000
.02502
.03004
.03506
.04010
.04514
.02752
.03304
.03856
.04409
.03252
.03903
.04555
.05208
.05861
.03502
.04203
.04905
.05607
.06310
.03751
.04503
.05254
.o6006
.04963
.03002
.03604
.04206
.04809
.05412
.06759
.04001oo
.04251
.04802 .05102
.05603 .05953
.o6405 .o68o4
.07207 .07655
.04501
.05401
.06302
.07203
.08104
.04750
.05701
.06651
.07601
.08552
.05000
.o6000ooo
.07000
.08000
.09000
.05019
.05525
.05518
.06017
.06516
.07014
.07512
.08010
.09005
.06542
.07053
.07190
.07751
.10000
.11000
.12000
.13000
.14000
.01600
.02400
.03201
.00000
.07838 .o8485
.08447 .09143
.09131 .09776
.09838 .10533
.10422
.11227
.uo66
.11921
.11711
.12614
.09503
.10453
.11404
.12356
.13307
.07565 .08313
.o8080 .08876
.08596 .09442
.09115 .10010l
.09636 .10580
.09059
.09671
.09803
.10465
.10286
.11128
.10903
.11521
.11794
.12461
.10547
.11258
.11970
.12683
.13398
.11291
.12050
.12810
.13571
.14334
.12033
.12841
.13649
.14458
.15269
.12775
.13631
.14487
.15344
.16202
.13517
.14421
.15325
.16230
.17135
.14259
.15210
.16163
.17115
.18068
.15000
.16oo000
.17000
.18000
.1goo9000
.1o160
.10686
.11215
.11748
.12284
.11152
.11728
.12306
.12887
.13471
.12142
.12766
.13392
.14021
.14653
.13130
.13801
.14475
.15151
.15830
.14115
.14834
.15555
.16278
.17004
.15099
.15865
.16633
.17403
.18175
.16081
.16894
.17709
.18525
.19343
.17062
.17922
.18783
.19645
.20509
.18041
.18948
.19856
.20764
.21673
.19021
.19974
.20.928
.21882
.22837
.20000
.21000
.22000
.23000
.24000
.12824
.13367
.13915
.14468
.15026
.14059
.14650
.15246
.15846
.16451
.15288
.15927
.16569
.17216
.17866
.16512
.17198
.17886
.18578
.19274
.17732
.18464
.19198
.19935
.20675
.18949
.19725
.20504
.21286
.22071
.20162
.20984
.21808
.22633
.23462
.21374
.22240
.23108
.23978
.24849
.22583
.23494
.24406
.25319
.26234
.23792
.24747
.25703
.26660
.27617
.25000
.26000
.27000
.28000
.29000
.15589
.16157
.16732
.17314
.17903
.17060
.17675
.18296
.18924
.19558
.18522
.19182
.19848
.20519
.21197
.19974
.20679
.21389
.22103
.22824
.21420
.22168
.22920
.23677
.24439
.22858
.23649
.24444
.25242
.26045
.24292
.25125
.25962
.26801
.27643
.25722
.26597
.27475
.28354
.29236
.27149
.28066
.28984
.29904
.30826
.28575
.29533
.30492
.31452
.32413
.30000
.31000
.32000
.33000
.34000
.18500 .20200
.19106 .20849
.19722 .21508
.20348 .22177
.20986 .22857
.21882
.22574
.23275
.23984
.24703
.23550
.24283
.25023
.25772
.26529
.25206
.25979
.26758
.27544
.28337
.26852
.27663
.28480
.29303
.30132
.28489
.29339
.30193
.31052
.31915
.30121
.31009
.31899
.32793
.33691
.31749
.32674
.33601
.34530
.35462
.33374
.34337
.35300
.36265
.37231
.35000
.36000
.37000
.38000
.39000
.21638
.22305
.22989
.23693
.25434
.26176
.26933
.27707
.24421
.23549
.24255
.24977
.25717
.26478
.28499
.27296
.28073
.28863
.29668
.30488
.29139
.29950
.30772
.31605
.32453
.30967
.31811
.32663
.33525
.34398
.32784
.33659
.34541
.35430
.36328
.34593
.35499
.36410
.37327
.38250
.36396
.37333
.38274
.39218
.40166
.38198
.39166
.40136
.41108
.42081
.4oooo0000
.41000
.42000
.43000
.44000
.25177
.25968
.26805
.27707
.28712
.27265
.28083
.28942
.29857
.30861
.29313
.30155
.31032
.31957
.32957
.31328
.32191
.33084
.34017
.35011
.33316
.34199
.35107
.36047
.37036
.35284
.36186
.37107
.38054
.39039
.37237
.38157
.39092
.40047
.41029
.39180
.40119
.41068
.42031
.43014
.41119
.42077
.43042
.44015
.45000
.43057
.44036
.45017
.46003
.46993
.45000
.46000
.47000
.48000
.49000
.30011
.32101
.34144
.36125
.38135
.40105
.42069
.44034
.46007
.47995
.50000
.08507
.06074 .06623 .07171 .07718 .08266 .08813 .09360 .09907
.06033 .06632 .07230 .07827 .08424 .09021 .09617 .10213 .10809
17
-.- -------
I(a, 1; P)
Table III.
Values of
p
for
between
0 and
1 in
-~~~~~~~
a
= .00
=
.00
.05
.10
.15
.20
.25
.30
.35
.40
.45
.00000
.00000
.00000
.00000
.00000
.02500
.02500
.02501
.02502
.02504
.05000
.05000
.05002
.05o04
.05007
.07500
.07501
.07502
.07505
.07509
.10000
.10001
.10003
.1ooo6
.10012
.12500
.12501
.12503
.12507
.12513
.15000
.15001
.15004
.15008
.15014
.17500
.17501
.17504
.17508
.17515
.20000
.20001
.20004
.20009
.20015
.22500
.22501
.22504
.22509
.22515
.00000
.00000
.00000
.00000
.00000
.02506
.02508
.02511
.02515
.02519
.05o11
.05015
.05021
.05028
.07515
.07521
.07529
.07538
.07548
.10018
.10026
.10036
.10047
.10059
.12521
.12530
.12541
.12553
.12567
.15022
.15032
.15044
.15058
.15073
.17524
.17534
.17546
.17561
.17577
.20024
.20035
.20047
.20062
.20079
.22524
.22535
.22547
.22562
.22579
.00000
.00000
.00000
.00000
.00000
.02524
.02529
.07560
.02540
.02547
.05o44
.05053
.05063
.05075
.05087
.07573
.07587
.07603
.07620
.10073
.10089
.10106
.10125
.10146
.12583
.12601
.12621
.12643
.12666
.15091
.15110
.15132
.15155
.15181
.17595
.17616
.17638
.17663
.17690
.20098
.20118
.20141
.20167
.20194
.22597
.22618
.22641
.22666
.22694
.36
.38
.00000
.00000
.00000
.00000
.00000
.02555
.02562
.02571
.02580
.02590
.05100
.05115
.05131
.05148
.05166
.07638
.07658
.07680
.07703
.07728
.10169
.10193
.10220
.10248
.10278
.12692
.12720
.12750
.12782
.12816
.15209
.15239
.15271
.15306
.15343
.17719
.17751
.17785
.17821
.17860
.20224
.20256
.20291
.20328
.20368
.22723
.22756
.22790
.22827
.22867
.40
.42
.44
.46
.48
.00000
.00000
.00000
.00000
.00000
.02601
.02612
.02624
.07755
.07814
.07847
.07881
.10311
.10345
.10382
.10422
.1o464
.12853
.12892
.12934
.12979
.13027
.15383
.15426
.15471
.15519
.15571
.17902
.17946
.17994
.18044
.18098
.20410
.20456
.20504
.20555
.20610
.22909
.22954
.23002
.02651
.05185
.05206
.05228
.05252
.05277
.50
.52
.54
.56
.58
.00000
.00000
.00000
.00000
.00000
.02666
.02681
.02698
.02716
.02735
.05304
.05333
.05364
.05397
.05432
.07918
.07958
.08000
.08044
.08092
.10509
.10556
.10607
.10661
.10719
.13077
.13131
.13188
.13249
.13314
.15626
.15684
.15745
.15811
.15881
.18155
.18216
.18280
.18349
.18421
.20668
.20729
.20795
.20864
.20938
.23165
.23226
.23291
.23359
.23432
.60
.62
.64
.66
.68
.00000
.00000
.00000
.00000
.00000
.02756
.02778
.02801
.02826
.02853
.05469
.05509
.05551
.05597
.o5646
.08143
.08197
.08255
.08317
.08384
.10780
.10845
.10915
.10990
.11070
.13383
.13457
.13535
.13619
.13709
.15955
.16034
.16119
.16209
.16304
.18499
.18581
.18668
.18761
.18861
.21016
.21099
.21187
.21281
.21381
.23509
.23591
.23678
.23771
.23869
.70
.00000
.00000
.00000
.00000
.00000
.02882
.02914
.02948
.02985
.03026
.05699
.05756
.05818
.05885
.05958
.08455
.08532
.08706
.08804
.11156
.11248
.11347
.11455
.11572
.13805
.13908
.14019
.14139
.14269
.16407
.16517
.16635
.16763
.16901
.18967
.19080
.19202
.19333
.19475
.21488
.21602
.21724
.21855
.21996
.23974
.24086
.24205
.24334
.24472
.00000
.00000
.00000
.00000
.00000
.03070
.03120
.03175
.03236
.03307
.06038
.06126
06224
.06334
.06458
.08911
.09029
.09160
.09306
.09471
.11699
.11839
.11993
.12165
.12358
.1441o
.14564
.14734
.14923
.15135
.17050
.17214
.17393
.17591
.17813
.19628
.19795
.19977
.20179
.20403
.22149
.22315
.22496
.22696
.22917
.24620
.24782
.24957
.25150
.25363
.98
.00000
.00000
.00000
.00000
.00000
.03388
.03485
.03603
.03756
.03980
.06602
.06771
06976
.07240
.07617
.09660
.09881
.10148
.10488
.10964
.12579
.12836
.13144
.13531
.14065
.15375
.15654
.15985
.16398
.16958
.18063
.18352
.18693
.19113
.19676
.20656
.20945
.21285
.21699
.22246
.23164
.23447
.23776
.24175
.24693
.25601
.25870
.26183
.26557
.27037
1.00
.00000
.04617
.o8605
.12119
.15270
.18142
.20794
.23274
.25618
.27856
.02
.04
.06
.08
.10
.12
.14
.16
.18
.20
.22
.24
.26
.28
.30
.32
.34
.72
.74
.76
.78
.80
.82
.84
.86
.88
.90
.92
.94
.96
.02534
.02637
.05035
.07784
.o8616
18
.23053
.23107
steps of
0. 02,
and a
between
0 and
1 in steps of
0.05.
1.00
.50
.55
.60
.65
.70
.75
.80
.85
.90
.95
.25000
.25001
.25004
.27500
.27501
.27504
.27508
.27514
.30000
.30001
.30003
.30008
.30013
.32500
.32501
.32503
.32507
.32512
.35000
.35001
.35003
.35006
.35011
.37500
.37501
.37502
.37505
.37509
.40000
.40000
.40002
.4000oo4
.40008
.42500
.42500
.42501
.42503
.42506
.45000
.45000
.45001
.45002
.45oo004
.47500
.47500
.47500
.47501
.47502
.50000
.50000
.50000
.50000
.50000
.25024
.25034
.25046
.25061
.25077
.27522
.27532
.27544
.32519
.32528
.32538
.32550
.32563
.35017
.35025
.35034
.35044
.35056
.37515
.37521
.37529
.37538
.37548
.40012
.40017
.40024
.40031
.40039
.42509
.42513
.42518
.42524
.45006
.45009
.45012
.45016
.27574
.30021
.30030
.30041
.30054
.30069
.42530
.45020
.47503
.47504
.47506
.47508
.47510
.50000
.50000
.50000
.50000
.50000
.25095
.25115
.25138
.25162
.25189
.27591
.27610
.27632
.27655
.27681
.30085
.30103
.30123
.30145
.30169
.32578
.32594
.32613
.32633
.32655
.35069
.35084
.35100
.35118
.35137
.37559
.37572
.37586
.376l01
.37618
.40049
.40059
.40070
.40083
.40096
.42537
.42545
.45025
.45030
.45036
.45043
.45050
.47513
.47515
.47518
.47521
.47525
.50000
.50000
.50000
.50000
.50000
.25218
.25249
.27709
.30195
.32678
.32703
.25283
.25319
.25358
.27770
.27805
.30253
.30285
.30319
.32731
.32760
.32791
.40111
.40127
.40144
.40162
.40181
.42585
.42597
.42610
.42623
.42638
.45057
.30223
.35158
.35181
.35205
.35231
.35258
.37636
.27738
.47529
.47533
.47537
.47547
.50000
.50000
.50000
.50000
.50000
.25399
.25443
.27881
.32825
.25539
.25592
.27967
.28014
.28064
.32898
.32938
.32980
.35288
.35319
.35353
.35388
.35425
.37747
.37774
.37802
.37832
.37864
.40202
.25489
.30356
.30395
.30436
.30480
.30527
.40224
.40247
.40271
.40297
.42654
.42670
.42688
.42707
.42726
.45104
.45115
.45127
.45139
.45152
.47552
.47558
.47564
.47570
.47577
.50000
.50000
.50000
.50000
.50000
.25647
.25707
.25769
.25836
.25906
.28117
.28174
.28233
.28296
.28363
.30576
.30628
.30684
.30742
.30804
.33025
.33072
.33122
.33175
.33232
.35465
.35507
.35551
.35598
.35647
.37898
.40325
.42747
.50000
.40354
.42769
.47591
.50000
.37971
.38011
.38053
.40385
.40417
.40451
.42793
.42817
.42843
.45166
.45181
.45197
.45213
.45231
.47584
.37934
.47599
.47607
.47616
.50000
.50000
.50000
.25981
.26060
.26144
.26233
.26328
.28434
.28509
.28588
.28672
.28762
.30870
.30939
.31012
.31090
.31173
.33291
.33353
.33420
.33490
.33564
.35699
.35754
.35813
.35874
.35939
.38097
.38144
.38194
.38246
.38301
.40487
.40525
.40565
.40607
.40652
.42870
.42899
.42929
.42961
.42995
.45249
.45268
.45288
.45309
.45332
.47625
.47634
.47644
.47655
.47666
.50000
.50000
.50000
.50000
.50000
.26429
.26536
.26651
.26774
.26906
.28857
.28958
.29065
.29180
.29304
.31260
.31353
.31452
.31558
.31671
.33643
.33727
.33816
.33910
.34011
.36008
.36081
.36159
.36242
.36330
.38359
.38421
.38487
.38556
.38630
.40699
.40749
.40802
.40857
.40917
.43030
.43067
.43107
.43149
.43194
.45355
.45380
.45407
.45434
.45464
.47678
.47690
.47703
.47717
.47732
.50000
.50000
.50000
.50000
.50000
.27048
.27201
.27368
.27550
.27751
.29436
.29579
.29734
.29903
.30089
.31792
.31923
.32064
.32217
.32385
.34120
.34236
.34361
.34497
.34646
.36424
.36525
.36633
.36750
.36878
.38709
.38794
.38884
.38982
.39088
.40980
.41048
.41120
.41198
.41282
.43241
.43291
.43345
.43403
.43466
.45495
.45529
.45564
.45602
.45643
.47747
.47763
.47781
.47800
.47820
.50000
.50000
.50000
.50000
.50000
.27974
.28226
.28516
.28861
.29298
.30295
.30525
.30790
.31101
.31491
.32570
.32777
.33013
.33289
.33629
.34809
.34991
.35196
.35435
.35726
.37018
.37173
.37348
.37549
.37791
.39204
.39332
.39476
.39640
.39835
.41374
.41475
.41587
.41715
.41866
.43533
.43608
.43690
.43783
.43892
.45687
.45736
.45789
.45849
.45918
.47841
.47865
.47891
.47919
.47952
.50000
.50000
.50000
.50000
.50000
.30011
.32101
.34144
.36125
.38135
.40105
.42069
.44034
.46007
.47995
.50000
.25008
.25015
.27558
.27842
.27923
.32860
.37655
.37676
.37698
.37721
.42554
.42563
.42573
19
----------
.45065
.45074
.45083
.45093
.47542
Acknowledgment
The author
is
indebted to Miss
Elizabeth J.
Campbell and to Miss Mida N.
Karakashian of the Joint Computing Group, M. I. T., for computing the values given in
the tables.
This work was supported in part by a Ford Foundation Postdoctoral Fellowship.
20