SMITHSONIAN MISCELLANEOUS COLLECTIONS V26

A"

SMITHSONIAN MISCELLANEOUS COLLECTIONS
VOLUME

109

(WHOLE VOLUME)

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SMITHSONIAN
ELLIPTIC FUNCTIONS TABLES
Prepared by

G. W. AND R. M. SPENCELEY
Miami University, Oxford, Ohio

(Publication 3863)

CITY OF WASHINGTON
PUBLISHED BY THE SMITHSONIAN INSTITUTION

NOVEMBER

L

1947

PREFACE
These tables were inspired by the similar work of Sir George Greenhill
and Col. R. L. Hippisley as published in Smithsonian Mathematical Formulae
and Tables of Elliptic Functions. The pattern they designed has been followed
closely by us. The chief difference is our inclusion of the three elliptic functions,

sn(u,k), cn(u,k), dn(u,k).

Columns A and D were computed first. From them were computed columns
sn(u) and cn(u), checked by means of
Check I: sn^(u) +cn-(u) =1, with a maximum error of ±2 in the 15th
decimal. Practically all errors in these first four columns were eliminated at
this time, but some few managed to elude us.
Column dn(u) was then computed, and checked by means of
Check II: sn(K— u). dn(u) =:cn(u), with a maximum error of ±2 in
the 15th decimal. Those few errors that had eluded us in Check I were discovered at this time.

Column E(u,k)

followed,

computed by the formula

E(u + v)=E(u)+E(v)— k2sn(u+v)sn(u)sn(v),
where u was taken as r/90
i.e.

1°.

K

and tabulated r°, while v was taken as 1/90 K,
Independent computations for r° = 15°, 30°, 45°, 60°, 75°, 90° were

maximum

used as Check III, with a

divergence of

±3

in the 15th decimal.

In this method of computing the E(u,k) column the errors are additive, and
is more sensitive than Checks I and II.
The ^ column was computed last by means of sin ^ = sn(u), using Andoyer's

hence Check III

and Cosines. This was checked by computing

15-place Table of Natural Sines

cos <^=:cn(u).

we

Interpolation into Andoyer's Tables

developed a rapid method,

fitted to

was not used. Instead

machine computation, based on the

two formulas
sin(<^+c) =:sin ^4-«cos<^

sin

^

cos(^4-£) =cos

<^

— € sin

cos

(f>

<f>-\

Maximum

divergence here was kept to

are

all,

±4

<^

sin

<^

6

2

Our manuscript columns

cos

6

2

in the 15th decimal.

thus, correct to

±4

in the 15th decimal.

In printing, these have been cut to 12 decimal figures, assuring the accuracy
of the 12th decimal digit.

Galley proofs were checked by adding entries in each column of our
script in

in the galley proofs

To

manu-

groups of ten and subtracting the entries of the corresponding groups

and demanding that the remainder be equal

to zero.

teachers of mathematics and amateur computers, elliptic functions

pleasant field of endeavor.

Our

present tables are not exhaustive. Three

is

a

more


columns, snh(u,k), cnh(u,k), dnh(u,k), added to ours, would open up the
territory. In the popular field of undergraduate mathe-

complex argument

new elementary text on elliptic functions with a wide variety of
problems and applications, supplemented by an adequate set of 5-place tables,

matics, a

would be a boon

to teachers

and students.

a pleasant task to acknowledge our indebtedness to Messrs. John Hook,
Arthur Gels, Earl Boland, John Brinko, Edward Murphy, and Victor Sutcliffe,
It is

who, as N. Y. A. students, shared the work with us in the earlier years. These
young men became expert and dependable computers; our thanks to them
and to the Federal Government which paid them for their splendid and
efficient

The

work.

considerable cost of printing these tables has been defrayed mainly by a

grant from the Research Corporation of

ment

is

made

for that grant, without

New

York.

which

this

Grateful acknowledge-

volume could not have

appeared.

We wish to acknowledge here our deep indebtedness to Dr. C. G. Abbot,
former Secretary of the Smithsonian, for his invaluable advice and encouragement throughout. Dr. A. Wetmore, who has succeeded Dr. Abbot as Secretary, has assumed full responsibility for the publication of this volume, and
for the work of seeing it through the press. We can never repay these two
great men, and their corps of advisers, for their critical advice and practical
assistance. To have been associated with them, however humbly, has been for
us the experience of a lifetime.

Oxford, Ohio,
April 25, 1946.

G.

W.

S.

R.

M.

S.


SMITHSONIAN ELLIPTIC
FUNCTIONS TABLES


ELLIPTIC FUNCTIONS

K= 1.57091

K'= 5.43490
r

59581 27243
98296 25564

TABLE

=
E = 1.57067
^

E'=

1'

67091 27960
1.00075 15777 01834


ELLIPTIC FUNCTIONS
q

= 0.00001

90395 55386 95351

q'= 0.40330 93063 38377 8
r

= 1»
D(90) = 1.00007

TABLE

^

61611 21696
1/D (90) =0.99992 38446 78379


ELLIPTIC FUNCTIONS

K = 1.57091
K'= 5.43490
r

59581 27243
98296 25563 5

TABLE

=V
E = 1.57067
E'=

67091 27960
1.00075 15777 01834

ELLIPTIC FUNCTIONS
q

= 0.00001

90395 55386 95351

q'= 0.40330 93063 38377 8
r

=
D(90) = 1.00007

TABLE

^

1'

61611 21696
1/D(90) =0.99992 38446 78379


ELLIPTIC FUNCTIONS

K= 1.57127

49523 72225

K'=4.74271 72652 78886

r

TABLE

= 2"
E = 1.57031
6

E'=

79198 97448
1.00258 40855 27552


ELLIPTIC FUNCTIONS
q

= 0.00007

61698 24679 77407

q'= 0.35316 56482 96037
r

1

= 2°
D(90) = 1.00030

TABLE

^

47257 20529

1/D(90)= 0.99969 53671 08948


ELLIPTIC FUNCTIONS

K= 1.57127
K'= 4.74271

49523 72225
72652 78886

r

u=(r/90)K=F(<^,k)

45
46

0.78563 74761 86

TABLE

= 2"
E = 1.57031

79198 97448
E'=: 1.00258 40855 27552

u
0.70721 44933 79

0.70699 90525 84

dnu
0.99969 53671 09


ELLIPTIC FUNCTIONS
q

= 0.00007

61698 24679 77407

q'= 0.35316 56482 96037
r

= 2"
D(90) = 1.00030

TABLE

47257 20529
1/D (90) =0.99969 53671 08948

1

A(r)

E(<^,k)

4>

D(r)

0.78555
0.80300
0.82045
0.83790
0.85536
0.87281

05030
37023
67161
95448
21889
46492

46
60
86
67
67
79

0.78546
0.80291
0.82035
0.83780
0.85525
0.87269

35464 39
13361 63
87550 28
58037 22
24833 81
87955 90

0.70710
0.71933
0.73135
0.74314
0.75470
0.76604

67729
97922
36937
48177
95726
44357

82

83
25
56
84
74

1.00015
1.00015
1.00016
1.00016
1.00017
1.00017

23628
76802
29911
82891
35676
88203

60
47
56
16
72
93

52
53
54

55

0.89026
0.90771
0.92517
0.94262
0.96007

69268
90228
09387
26762
42373

19
23

47423
03262
55500
04172
49315

83
69
92

0.77714
0.78801
0.79863

0.80901
0.81915

59543
07466
55033
69878
20380

14
75
35

11

0.89014
0.90759
0.92503
0.94248
0.95992

88
35

1.00018
1.00018
1.00019
1.00019
1.00020

40408
92227
43597
94455
44740

80
72
56
7Z
28

56
57
58
59
60

0.97752
0.99497
1.01242
1.02987
1.04732

56239
68385
78835
87617
94761

49
19
54
93
78

0.97736
0.99481
1.01225
1.02969
1.04714

90972
29189
64017
95510
23728

75
66
33
53
13

0.82903
0.83867
0.84804
0.85716
0.86602

75665
05621
80906
72954
53987

39
72
30
25
60

1.00020
1.00021
1.00021
1.00022
1.00022

94389
43344
91543
38928
85442

92
19
42
90
90

61

1.06478
1.08223
1.09968
1.11713
1 13458

00298
04261
06686
07609
07070

51

62
6Z
64
65

1.06458
1.08202
1.09946
1.11691
1.13435

48732
70591
89375

05157
18016

95
76
12

0.87461
0.88294
0.89100
0.89879
0.90630

97023
75883
65199
40422
77832

69
43
26
91
80

1.00023
1.00023
1.00024
1.00024
1.00025

31028
75630
19195
61668
02998

90
03
04

66
67
68
69
70

1.15203
1.16948
1.18692
1.20437
1.22182

05109
01770
97095
91132
83927

68

10
70
28

28033
35294
39887
41903
41437

95

0.91354
0.92050
0.92718
0.93358
0.93969

54541
48501
38515
04237
26182

35

47

21

1.15179
1.16923
1.18667
1.20411
1.22155

35

1.00025
1.00025
1.00026
1.00026
1.00026

43135
82029
19635
55905
90795

71

72
73
74
75

1.23927
1.25672
1.27417

1.29162
1.30907

75529
65989
55358
43689
31037

39
17
27
76
96

1.23899
1.25643
1.27387
1.29131
1.30875

38586
33453
26139
16750
05397

90
03

2>7

0.94551
0.95105
0.95630
0.96126
0.96592

85732
65141
47540
16942
58247

73
88
66
43
87

1.00027
1.00027
1.00027
1.00028
1.00028

24264
56270
m772>
15738

43129

76
77
78
79
80

1.32652
1.34397
1.36141
1.37886
1.39631

17458
03007
87743
71724
55010

39
70

1.32618
1.34362
1.36106
1.37850
1.39594

92189

77240
60664
42581
23108

33
06
92
26
24

0.97029
0.97437
0.97814
0.98162
0.98480

57249
00636
75997
71826
77523

58

1.00028
1.00028
1.00029
1.00029
1.00029

68912
93056
15532
36312

81

82
83
84
85

1.41376
1.43121
1.44866
1.46610
1.48355

37662
19740
01307
82425
63157

46
59
28
09

02366
80479
57569
ZZ762
09184

75
18
34
28
14

0.98768
0.99026
0.99254
0.99452
0.99619

83400 34
80682 96
61512 99

13

1.41338
1.43081
1.44825
1.46569
1.48313

1.00029
1.00029
1.00030
1.00030
1.00030

72686
88234
01998
13962
24109

86
87
88
89
90

1.50100
1.51845
1.53590
1.55334
1.57079

43566
23718
03676
83504
63267

97
58
24
47
95

1.50056
1.51800
1.53544
1.55288
1.57031

83961 99
58223 69
32097 7Z
05713 07
79198 97

0.99756
0.99862
0.99939
0.99984

47

1.00030
1.00030
1.00030
1.00030
1.00030

32429
38910
43545
46329
47257

45

46
47
48
49
50
51

.

51

82

50
03
28

24

61

82
95

84
41

2>7

47

81

38
27
19

01

93

1

91

IS

41

53
18

23

18951 16

46979 16

40501
95346
08269
76951
.00000 00000

91
91

49
00

75

19

13

96

30
31

83

33

04

%

94
68
81

92
59
44

55371 16

51

42
92
26
86
Z6
61

72
OS
21

ELLIPTIC FUNCTIONS

K= 1.57187
K'= 4.33865
r

36105 14009
39759 99725

u=(r/90)K=F(
TABLE

= 3'
E = 1.56972
6

E'=
snu

u

01504 23979
1.00525 85872 09152

dnu


ELLIPTIC FUNCTIONS
q

= 0.00017

14256 42257 4727

q'=: 0.32040 03371 34866 4
r

= 3'
D(90) = 1.00068

TABLE

6

59377 43666
1/D(90) =0.99931 45324 44402


ELLIPTIC FUNCTIONS

K = 1.57187
K'= 4.33865
r

36105 14009
39759 99725

TABLE

= 3«
E = 1.56972
e

E'=

01504 23979
1.00525 85872 09152


ELLIPTIC FUNCTIONS
q

= 0.00017

14256 42257 4727

E(0,k)

4>

10146
40962
67604
90079
08400
22585

52
53

54
55

0.89045
0.90790
0.92535
0.94280
0.96025

32657
38643
40577
38497
32445

56
57
58
59
60

0.97770
0.99515
1.01259
1.03004
1.04749

22468
08620
90956

69540
44436

68
22
83

61

62
63
64
65

1.06494
1.08238
1.09983
1.11728
1.13472

15716
83455
47733
08632
66241

75
79
17
55

66
67
68
69
70

1.15217
1.16961
1.18706
1.20450
1.22195

71

6

59377 43666
1/D(90) =0.99931 45324 44402

q'= 0.32040 03371 34866 4

0.78574
0.80319
0.82064
0.83809
0.85555
0.87300

= 3'

D(90) = 1.00068

TABLE

69
83
50
44
43
37

0.78554
0.80298
0.82042
0.83786
0.85530
0.87274

53078
62172
62923
55346
39468
15324

13

0.89017
0.90761
0.92504

0.94248
0.95991

82961
42433
93808
37161
72578

0.97735
0.99478
1.01221
1.02964
1.04707

A(r)

D(r)

0.70710
0.71933
0.73135
0.74314
0.75470
0.76604

67396
97595
36616
47863

95420
44059

27
36
33
66
39

59252
07184
54760
69615
20126

78
97
47
20

01

0.77714
0.78801
0.79863
0.80901
0.81915

00153
19991

32207
36924
34274

00
36
24
04
30

0.82903
0.83867
0.84804
0.85716
0.86602

24399
07449
83584
52970
15782

50
90
39
24

71

1.06450

1.08193
1.09935
1.11678
1.13421

20652
71960
20264
65669
08280

43
37
97
34
10

1.15163
1 16906
1.18648
1.20391
1.22133

1.23939
1.25683
1.27428
1.29172
1.30916

48207

85564
20467
53035
83391

27

1.32661
1.34405
1.36149
1.37893
1.39638

11660
37969
62448
85230
06447

80
88
97

82
83
84
85

1.41382
1.43126

1.44870
1.46614
1.48358

26237 15
44736 84
62085 92

86
87
88
89
90

1.50103
1.51847
1.53591
1.55335
1.57079

51

72

73
74
75
76
77
78

79

80
81

1.00034
1.00035
1.00036
1.00037
1.00039
1.00040

29688
49383
68931
88188
07009
25247

11

1.00041
1.00042
1.00043
1.00044
1.00046

42761
59405
75039

89520
02711

05
82
35

75420
05387
80681
72739
53783

82
03
63
75
35

1.00047
1.00048
1.00049
1.00050
1.00051

14472
24668
33165
39830
44533

73
79
29
04
08

96829
75699
65026
40259
77680

75

84
02
96
09

1.00052
1.00053
1.00054
1.00055
1.00056

47146
47546
45609
41215

34250

84

93

0.87461
0.88294
0.89100
0.89879
0.90630

72205
22430
66655
05086
37937

93
47
31
52
18

0.91354
0.92050
0.92718
0.93358
0.93969

54398
48369
38392
04124
26078

77
33
73
04
65

1.00057
1.00058
1.00058
1.00059
1.00060

24598
12150
96800
78444
56982

41

1.23875
1.25617
1.27360
1.29102

1.30844

65427
87782
05236
18027
26399

16
83
82
68
66

0.94551
0.95105
0.95630
0.96126
0.96592

85638
65056
47463
16873
58186

19

1.00061
1.00062

1.00062
1.00063
1.00063

32320
04365
73029
38229
99886

31

20
54
54
83

1.32586
1.34328
1.36070
1.37812
1.39554

30602
30890
27523
20764
10882

34

37
17
58
55

0.97029
0.97437
0.97814
0.98162
0.98480

57196
00589
75957
71792
77495

89
64
47
22

1.00064
1.00065
1.00065
1.00066
1.00066

57924
12272

62865
09640
52541

12
52
27
72
90

98148
82838
65230
45605
24246

85
67
37
07
34

0.98768
0.99026
0.99254
0.99452
0.99619

83377
80664

61499
18940
46972

54
87
09

1.00066
1.00067
1.00067
1.00067
1.00068

91516
26517
57501
84430
07272

52

78425 13
93896 45

1.41295
1.43037
1.44779
1.46521
1.48263

08642
22808
36537
49975
63267

1.50005
1.51746
1.53488
1.55230
1.56972

01439
77473
52634
27215
01504

87
09
83
02
24

0.99756
0.99862
0.99939
0.99984
1.00000

40496
95344
08268
76951
00000

89
33
76

1.00068
1.00068
1.00068
1.00068
1.00068

25999
40589
51022
57288
59377

94
24
89

65
80
41

18

21

60

13
12

62
87

13
51

91

43
62
62
95

.

35
20
04
51

40
62
12

88
81

68

13

14

01

91

02

21

00

72
13
71

81
13
91

10
51

31

17
93
13

69
32
14
70

18

53
71

28

11

02
51

76

16

44


ELLIPTIC FUNCTIONS

K = 1.57271
K'= 4.05275
r

6

= A"

Ez=
E'=

24349 95227
81695 49437

u=(r/9O)K=F(0,k)

TABLE

snu

cnu

1.56888 37196 07763
1.00864 79569 07096

dnu


ELLIPTIC FUNCTIONS
q

= 0.00030

48651 48813 9401

q'= 0.29548 83855 58691 4
r

= 4°
D(90) = 1.00122
1/D(90) = 0.99878

TABLE

5

02045 90964
12824 93732


ELLIPTIC FUNCTIONS

K = 1.57271
K'= 4.05275

r

24349 95227
81695 49437

TABLE

= 4«
E = 1.56888

E'=

37196 07763
1.00864 79569 07096


ELLIPTIC FUNCTIONS
q

= 0.00030

48651 48813 9401

= 4«
D(90) = 1.00122

TABLE

5

02045 90964
1/D (90) =0.99878 12824 93732

q'= 0.29548 83855 58691 4
E(0,k)

A(r)

D(r)

0.78565
0.80309
0.82052
0.83794
0.85537
0.87280

98346 74
11081 44
08987 31
92092 71
60444 02
14105 59

0.70710
0.71933
0.73135
0.74314
0.75470

0.76604

66497
96712
35751
47017
94594
43254

46
90
54

0.89022
0.90764
0.92506
0.94248
0.95990

53159 65
nn(y 20
87862 89
83764 87
65564 62

0.77714
0.78801
0.79863
0.80901
0.81915

53
23
70
84

0.97732
0.99473
1.01215
1.02956
1.04697

33431
87552
28131
55386
69554

11
87
18

78407
95393
06224
11050
10027

52
44

95
89
38

1.15237
1.16980
1.18724
1.20468
1.22212

03317
91091
73527
50806
23119

72
1Z
74
75

1.23955
1.25699
1.27443
1.29186
1.30930

90661
53635
12246

66707
17236

76
11
78
79
80

1.32673
1.34417
1.36160
1.37903
1.39647

64054
07389
47471
84534
18818

91

81

82
83
84
85

1.41390
1.43133
1.44877
1.46620
1.48363

86
87
88
89
90

1.50106
1.51850
1.53593
1.55336
1.57079

0.78600
0.80346
0.82091
0.83836
0.85581
0.87326

78936
08082
29804
44116
51042

50612

20
25
52
90
30

0.89071
0.90816
0.92561
0.94305
0.96050

42868
27860
05646
76293
39878

71

56
57
58
59
60

0.97794
0.99539

1.01283
1.03028
1.04772

96485
46208
89148
25414
55125

81

61

62
63
64
65

1.06516
1.08260
1.10005
1.11749
1.13493

66
67
68
69
70

71

11
58
49

1.00061
1.00063
1.00065
1.00067
1.00069
1.00071

01022
13945
26608
38753
50121
60454

24
47

58470
06425
54025
68904
19440

32

64
12
62
99

1.00073
1.00075
1.00077
1.00079
1.00081

69496
76993
82692
86342
87695

95
99
79
1Z
70

74761
04754
80076
72161
53232

78

58

1.00083
1.00085
1.00087
1.00089
1.00091

86506
82532
75535
65279
51534

38

44

0.82903
0.83867
0.84804
0.85716
0.86602

1.06438
1.08179
1.09920
1.11661
1.13401

70887 07
59651 63
36130 76
00621 97
53437 31

0.87461
0.88294
0.89100
0.89879
0.90630

96307
75205
64559
39820
77268

87
57

1.00093
1.00095
1.00096
1.00098
1.00100

34072 55
12671 69
87114 27

57187 75
22684 92

67
92
00
24
26

1.15141
1.16882
1.18622
1.20362
1.22102

94903
25359
45158
54668
54266

0.91354
0.92050
0.92718
0.93358
0.93969

54014
48012
38061

03819
25799

56
06
95
23
20

1.00101
1.00103
1.00104
1.00106
1.00107

83404
39149
89731
34966
74677

14
62
59
59
69

71

1.23842

1.25582
1.27321
1.29061
1.30801

44344
25303
97558
61531
17657

47
82

0.94551
0.95105
0.95630
0.96126
0.96592

85383
64825
47255
16687
58022

40
32

1.00109

1.00110
1.00111
1.00112
1.00113

08694
36854
59000
74983
84663

65
21
22
86
82

59
35
98
94

1.32540
1.34280
1.36019
1.37758
1.39497

66378
08149

43428
72686
96397

97
29

57051 64
00464 55
75850 14
71701 61
77419 72

1.00114
1.00115
1.00116
1.00117
1.00118

87906
84586
74584
57792
34109

49
06

16

0.97029
0.97437
0.97814
0.98162
0.98480

93
20

50565
80018
07426
33039
57109

07
29
30
28
62

1.41237
1.42976
1.44715
1.46454
1.48193

15044
29116
39108

45519
48851

52
95
74
10
57

0.98768
0.99026
0.99254
0.99452
0.99619

83316
80616
61461
18913
46952

61

28
78

1.00119
1.00119
1.00120
1.00120

1.00121

03440
65702
20819
68723
09357

59
63
45
92
66

79891
01640
22614
43070
63267

53
82

1.49932
1.51671
1.53410
1.55149
1.56888

49613

48315
45469
41591
37196

44
08
36
03
08

0.99756
0.99862
0.99939
0.99984
1.00000

40484
95337
08265
76950
00000

55
38
67
43
00

1.00121

1.00121
1.00121
1.00121
1.00122

42671
68623
87184
98329
02045

17
87

61

36
18
54
53

04
23
59

44

55
72

95

41

01

06
80
46
78

13
25
07

81
13

17

21

70
94
14
11

15

72

87
34

09
11

95
58
80
51

55
38
74
43

1(y

12
33
91


ELLIPTIC FUNCTIONS TABLE

K= 1.57379
K'= 3.83174
r

21309 24768

19997 84146

u=(r/9O)E:=F(0,k)

= 5'
E = 1.56780
E'=

snu

u

90739 77622
1.01266 35062 34396

dnu


ELLIPTIC FUNCTIONS TABLE
q

= 0.00047

65699 16866 8574

q'= 0.27517 98048 73562 8
r

d

= S°

D(90) =1.00190 80983 52013
1/D (90) =0.99809 55355 53459


ELLIPTIC FUNCTIONS TABLE

K = 1.57379
K'= 3.83174
r

21309 24768
19997 84146

= 5'
E = 1.56780
6

E'=

90739 77622
1.01266 35062 34396

dnu

cnu

u=(r/90)K=F(

38
90
05
99

0.70643
0.69397
0.68130
0.66842
0.65534
0.64206

24886
28532
26909
59004
64411
83314

58
72
45
95
99
89

0.99809
0.99802
0.99796
0.99789

0.99783
0.99776

55355
91362
28222
66743
07730
51986

S3
62
77
66
82
57

22879
97314
64953
94047
53604

86
68
07
57
83

0.62859

0.61493
0.60108
0.58705
0.57284

56476
25224
31437
17534
26457

34

0.99770
0.99763
0.99757
0.99750
0.99744

00309
53491
12320
77577
50032

13
58
97
30
62

0.82953
0.83914
0.84850
0.85760
0.86643

13393
43951
16594
03420
77320

44
54
22

0.55846
0.54390
0.52919
0.51431
0.49928

01660
87093
27191
66858
51454

34

27
09

0.99738
0.99732
0.99726
0.99720
0.99714

30450
19582
18173
26953
46640

05
93
82
69
97

13331 82
15
47
80
12

0.87501
0.88331
0.89135

0.89912
0.90661

11981
81895
62362
29499
60246

67

26782
39072
34967
61512
66137

43
05
42

80
49

0.48410
0.46877
0.45330
0.43769
0.42195

77940
21543
78125
48347
32850

73
79
93
05

06

0.99708
0.99703
0.99697
0.99692
0.99687

1.15411
1.17160
1.18908
1.20657
1.22406

42293
08085
73878
39670
05462

45
77
10
42
75

0.91383
0.92077
0.92743
0.93380
0.93990

32367
24460
15958
87139
19122

53
22
76
00
90

0.40608
0.39010
0.37399
0.35777
0.34144

96642
01187
28274
26734
45713

43
48
33
17
04

0.99682
0.99677
0.99672
0.99668
0.99663

32261
47188
78219
25924
90851

64
87
44

1.24154

1.25903
1.27652
1.29400
1.31149

71255
37047
02839
68632
34424

07

72
73
74
75

40
72
05
37

0.94570
0.95122
0.95646
0.96140
0.96604

93882

94245
03895
07380
90111

78
36
60
27
39

0.32501
0.30848
0.29186
0.27515
0.25835

34657
43300
21647
19960
88745

56
66
34
34
86

0.99659

0.99655
0.99651
0.99648
0.99644

73527
74460
94132
33006
91519

65
26
92
83
86

76
77
78
79
80

1.32898
1.34646
1.36395
1.38143
1.39892

00216

66009
31801
97593
63386

70
02
35
67
00

0.97040
0.97446
0.97822
0.98169
0.98486

38369
39306
80947
52196
42834

40

78739
40890
26351
86457
72717

25
74
04

71

0.24148
0.22454
0.20753
0.19045
0.17332

81

0.99641
0.99638
0.99635
0.99633
0.99630

70086
69094
88911
29875
92300

02
97
56

37
36

81
82
83

29178
94970
60762
26555
92347

32
65
97
30
62

0.98773
0.99030
0.99257
0.99454
0.99620

43525
45815
42135
25804
91029

52
35
75
94
23

0.15614
0.13891
0.12164
0.10433
0.08699

36799
30511
05793
14697
09377

52
92
59
68
65

0.99628
0.99626
0.99625
0.99623
0.99622

76474
82659
11089
61972
35490

41
07

84
85

1.41641
1.43389
1.45138
1.46887
1.48635

86
87
88
89
90

1.50384
1.52133
1.53881
1.55630
1.57379

58139
23932
89724
55516
21309

95
27
60
92
25

0.99757
0.99863
0.99939
0.99984
1.00000

32904
47415
31440
82748
00000

20
76
95
66
00

0.06962
0.05223
0.03483
0.01741
0.00000

42072
65094
30810
91632
00000

88
38
44
34
00

0.99621
0.99620
0.99619
0.99619
0.99619

31794
51012
93240
58549
46980

67
17
38
29
92

04313
11659
18055
87409
84332
74141

26

95
27
60
92
25

0.70778
0.72000
0.73200
0.74377
0.75532
0.76664

55408

21200
86993
52785
18577

57
90
22
55
87

0.77773
0.78857
0.79918
0.80954
0.81966

0.97924
0.99673
1.01422
1.03170
1.04919

84370
50162
15954
81747
47539

20

52
85
17
50

65

1.06668
1.08416
1.10165
1.11914
1.13662

79124
44916
10708
76501

66
67
68
69
70
71

45
46
47
48
49

50

0.78689
0.80438
0.82186
0.83935
0.85684
0.87432

60654
26446
92239
58031
23823
89616

62

51

52
53
54
55

0.89181
0.90930
0.92678
0.94427
0.96176

56
57
58
59
60
61

62
63
64

17

58
58

13
16

11

54
45

20

10

66

74

80

10

49

51

12

38
71

11

18
61

03


= 5°
D(90) = 1.00190

ELLIPTIC FUNCTIONS TABLE
q

= 0.00047

65699 16866 8574

D(r)

A(r)

E(<^,k)

r

80983 52013

1/D (90) =0.99809 55355 53459

q'= 0.27517 98048 73562 8

43
93
75
87
37
40

0.78580
0.80322
0.82064
0.83805
0.85546
0.87287

71879
60664
26286
68789
88248
84764

43
72
39
84
63
36

0.70710
0.71933
0.73135
0.74314
0.75470
0.76604

64599
94849
33925
45231
92851
41555

92

91
84
98
16
76

1.00095
1.00098
1.00102
1.00105
1.00108
1.00111

40491
73450
06002
37744
68271
97180

11

0.89028
0.90769
0.92509
0.94249
0.95989

58466
09512

38086
44401
28696

53
39
72
60
15

0.77714
0.78801
0.79863
0.80901
0.81915

56818
04822
52472
67404
17994

43
57
70
48
61

1.00115
1.00118

1.00121
1.00124
1.00128

24071
48545
70207
88665
03532

0.97728
0.99468
1.01207
1.02946
1.04685

91236
32314
52247
51381
30084

22
07
97
86
88

0.82903
0.83867

0.84804
0.85716
0.86602

73370
03419
78798
70941
52070

44
38
08
37
94

1.00131
1.00134
1.00137
1.00140
1.00143

14422
20959
22768
19481
10737

88

88750
27798
47668
48827
31761

93
19
57
23
95

0.87461
0.88294
0.89100
0.89879
0.90630

95203
74160
63573
38893
76399

83
65
53
87
80

1.00145
1.00148
1.00151
1.00154
1.00156

96182
75467
48252
14204
73001

13
05

15852 01
39374 71

1.06423
1.08162
1.09900
1.11638
1.13376

99
65

1.15262
1.17005
1.18748

1.20491
1.22234

54012
60031
57708
47329
29191

58
66
43
50
26

1.15113
1.16851
1.18588
1.20325
1.22062

96982
45020
76427
91775
91656

58
36
33

63
78

0.91354
0.92050
0.92718
0.93358
0.93969

53203
47257
37363
03175
25209

43
63
73
23

1.00159
1.00161
1.00164
1.00166
1.00168

24326
67874
03347
30458

48932

80
22
20
84
46

1.23977
1.25719
1.27462
1.29204
1.30947

03599
70869
31323
85294
33121

52
12
55
59
86

1.23799
1.25536
1.27273
1.29009

1.30745

76681
47476
04688
48979
81025

02

72
73
74
75

59
00
08

0.94551
0.95105
0.95630
0.96126
0.96592

84845
64337
46816
16295
57675

52
89
99
90
06

1.00170
1.00172
1.00174
1.00176
1.00178

58501 86
58911 73
49917 89
31287 63
02799 99

76
77
78
79
80

1.32689
1.34432
1.36174
1.37916
1.39658

75152
11740
43246
70037
92486

43
50
56
17

1.32482
1.34218
1.35954
1.37690
1.39425

01518
11166
10687
00812
82283

93
59
19
06
87

0.97029
0.97437
0.97814
0.98162
0.98480

56746
C0199
75623
71509
77260

86
92
21
80
34

1.00179
1.00181
1.00182
1.00183
1.00185

64245
15428
56164
86281
05621

99
95
67
68
47

81

1.41401
1.43143
1.44885
1.46627
1.48369

10970
25872
37579
46482
52975

16
18
21
09
10

1.41161
1.42897
1.44632
1.46368

1.48103

55855
22290
82358
36839
86519

70
18

83186
80513
61382
18854
46912

1.00186
1.00187
1.00187
1.00188
1.00189

14038
11401
97590
72500
36042

62

05

04

0.98768
0.99026
0.99254
0.99452
0.99619

37

82
83
84
85

86
87
88
89
90

1.50111
1.51853
1.53595
1.55337
1.57079

57455

60322
61978
62825
63267

38
56
20
32
95

1.49839
1.51574
1.53310
1.55045
1.56780

32188
74643
14684
53115
90739

26
48
80
25
78

0.99756

0.99862
0.99939
0.99984
1.00000

40458
95322
08259
76948
00000

50
70

1.00189
1.00190
1.00190
1.00190
1.00190

88136 11
28719 71
57743 36

0.78635
0.80380
0.82125
0.83870
0.85615
0.87360

13029
39989
55345
59118
51345
32075

0.89105
0.90849
0.92594
0.94338
0.96082

01373
59316
05997
41521
66008

56
57
58
59
60

0.97826
0.99570
1.01314
1.03058

1.04797

79589
82412
74633
56427
43162

61

62
63
64
65

1.06545
1.08289
1.10032
1.11776
1.13519

89478 29
41142 25
83189 24

66
67
68
69
70

71

45
46
47
48
49
50

55

59
81
37

82
00
83
10
60

41

51

51

59

21

81

16

49
96
17

13

80
00

76
12
82
70
58
77
53
61

68
85
12

35

07

33
64

11

14

88
01

75171 71
80983 52