Smithsonian miscellaneous collections

A". SMITHSONIAN MISCELLANEOUS COLLECTIONS VOLUME. 109. (WHOLE VOLUME). Eesearcf) Corporation Jfunb. 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(</.,k). 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(</.,k). 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.

ELLIPTIC FUNCTIONS. =. K 1.57511 36077 77251 K'=: 3.65185 59694 78752 r. u=(r/90)K=F(</>,k). TABLE. = 6° E = 1.56649 5. E'=. 67877 60132 1.01723 69183 41019. dnu.

ELLIPTIC FUNCTIONS q. = 0.00068. 66451 27304 8341. q'= 0.25794 01957 66336 r. 7. E(^,k). <p. = 6' D(90) = 1.00275 1/D(90) = 0.99725. TABLE. 6. A(r). 03575 48436 71861 70284 D(r). 0.00000 0.01750 0.03500 0.05250 0.07000 0.08750. 00000 1252S 24465 35235 44250 50932. 00 48 43 05 99 02. 0.00000 0.01750 0.03500 0.05250 0.07000 0.08750. 00000 12427 23684 32600 38009 38748. 00 86 69 87 73 90. 0.00000 0.01745 0.03489 0.05233 0.06975 0.08715. 00000 23735 94309 58578 63428 55796. 00 33 65 12 29 27. 1.00000 1.00000 1.00000 1.00000 1.00001 1.00002. 00000 08377 33498 7S233 33831 08920. 00 22 66 73 45 56. 6 7 8 9 10. 0.10500 0.12250 0.14000 0.15750 0.17500. 54699 54979 51200 42798 29211. 80 56 83. 0.10500 0.12250 0.14000 0.15749 0.17499. 33661 21599 01420 71994 32201. 83 16 16 16 94. 0.10452 0.12186 0.13917 0.15643 0.17364. 82682 91169 28435 41772 78600. 89 82 74 36 54. 1.00003 1.00004 1.00005 1.00006 1.00008. 00509 08486 32720 73060 29334. 56 87 93 40 28. 11. 0.19250 0.20999 0.22749 0.24499 0.26248. 09887 77 84280 29 51850 43. 80937 17107 39636 47463 39548. 09. 0.19080 0.20791 0.22495 0.24192 0.25881. 86486 13156 06515 14660 85896. 27 62. 12067 88 64411 34. 0.19248 0.20998 0.22747 0.24496 0.26245. 71 55 87. 1.00010 1.00011 1.00013 1.00016 1.00018. 01352 88904 91762 09679 42390. 0.27998 0.29747 0.31496 0.33245 0.34994. 08369 43440 69133 84970 90484. 17 03 48 58 48. 0.27994 0.29742 0.31491 0.33239 0.34987. 14867 72420 11228 30332 28803. 94 99 02. 68754 12004 64672 76055 95737. 80 65 43 40 55. 1.00020 1.00023 1.00026 1.00029 1.00032. 89610 33 51039 01 26357 75. 04. 0.27563 0.29237 0.30901 0.32556 0.34201. 0.36743 0.38492 0.40241 0.41990 0.43738. 85321 68739 40611 00425 47782. 00. 0.36735 0.38482 0.40229 0.41976 0.43723. 05732 60240 91476 98614 80862. 44 94 22 64 25. 0.35836 0.37460 0.39073 0.40673 0.42261. 73604 59860 05041 60029 76070. 89 76. 1.00035 1.00038 1.00041 1.00045 1.00049. 32217 59579 98993 50045 12308. 84 49 27 67 97. 0.45486 0.47235 0.48983 0.50731 0.52478. 82300 03611 11364 05225 84876. 37455 67663 70786 46158 93147. 77 50. 04789 98200 08725 89208 92927. 1.00052 1.00056 1.00060 1.00064 1.00068. 85341 68689 61885 64450 75893. 81. 00 09. 0.43837 0.45398 0.46947 0.48480 0.49999. 27. 36 02 00. 0.45470 0.47216 0.48962 0.50708 0.52453. 50017 00366 35659 55650 60112. 24 48 53 60 63. 0.54199 0.55943 0.57688 0.59432 0.61176. 11156 99625 58027 85876 82719. 56 10 68 15 83. 0.51503 0.52991 0.54463 0.55919 0.57357. 73612 85454 83125 21786 57105. 44 88 43 05. 1.00072 1.00077 1.00081 1.00086 1.00090. 95714 23400 58431 00277 48399. 45 77. 35. 0.54226 0.55974 0.57721 0.59468 0.61215. 36 37 38 39 40. 0.62962 0.64709 0.66455 0.68202 0.69948. 48837 21636 78339 18798 42881. 51. 19. 0.62920 0.64663 0.66406 0.68149 0.69891. 48146 81780 83288 52373 88779. 07 68 43 38 26. 0.58778 0.60181 0.61566 0.62931 0.64278. 45267 42992 07543 96742 68983. 57 42 33 44 09. 1.00095 1.00099 1.00104 1.00108 1.00113. 02251 61281 24929 92630 63814. 62 32 18 29 85. 41. 0.71694 0.73440 0.75186 0.76931 0.78677. 50479 41502 15881 73565 14527. 25 42 07 88 89. 0.71633 0.73375 0.75116 0.76858 0.78598. 92289 62728 99960 03889 74461. 77 76 50 76 96. 0.65605 0.66912 0.68199 0.69465 0.70710. 83242 99094 76721 76925 61144. 53 40 03 62. 1.00118 1.00123 1.00127 1.00132 1.00137. 37908 14334 92511 71857 51787. 79 49. 1. 2 3 4. 12 13. 14 15. 16. 17 18 19. 20 21. 22 23 24 25 26 27 28 29 30. 31. 32 33 34. 42 43. 44 45. 11. 57. 14. 69 70 97 61 41. 37 81. 06. 41. 24 77 39. 71. 21. 23. 01. 02 68. 15. 40 33 78. 51. 11. 17 51. 79 86 21. 15231 12 17307 16. 71. 62 49 86. 75. 38 32. 51. 26 73.

ELLIPTIC FUNCTIONS. K = 1.57511 K'= 3.65185 r. 36077 77251 59694 78752. TABLE. = 6' E = 1.56649 E'=. 67877 60132 1.01723 69183 41019.

ELLIPTIC FUNCTIONS q. = 0.00068. 66451 27304 8341. q'= 0.25794 01957 66336 r. 45 46 47 48 49 50. 0.78677 0.80422 0.82167 0.83912 0.85657 0.87401. 14527 38758 46269 37093 11281 68908. 89 50. 51. 52 53 54 55. 0.89146 0.90890 0.92634 0.94378 0.96122. 56 57 58 59 60 61. 6. 03575 48436. 1/D(90)= 0.99725 71861 70284. 7. E(0,k). <P. = 6" D(90) = 1.00275. TABLE. 09 69 06. 0.78598 0.80339 0.82079 0.83818 0.85558 0.87297. 74461 11663 15520 86099 23510 27900. 10065 34865 43442 35948 12555. 08 66 62 50 37. 0.97865 0.99609 1.01352 1.03095 1.04838. 73454 18856 48991 64105 64465. 62 63 64 65. 1.06581 1.08324 1.10066 1.11809 1.13551. 50356 22078 79950 24308 55504. 66 67 68 69 70. 1.15293 1.17035 1.18777 1.20519 1.22261. 71. 72 73 74 75. A(r). 05 98 76 54. 0.70710 0.71933 0.73135 0.74314 0.75470 0.76604. 61144 91457 30600 41979 89676 38462. 0.89035 0.90774 0.92512 0.94250 0.95987. 99457 61 38410 16 45025 95 19611 98 62514 11. 0.77714 0.78801 0.79863 0.80901 0.81915. 62 74 02 31 68. 0.97724 0.99461 1.01198 1.02934 1.04670. 74116 54841 05148 25534 16530. 58 56 62. 11. 48 66 58. 1.06405 1.08141 1.09876 1.11610 1.13345. 78706 12663 19040 98506 51764. 24 83 42 29. 73906 79896 73874 56252 27458. 10 58 43 59 08. 1.15079 1.16813 1.18547 1.20281 1.22014. 79548 82624 61786 17858. 44. 1.24002 1.25744 1.27485 1.29227 1.30968. 87931 38126 78508 09555 31757. 42 14 20 44 00. 1.23747 1.25480 1.27213 1.28945 1.30678. 64164 56180 28668 82580. 18 45 67 79. 76 77 78 79 80. 1.32709 1.34450 1.36191 1.37932 1.39673. 45612 51632 50335 42250 27914. 72 54 86 95 28. 81. 82 83 84 85. 1.41414 1.43154 1.44895 1.46636 1.48376. 07869 82668 52867 19029 81722. 86 87 88 89 90. 1.50117 1.51857 1.53598 1.55339 1.57079. 41518 98991 54720 09285 63267. 27 23 25 95. D(r). 87 69 04 03. 1.00137 1.00142 1.00147 1.00151 1.00156 1.00161. 51787 31718 11063 89240 65666 39760. 53809 01903 49645 64672 15360. 99 06 42 43 46. 1.00166 1.00170 1.00175 1.00180 1.00184. 10945 78646 42294 01323 55176. 0.82903 0.83867 0.84804 0.85716 0.86602. 70836 00987 76470 68718 49954. 53 72 34 88 70. 1.00189 1.00193 1.00197 1.00202 1.00206. 03298 45143 80174 07861 27681. 09 72 70 02 61. 0.87461 0.88294 0.89100 0.89879 0.90630. 93194 72258 61778 37205 74817. 48 49. 1.00210 1.00214 1.00218 1.00222 1.00225. 39124 41689 34885 18233 91266. 98 85 76 67. 0.91354 0.92050 0.92718 0.93358 0.93969. 51726 45884 36091 02003 24134. 1.00229 1.00233 1.00236 1.00239 1.00242. 53529 04582 43995 71357 86268. 81. 83865 63450 46017 15582 57042. 92 20 96 04 61. 1.00245 1.00248 1.00251 1.00254 1.00256. 88344 77217 52536 13965 61185. 36 73 47. 18891 11. 0.94551 0.95105 0.95630 0.96126 0.96592. 1.32410 1.34142 1.35874 1.37606 1.39337. 38595 42708 32264 08314 71928. 07 05 14 92. 0.97029 0.97436 0.97814 0.98162 0.98480. 56191 99717 75209 71160 76970. 79 98 92 46 09. 1.00258 1.00261 1.00263 1.00265 1.00266. 93895 11812 14670 02223 74241. 62 69 97 31 21. 87 65 77 97 85. 1.41069 1.42800 1.44531 1.46263 1.47994. 24186 66186 99036 23856 41774. 60. 0.98768 0.99026 0.99254 0.99452 0.99619. 82950 80325 61238 18748 46838. 11. 75 21. 1.00268 1.00269 1.00270 1.00272 1.00272. 30515 70854 95088 03065 94654. 08 55 62 93 93. 18. 1.49725 1.51456 1.53187 1.54918 1.56649. 53930 61466 65532 67284 67S77. 06 00. 0.99756 0.99862 0.99939 0.99984 1.00000. 40411 95295 08247 76945 00000. 05 96 23 82 00. 1.00273 1.00274 1.00274 1.00274 1.00275. 69744 03. 51. 14. 96 15. 13. 64. 17. 16 16. 08 51691 35. 18. 51. 54 29 96. 71. 28 60. 25. 11. 01. 51. 61. 58 19. 15. 86 80 79. 68 40. 28241 70076 95198 03575. 73 20 95 97 68 61. 17. 29 14 85 15. 51. 20 99 64 32. 15. 27. 75. 82 27 48.

ELLIPTIC FUNCTIONS. K= 1.57667 K'= 3.50042 r. 79815 92838 24991 71838. u=(r/9O)K=:F(0,k). TABLE. = 1" E = 1.56494 6. E'= snu. 75629 69419. 1.02231 25881 67584. dnu.

ELLIPTIC FUNCTIONS q. = 0.00093. 52197 97815 9662. q'= 0,24291 29743 06665 r. 4. = 7° D(90) = 1.00374. TABLE. <?. 78893 91977. 1/D(90)= 0.99626 61048 34106.

ELLIPTIC FUNCTIONS. K = 1.57667 K'= 3.50042 r. 79815 92838 24991 71838. TABLE. = 1" E = 1.56494 E'= 1.02231. 75629 69419 25881 67584.

ELLIPTIC FUNCTIONS q. = 0.00093. E(^,k). 55442 85859 25115 36613 84437 33357. 0.77714 0.78800 0.79863 0.80901 0.81915. 48846 97086 44980 60165 11014. 62 38. 66656 96975 72629 65052 46463. 03. 26 30. 0.82903 0.83866 0.84804 0.85716 0.86602. 90 87 22 02 34. 0.87461 0.88294 0.89100 0.89879 0.90630. 89879 69120 58817 34420 72207. 41. 03. 0.91354 0.92050 0.92718 0.93358 0.93969. 49289 43617 33993 00070 22362 82249 61985 44699 14404 55999. 76 65. 92. 0.94551 0.95105 0.95630 0.96126 0.96592. 80754 06156 11787 99075 69470. 80 86 33 02 62. 0.97029 0.97436 0.97814 0.98162 0.98480. 55276 98922 74528 70584 76491. 02 86 06. 1.40960 1.42686 1.44412 1.46139 1.47865. 24445 65487 94103 11814 20153. 01. 0.98768 0.99026 0.99254 0.99452 0.99619. 82560 80016 61000 18573 46716. 34 37 68. 1.49591 1.51317 1.53043 1.54768 1.56494. 20665 14905 04433 90818 75629. 89 33 65. 0.99756 0.99862 0.99939 0.99984 1.00000. 40332 95251 08227 76940 00000. 78 85 61. 86008 06929 05078 80500 33268 63481. 12 03 45 97 83 86. 0.78620 0.80358 0.82096 0.83834 0.85571 0.87308. 07056 64984 77545 44834 67000 44248. 0.89194 0.90938 0.92682 0.94425 0.96168. 71267 56779 20200 61738 81628. 32 74 68 68. 0.89044 0.90780 0.92516 0.94251 0.95985. 76838 65082 09349 10057 67681. 0.97911 0.99654 1.01397 1.03139 1.04881. 80131 57535 14153 50322 66405. 99 89 23 08 29. 0.97719 0.99453 1.01186 1.02919 1.04652. 82746 55829 87557 78607 29706. 91. 1.06623 1.08365 1.10106 1.11848 1.13589. 62790 39887 98131 37980 59913. 07. 62 63 64 65. 93 69 26. 1.06384 1.08116 1.09847 1.11578 1.13309. 41628 15196 51278 50786 14677. 66 67 68 69 70. 1.15330 1.17071 1.18812 1.20552 1.22293. 64430 52055 23330 78817 19098. 83 66 43 59 65. 1.15039 1.16769 1.18499 1.20228 1.21957. 43952 39651 02857 34690 36309. 71. 72 73 74 75. 1.24033 1.25773 1.27513 1.29253 1.30993. 44773 56459 54791 40420 14010. 52 74 74 08 64. 1.23686 1.25414 1.27142 1.28870 1.30598. 08907 53715 71995 65040 34175. 83 58 00. 76 77 78 79 80. 1.32732 1.34472 1.36211 1.37951 1.39690. 76243 27813 69427 01803 25673. 84 79 49 92 21. 1.32325 1.34053 1.35780 1.37506 1.39233. 81. 82 83 84 85. 1.41429 1.43168 1.44907 1.46646 1.48385. 41775 50861 53688 51021 43634. 74 27 00 68 68. 86 87 88 89 90. 1.50124 1.51863 1.53592 1.55340 1.57079. 32305 17815 00952 82506 63267. 05 62 99. 56 57 58 59. 60 61. 71. 47. 64 95. D(r). A(r) 0.70710 0.71933 0.73135 0.74314 0.75470 0.76604. 0.78726 0.80472 0.82217 0.83961 0.85706 0.87450. 52 53 54 55. ». 78893 91977 1/D(90) =0.99626 61048 34106. 52197 97815 9662. q'= 0.24291 29743 06665 4. 51. = 7" D(90) = 1.00374. TABLE. 64 58 52 17. 35 68. 36 78 09 74 91. 50 15. 51. 49 64 26. 31. 42 72 59 70. 11. 69. 29. 65 33. 26 99 67 92. 91. 03 57. 91. 99 18 28. 26 06 29 20 01. 88 65 31 15. 71. 30 17. 13 21. 52 18. 91. 00. 1.00187 1.00193 1.00200 1.00206 1.00213 1.00219. 39446 93444 46644 98252 47473 93517. 93 20 67. 1.00226 1.00232 1.00239 1.00245 1.00251. 35597 72929 04738 30254 48715. 03 46 54. 1.00257 1.00263 1.00269 1.00275 1.00281. 59367 61466 54279 37084 09170. 32 68 79 39 42. 1.00286 1.00292 1.00297 1.00302 1.00307. 69840 18412 54217 76602 84931. 88 69 50 50 26. 1.00312 1.00317 1.00322 1.00326 1.00330. 78584 56960 19476 65569 94696. 45 63 98 99. 1.00335 1.00338 1.00342 1.00346 1.00349. 06332 99978 75152 31399 68284. 67. 1.00352 1.00355 1.00358 1.00361 1.00363. 85396 82350 58783 14359 48766. 54 31 60. 1.00365 1.00367 1.00369 1.00370 1.00371. 61720 52959 22252 69392 94199. 10 52 26 04 61. 1.00372 1.00373 1.00374 1.00374 1.00374. 96522 76237 33245 67478 78893. 27 57 35 92. 52 87 75. 51. 27. 16. 01. 57 27 07. 62 98. 91.

ELLIPTIC FUNCTIONS. K= 1.57848 K'= 3.36986 r. 65776 88648 80266 68445. u=(r/9O)K=:F(0,k). TABLE. = 8" E = 1.56316 ^. E'= u. 22295 18261 1.02784 36197 40833. dnu.

ELLIPTIC FUNCTIONS q. = 0.00122. 24470 64293 800. q'= 0.22956 71598 81194 r. 2. = 8° D(90) = 1.00490. TABLE. 17725 72135 1/D(90) =0.99512 21375 99988.

ELLIPTIC FUNCTIONS. K= 1.57848. 65776 88648 K'= 3.36986 80266 68445 r. TABLE. = 8' £ = 1.56316 6. E'= 1.02784. 22295 18261 36197 40833.

ELLIPTIC FUNCTIONS q. = 0.00122. 24470 64293 800. q'= 0.22956 71598 81194 r. 2. = 8' D(90) = 1.00490. TABLE. 6. 17725 72135 1/D(90) =0.99512 21375 99988.

ELLIPTIC FUNCTIONS. K=: 1.58054 K'= 3.25530 r. 09338 95721 29421 43555. u=(r/9O)K=F(0,k). TABLE. = 9" E = 1.56114 6. E'= u. 17453 51334 1.03378 94623 90754. dnu.

ELLIPTIC FUNCTIONS q. = 0.00154. 85045 16579 192. q'= 0.21754 89496 99726 r. 3. = 9° D(90) = 1.00621. TABLE. <?. 32605 90419 1/D (90) =0.99382 51056 37374.

ELLIPTIC FUNCTIONS. K= 1.58054 K'= 3.25530 r. TABLE. 09338 95721 29421 43555. = 9' E = 1.56114 6. E'=. 17453 51334 1.03378 94623 90754. dnu. u=(r/90)K=F(<^,k). 19095 32 76366 20 08213 54. 0.70491 0.69242 0.67973 0.66683 0.65374 0.64044. 34845 87297 58692 88966 18591 88569. 29 47 98 09 68 74. 0.99382 0.99361 0.99339 0.99318 0.99296 0.99275. 51056 02955 57935 18606 87570 67418. 37 57 30 25 50 36. 81957 65687 28264 39331 69310. 52 06 77. 0.62696 0.61329 0.59943 0.58540 0.57119. 40409 16116 58171 09522 13564. 91. 37 70. 0.77904 0.78985 0.80042 0.81074 0.82081. 0.99254 0.99233 0.99212 0.99192 0.99172. 60725 70048 97924 46866 19358. 26 67 96 45. 76922 92581 08240 23899 39559. 02 34 66 98 30. 0.83063 0.84020 0.84951 0.85857 0.86736. 89411 71634 88772 14414 22948. 85 88. 0.55681 0.54226 0.52755 0.51269 0.49767. 14127 55460 82215 39437 72543. 66. 17855 44781 02520 93421 19790. 81. 90 34 64. 0.99152 0.99132 0.99113 0.99093 0.99075. 1.07125 1.08881 1.10637 1.12394 1.14150. 55218 70877 86537 02196 17855. 63 95 27 59 91. 0.87588 0.88414 0.89214 0.89986 0.90730. 89563 90254 01818 01861 68799. 81. 0.48251 0.46720 0.45175 0.43617 0.42046. 27315 49880 86700 84558 90542. 32 43 82 46 00. 0.99056 0.99038 0.99021 0.99004 0.98987. 83889 87934 34090 24472 61138. 87 62 74. 66 67 68 69 70. 1.15906 1.17662 1.19418 1.21174 1.22930. 33515 49174 64833 80493 96152. 24. 0.91447 0.92137 0.92798 0.93432 0.94037. 81857 21071 67289 02174 08202. 50 20 65. 0.40463 0.38868 0.37261 0.35643 0.34015. 52033 16695 32456 47502 10259. 47 21 92 95. 0.98971 0.98955 0.98940 0.98926 0.98912. 46092 81278 68577 09809 06727. 80 42. 71 72 73 74 75. 1.24687 1.26443 1.28199 1.29955 1.31711. 11811 27471 43130 58789 74449. 84. 68662 67660 90115 21764 49158. 33 00 50 49 27. 0.32376 0.30728 0.29071 0.27406 0.25732. 69383 73747 72430 14706 50027. 29 28 68 03 71. 0.98898 0.98885 0.98873 0.98861 0.98850. 61016 74292 48099 83910 83119. 36 38 95. 13. 0.94613 0.95161 0.95680 0.96171 0.96632. 76 77 78 79 80. 1.33467 1.35224 1.36980 1.38736 1.40492. 90108 05767 21427 37086 52745. 45 77 10 42 74. 0.97064 0.97467 0.97840 0.98184 0.98499. 59663 41463 83556 75757 08694. 83 90 95 08 02. 0.24051 0.22362 0.20668 0.18967 0.17260. 28020 98467 11300 16585 64517. 37 52 22 99 79. 0.98840 0.98830 0.98821 0.98813 0.98805. 47048 76937 73949 39167 73588. 20 39 77. 81. 1.42248 1.44004 1.45760 1.47517 1.49273. 68405 84064 99723 15383 31042. 06 38 70 03 35. 0.98783 0.99038 0.99263 0.99458 0.99624. 73812 63374 70453 88941 13544. 90 23 68 97 72. 0.15549 0.13832 0.12112 0.10388 0.08662. 05403 89653 67772 90347 08037. 10. 0.98798 0.98792 0.98787 0.98782 0.98778. 78131 53628 00826 20386 12883. 79. 82 83 84 85 86 87 88 89 90. 1.51029 1.52785 1.54541 1.56297 1.58054. 46701 62360 78020 93679 09338. 67 99. 0.99759 0.99864 0.99939 0.99984 1.00000. 39782 63989 83316 95727 00000. 33 84 85 42 00. 0.06932 0.05201 0.03468 0.01734 0.00000. 71562 31694 39245 45059 00000. 0.98774 0.98772 0.98770 0.98769 0.98768. 78805 18552 32434 20674 83405. 69 10 05 12 95. 0.79027 0.80783 0.82539 0.84295 0.86051 0.87807. 04669 20328 35988 51647 67306 82966. 48 80. 0.89563 0.91320 0.93076 0.94832 0.96588. 98625 14284 29944 45603 61262. 41. 52 53 54 55 56 57 58 59 60. 0.98344 1.00100 1.01857 1.03613 1.05369. 61. 62 63 64 65. 45 46 47 48. 49 50 51. 12. 44 77 09. 73 05. 56. 88 20 52. 17 49 81. 31. 64 96. 0.70929 0.72148 0.73345 0.74520 0.75671 0.76800. 32957 03 62813 81 69844 17. 51. 63. 67 41. 27. 18 13 71. 81. 75. 25. 36. 11. 27 19 47. 41. 71. 17 35. 47 38 50. 48 86 83 00. 35. 02 34 44 58. 21. 63. 81. 78 47. 31. 78. 01. 81. 41. 06 19 59.

ELLIPTIC FUNCTIONS q. = 0.00154. r. 0.78849 0.80594 0.82339 0.84083 0.85827 0.87571. 51625 64011 38697 75759 75320 37549. 29 87. 51. 52 53 54 55. 0.89314 0.91057 0.92800 0.94542 0.96283. 56 57 58 59 60 61. 1.00621 32605 90419. A(r). E(</.,k). 45 46 47 48 49 50. = 9°. 1/D(90)= 0.99382 51056 37374. 3. 4>. 6. D(90)=. 85045 16579 192. q'= 0.21754 89496 99726. TABLE. 65 68 76 30. 0.78672 0.80406 0.82140 0.83872 0.85604 0.87336. 67023 83069 24195 90570 82455 00199. 62659 50908 02602 18087 97756. 08 93 45 54 00. 0.89066 0.90796 0.92525 0.94253 0.95980. 44244 15118 13440 39915 95335. 0.98025 0.99766 1.01507 1.03247 1.04987. 42043 51426 26427 67606 75566. 03 63 06 13 53. 0.97707 0.99433 1.01159 1.02884 1.04608. 80579 96606 44465 25280 40260. 00 96. 62 63 64 65. 1.06727 1.08466 1.10206 1.11944 1.13683. 50951 94441 06759 88662 40944. 06 85 48. 1.06331 1.08054 1.09777 1.11498 1.13219. 90690 77935 03433 68696 75308. 97 66 22 09. 66 67 68 69 70. 1.15421 1.17159 1.18897 1.20634 1.22371. 64437 60006 28548 70997 88313. 60 05 80 62. 1.14940 1.16660 1 18379 1.20098 1.21816. 71. 72 73 74 75. 1.24108 1.25845 1.27581 1.29318 1.31054. 81491 51551 99544 26545 33656. 15 53 33 60 59. 76 77 78 79 80. 1.32790 1.34525 1.36261 1.37996 1.39732. 22002 92730 47009 86028 10993. 38 56 82 58. 81. 82 83 84 85. 1.41467 1.43202 1.44937 1.46671 1.48406. 23128 23671 13875 95006 68339. 86 87 88 89 90. 1.50141 1.51875 1.53610 1.55345 1.57079. 35160 96762 54444 09510 63267. D(r). 78 39 21 45 16 76. 0.70710 0.71933 0.73135 0.74314 0.75470 0.76604. 33901 64709 04389 16340 64645 14073. 00 80. 52. 0.77714 0.78800 0.79863 0.80901 0.81914. 30093 78887 27357 43134 94594. 63 69. 0.82903 0.83866 0.84804 0.85716 0.86602. 50860 81818 58120 51198 33271. 95 22. 77354 57263 47627 23896 62344. 18 18 87. 51. 0.87461 0.88294 0.89100 0.89879 0.90630. 24924 19265 60120 49338 88833. 54 99 38 67 06. 0.91354 0.92050 0.92718 0.93357 0.93969. 1.23534 1.25252 1.26969 1.28685 1.30402. 80574 26591 28964 89826. 67. 72 79 64 04. 1.00310 1.00321 1.00332 1.00343 1.00353 1.00364. 66302 50501 3337S 13615 89896 60909. 72 06 47 63 44 62. 1.00375 1.00385 1.00396 1.00406 1.00416. 25350 81921 29336 66318 91604. 31. 1.0O427 1.00437 1.00446 1.00456 1.00465. 03944 02106 84873 51048 99454. 67 54 62 57 26. 60 49. 1.00475 1.00484 1.00493 1.00501 1.00510. 28935 38358 26617 92628 35337. 40080 35055 26066 92765 15664. 68 27 07 07 64. 1.00518 1.00526 1.00534 1.00541 1.00548. 53716 96 46770 22 13531 00 53065 13 64471 60. 76143 56452 39719 09954 52056. 46. 96 11361 92. 0.94551 0.95105 0.95630 0.96126 0.96592. 00 49 77. 1.00555 1.00561 1.00568 1.00574 1.00579. 46883 67 99469 91 21435 27 12021 96 70510 45. 1.32117 1.33833 1.35548 1.37263 1.38978. 95797 45406 62502 49437 08597. 0.97029 0.97436 0.97814 0.98162 0.98480. 51816 95918 71951 68406 74681. 00 69 82 58 89. 1.00584 1.00589 1.00594 1.00598 1.00602. 96220 88511 46782 70477 59078. 04 86 44 56. 42403 53304 43778 16325 73466. 27. 79 69 98. 1.40692 1.42406 1.44120 1.45834 1.47547. 0.98768 0.99026 0.99254 0.99452 0.99619. 81087 66 78847 71 60102 50 17911 44 46255 11. 1.00606 1.00609 1.00612 1.00614 1.00616. 12112 29150 09803 53732 60638. 79 00 92 64 95. 96 73 62 64 95. 1.49261 1.50974 1.52687 1.54400 1.56114. 17742 51708 77929 98984 17453. 69 04 87 30. 0.99756 0.99862 0.99939 0.99984 1.00000. 40037 95085 08153 76922 00000. 1.00618 1.00619 1.00620 1.00621 1.00621. 30270 77 62421 44 56929 95. 15. 69. 11. 51. 13 31. .. 91. 85 83 99. 10. 44 34. 12 12. 26. 83 63 37 77. 51. 37. 13 93 66 21. 14. 85 65. 11. 40 76. 21. 02 19 44 35 00. 63 34 31. 14. 19. 94 52 72 43. 31. 13681 15 32605 90.

ELLIPTIC FUNCTIONS. K= 1.58284. 28043 38351. K'=3.15338 52518 87839 r. u=(r/90)K=F(.^,k). TABLE. = 10° E = 1,55888. 6. E'=. 71966 01596 1.04011 43957 06010. dnu.

ELLIPTIC FUNCTIONS q. = 0.00191. 35945 90170 284. q'= 0.20660 97552 00965 r. = 10° D(90) = 1.00768. TABLE. 6. 37856 61824 1/D(90) =0.99237 48047 04456.



ELLIPTIC FUNCTIONS q. = 0.00191. 37856 61824 1/D(90) =0.99237 48047 04456. 35945 90170 284. q'= 0.20660 97552 00965. A(r). E(<^,k). 0.78922 0.80667 0.82412 0.84156 0.85900 0.87643. 53338 60395 20865 34846 02492 24013. 92 89 99 77 32 07. 0.78703 0.80435 0.82166 0.83895 0.85624 0.87352. 97217 50494 11332 79950 56677 41956. 52 53 54 55. 0.89385 0.91128 0.92870 0.94611 0.96352. 99675 29801 14768 55008 51007. 43 46 46 40. 0.89079 0.90805 0.92530 0.94254 0.95978. 36338 40485 55169 81268 19767. 56 57 58 59 60. 0.98093 0.99833 1.01572 1.03312 1.05050. 03305 12494 79218 04174 88108. 0.97700 0.99422 1.01143 1.02863 1.04582. 71757 38432 21087 21120 40026. 69. 61. 62 63 64 65. 1.06789 1.08527 1.10265 1.12002 1.13739. 31815 36140 01973 30252 21961. 33 94 90. 1.06300 1.08018 1.09735 1.11451 1.13166. 66 67 68 69 70. 1.15475 1.17211 1.18947 1.20683 1.22418. 78127 99817 88144 44259 69350. 00 69 78 09 03. 71. 72 73 74 75. 1.24153 1.25888 1.27622 1.29356 1.31090. 64644 31403 70925 84537 73599. 18 74. 76 77 78 79 80. 1.32824 1.34557 1.36291 1.38024 1.39757. 81. 82 83 84 85. 86 87 88 89 90. 51. = 10' D(90) = 1.00768. TABLE. D(r). 95 62 70 36 73 33. 0.70710 0.71933 0.73134 0.74313 0.75470 0.76603. 16025 47159 87190 99518 48222 98070. 76 95. 39. 0.77714 0.78800 0.79863 0.80901 0.81914. 14532 63786 12732 29003 80969. 44 44 96 24. 60 64 59. 0.82903 0.83866 0.84804 0.85716 0.86602. 37754 69240 46079 39702 22325. 22 39 82. 79397 40919 26372 37626 76636. 43 80 86 06 81. 0.87461 0.88294 0.89100 0.89879 0.90630. 66960 47424 38343 15164 54160. 1.14881 1.16595 1.18308 1.20021 1.21733. 45448 46185 81055 52342 62405. 03 62 89 80 25. 0.91354 0.92050 0.92718 0.93357 0.93969. 13674 08649 49897 40047 81788. 16. 94. 1.23445 1.25156 1.26866 1.28576 1.30285. 57 37 00. 39502 83661 07520 12546 00228. 55 92 91 57 45. 1.31994 1.33703 1.35411 1.37119 1.38826. 77865 31078 44275 20354 62253. 08 94 77 90. 1.41489 1.43222 1.44954 1.46687 1.48419. 72076 29620 74406 07994 31960. 77 62. 70 95. 1.40533 1.42240 1.43947 1.45653 1.47359. 72952 55466 12843 48161 64524. 1.50151 1.51883 1.53615 1.55347 1.57079. 47891 57380 62033 63458. 06 74 38 12 63267 95. 1.49065 1.50771 1.52477 1.54183 1.55888. 65057 52904 31223 03185 71966. 41. 15. 08 72 88 71. 79 19. 11. 21. 15. 91. 82 75 92. 17. 55. 18. 53. 24 44 84 81. 79 62 91. 35 02 41. 1.00384 1.00397 1.00410 1.00424 1.00437 1.00450. 18927 59729 98896 34799 65809 90304. 77 03 74 31 17 68. 1.00464 1.00477 1.00490 1.00502 1.00515. 06672 13307 08619 91029 58975. 14 77 62 58 20. 1.00528 1.00540 1.00552 1.00564 1.00576. 10911 45313 60677 55522 28392. 70 79 54 20 06. 10 05 42. 1.00587 1.00599 1.00610 1.00620 1.00631. 77858 02520 01007 71982 14139. 32439 27950 19487 86703 10107. 62 02 77 20 06. 1.00641 1.00651 1.00660 1.00669 1.00678. 26208 06958 55192 69755 49534. 16 27 84 63. 0.94551 0.95105 0.95630 0.96126 0.96592. 71076 51860 35586 06262 48785. 46 57 02 04 38. 1.00686 1.00695 1.00702 1.00710 1.00716. 93456 00494 69663 00027 90696. 75 03 21 17 08. 0.97029 0.97436 0.97814 0.98162 0.98480. 48944 93425 69814 66599 73180. 88 84 06 65 51. 1.00723 1.00729 1.00735 1.00740 1.00745. 40828 49632 16365 40338 20911. 46 24 66 27 68. 0.98768 0.99026 0.99254 0.99452 0.99619. 79865 77877 59357 17362 45872. 63 96 19 05 52. 1.00749 1.00753 1.00756 1.00759 1.00762. 57500 49572 96650 98310 54186. 38 45 22 83 75. 0.99756 0.99862 0.99939 0.99984 1.00000. 39791 94946 08091 76906 00000. 60 90 90 96 00. 1.00764 1.00766 1.00767 1.00768 1.00768. 63966 27393 44270 14452 37856. 24. 61. 37 28 78. 41. 55. 46 75 20. 14 01. 43 06 09 81. 70 04 84 62.

ELLIPTIC FUNCTIONS. K= 1.58539 K'= 3.06172 r. 41637 75538 86120 38789. u=(r/9O)K=F(0,k). TABLED = 11' E= E'= cnu. 1.55639 97977 70947 1.04678 64993 44049. dnu.

ELLIPTIC FUNCTIONS q. = 0.00231. D(90). 79450 15820 765. q'= 0.19656 76611 43642. 00000 54575 07161 55770 98421 33139. 00 42 33. 0.10568 0.12329 0.14090 0.15851 0.17612. A(r). 00000 54243 04508 46820 77215 91746. 00. 53 20. 0.00000 0.01761 0.03523 0.05284 0.07045 0.08806. 57959 70928 70111 53586 19452. 10 95 34 10 72. 0.10567 0.12328 0.14089 0.15849 0.17608. 86484 57527 01002 13074 89945. 37 05 48. 15. 0.19372 0.21132 0.22892 0.24652 0.26412. 65832 90871 92742 69647 19817. 69 88 84 05 17. 0.19368 0.21127 0.22885 0.24643 0.26401. 16 17 18 19 20. 0.28171 0.29930 0.31688 0.33447 0.35205. 41519 33054 92763 19023 10254. 26 86 17 04 99. 21. 0.36962 0.38719 0.40476 0.42232 0.43988. 64923 81537 58654 94879 88871. 16. 22 23 24 25 26 27 28 29 30. 0.45744 0.47499 0.49254 0.51008 0.52761. 39337 45041 04802 17496 82055. 30 67. 31. 32 33 34 35. 0.54514 0.56267 0.58019 0.59771 0.61522. 97474 62807 77169 39738 49758. 36 37 38 39 40. 41. 6 7. 8 9 10 11. 12 13. 14. 42 43 44 45. = 1.00931. 1/D(90)= 0.99077. 2. E(0,k) 0.00000 0.01761 0.03523 0.05284 0.07045 0.08807. TABLED = 11' 49632 07826 10045 45280 D(r). 0.00000 0.01745 0.03489 0.05233 0.06975 0.08715. 00000 20314 87475 48345 49818 38838. 00 76 78 53 77 80. 1.00000 1.00000 1.00001 1.00002 1.00004 1.00007. 00000 28372 13453 55141 53262 07576. 00. 02 33. 0.10452 0.12186 0.13917 0.15643 0.17364. 62413 67631 01678 11853 45582. 54 67 74 24 63. 1.00010 1.00013 1.00018 1.00022 1.00028. 17771 83471 04229 79533 08804. 43 02 30 64 96. 27865 23132 72100 71182 16853. 24 52 57 09 58. 0.19080 0.20790 0.22494 0.24191 0.25881. 50439 74156 64644 70006 38553. 32 65 80 60 37. 1.00033 1.00040 1.00047 1.00054 1.00062. 91398 26604 13648 51694 39841. 42. 0.28158 0.29914 0.31669 0.33424 0.35179. 05659 34218 99222 97449 25757. 80. 0.27563 0.29236 0.30901 0.32556 0.34201. 18820 59583 09873 18991 36526. 63 75 57 89 90. 1.00070 1.00079 1.00088 1.00098 1.00108. 77131 62543 94997 73359 96436. 0.36932 0.38685 0.40437 0.42188 0.43939. 81096 60508 61132 80204 15067. 75 95 22 55. 12368 96723 40131 93478 08013. 56. 12. 0.35836 0.37459 0.39072 0.40672 0.42261. 75 70 15. 1.00119 1.00130 1.00142 1.00154 1.00166. 62981 71696 21229 10180 37100. 30 93. 0.45688 0.47437 0.49184 0.50931 0.52677. 63167 22062 89422 63030 40789. 48 66 09 30 61. 0.43836 0.45398 0.46946 0.48480 0.49999. 35360 27538 36969 16499 19407. 60 29 83 65 40. 1.00179 1.00191 1.00205 1.00218 1.00232. 00495 98825 30508 93922 87407. 36 28 63 96. 99 49 24 90 94. 0.54422 0.56166 0.57908 0.59650 0.61391. 20722 00973 79813 55637 26969. 51. 0.51502 0.52991 0.54463 0.55918 0.57356. 99422 10736 08020 46435 81648. 16. 95 47 14 35. 54 65 86. 1.00247 1.00261 1.00276 1.00291 1.00306. 09263 57759 31130 27581 45289. 46 58 74 87. 0.63273 0.65023 0.66772 0.68521 0.70269. 06536 52 09444 27. 0.63130 0.64869 0.66607 0.68343 0.70078. 92464 50907 01218 42447 73781. 40 93 22 28 72. 0.58777 0.60180 0.61565 0.62931 0.64277. 69843 67737 32591 22225 95030. 1.00321 1.00337 1.00353 1.00368 1.00384. 82405 37055 07347 91366 87182. 32 83. 0.72017 0.73764 0.75511 0.77257 0.79003. 72158 98643 68903 82784 40202. 0.71812 0.73546 0.75278 0.77008 0.78738. 94543 04190 02317 88655 63071. 57 82 82 54 62. 0.65605 0.66912 0.68199 0.69465 0.70709. 09980 26645 05203 06453 91828. 1.00400 1.00417 1.00433 1.00449 1.00465. 92853 06420 25919 49377 74814. 33 94 95 24 88. 71. 18. 02 70 05. 81. 57921 03 51472 41 89671 43. 92 87 81 91. 19. 75. 52 43 86 48. 15. 88 23 46. 43. 81. 51. 42 18 41. 76 75. 47 08 13. 72 43. 12 91. 70 89 07. 21. 45 07. 88 64 25 95 71. 54 98 61. 62 50 68. 15. 76. 16 18. 98.

ELLIPTIC FUNCTIONS. K= 1.58539 K'= 3.06172 r. 41637 75538 86120 38789. TABLE. = ll" E = 1.55639 <>. E'=. 97977 70947 1.04678 64993 44049.

ELLIPTIC FUNCTIONS q. = 0.00231. 79450 15820 765. q'= 0.19656 76611 43642 2 r. = ir D(90) = 1.00931. TABLE. e. 1/D(90)= 0.99077. 49632 07826 10045 45280.

ELLIPTIC FUNCTIONS. K = 1.58819 K'= 2.97856. 72125 27520 89511 81384. i=(r/9O)K=F(0,k). TABLE. = 12' E = 1.55368 6. E'= 1.05377 cnu. 08919 36509 69204 07046. dnu.

ELLIPTIC FUNCTIONS q. = 0.00276. 18093 29251 567. q'= 0.18728 51836 10216 r. 8. = 12° D(90) = 1.01110. TABLE. 6. 85969 69203 1/D(90) =0.98901 34482 06750.

ELLIPTIC FUNCTIONS. K= 1.58819 K'= 2.97856 r. 72125 27520 89511 81384. TABLE. = 12E = 1.55368 6. E'=. 08919 36509. 1.05377 69204 07046.

ELLIPTIC FUNCTIONS q. = 0.00276. = 12« D(90) = 1.01110. TABLE. 6. 85969 69203 1/D(90) =0.98901 34482 06750. 18093 29251 567. q'= 0.18728 51836 10216 8. D(r). A(r). E(0,k). 0.70709 0.71932 0.73134 0.74313 0.75469 0.76603. 59942 92097 33230 46737 96693 47863. 27 35 33 65 99 24. 1.00555 1.00574 1.00594 1.00613 1.00632 1.00651. 42982 81404 17465 48805. 51. 0.77713 0.78800 0.79862 0.80900 0.81914. 65709 16406 66849 84665 38220. 23 32 66 34 29. 1.00670 1.00689 1.00708 1.00727 1.00745. 91018 80046 52704 06708 39802. 50 82 16 97 42. 96631 29777 08303 03634 87981. 91. 1.00763 1.00781 1.00798 1.00816 1.00833. 49751 34350 91424 18834 14475. 18 10 92 92 51. 1.00849 1.00866 1.00881 1.00897 1.00912. 76280 02226 90330 38659 45325. 83 20 68 39 94. 08494 26382 97263 19465 91378. 68 98 35 58 77. 0.79092 0.80837 0.82581 0.84324 0.86067 0.87810. 17258 11215 37945 97595 90396 16660. 80 94 25 44 74 52. 0.78776 0.80502 0.82226 0.83948 0.85670 0.87390. 66456 10061 20739 98835 44855 59464. 01. 0.89551 0.91292 0.93033 0.94772 0.96511. 76778 71223 00547 65379 66427. 84 85 19 14 85. 0.89109 0.90826 0.92543 0.94258 0.95971. 43486 97902 23849 22619 95654. 02 37 96 60. 0.98250 0.99987 1.01724 1.03461 1.05197. 04478 80391 95103 49622. 39 67 38 75. 44546 34 71036 83. 4503126. 0.97684 0.99395 1.01105 1.02814 1.04522. 77011 52 64499 17 35668 61. 0.82902 0.83866 0.84804 0.85716 0.86601. 62 63 64 65. 1.06932 1.08667 1.10401 1.12135 1.13868. 82481 63194 88461 59636 78142. 32 73 23 85 22. 1.06228 1.07934 1.09638 1.11342 1.13044. 92825 38411 74996 05281 32088. 97 60 97 40 65. 0.87461 0.88294 0.89100 0.89878 0.90630. 34351 16554 09212 87765 28482. 43 42 16. 66 67 68 69 70. 1.15601 1.17333 1.19065 1,20796 1.22527. 45460 63137 32775 56035 34633. 85. 1.14745 1.16445 1.18145 1.19843 1.21541. 58363 87167 21676 65174 21052. 35. 0.91354 0.92050 0.92717 0.93357 0.93968. 08465 05657 98848 67684 92670. 82 35 39 09 16. 1.00927 1.00941 1.00954 1.00968 1.00980. 71. 1.24257 1.25987 1.27717 1.29446 1.31175. 70339 64973 20402 38543 21355. 87 28 89 96. 1.23237 1.24933 1.26628 1.28323 1.30017. 92803 84015 98370 39640 11679. 28 56 48 90. 0.94551 0.95105 0.95630 0.96125 0.96592. 55178 37454 22618 94676 38521. 78 32 79 99 38. 1.00993 1.01004 1.01015 1.01026 1.01036. 11453 78202 90205 46106 44620. 31. 72 73 74 75. 76 77 78 79 80. 1.32903 1.34631 1.36359 1.38087 1.39814. 70840 89036 78024 39914 76852. 01. 1.31710 1.33402 1.35094 1.36785 1.38476. 18423 63880 52131 87321 73657. 45 75 72 82 28. 0.97029 0.97436 0.97814 0.98162 0.98480. 39936 85604 63106 60930 68469. 75 52 85 41 96. 1.01045 1.01054 1.01062 1.01070 1.01077. 84529 64688 84025 41543 36317. 10 49 97 30 57. 81. 1.41541 1.43268 1.44995 1.46722 1.48448. 91010 84590 59816 18936 64214. 1.40167 1.41857 1.43546 1.45236 1.46925. 15400 16863 82406 16428 23366. 18 54 39 73 51. 0.98768 0.99026 0.99254 0.99452 0.99619. 76031 74835 57018 15638 44672. 51. 67 95 23 07. 37 78 35. 1.01083 1.01089 1.01094 1.01098 1.01102. 67502 34328 36105 72222 42146. 29 48 53. 1.50160 1.51901 1.53627 1.55353 1.57079. 43491 22386 39883 52737 63267. 32 64 32 06 95. 1.48614 1.50302 1.51991 1.53679 1.55368. 07686 55 73881 41 26464 30 69963 94 08919 37. 0.99756 0.99862 0.99939 0.99984 1.00000. 39021 94513 07898 76858 00000. 1.01105 1.01107 1.01109 1.01110 1.01110. 45429 81699 50669 52134 85969. 14 38 76 40 69. 51. 52 53 54 55 56. 57 58 59 60. 61. 82 83 84 85. 86 87 88 89. 90. 27 16 37 91. 91. 97 53 93 44 81. 38 54 57 25 05. 12. 38 08 39 80. 12. 49. 49 48 36 47. 71. 95. 15. 60 00 81. 65 00. 73 28 38 73071 99 87920 68. 70 44 74 14. 11. 88.

ELLIPTIC FUNCTIONS. K= 1.59125 K'= 2.90256 r. 43820 13687 49406 70027. u=(r/90)K=F(</.,k). = 13". TABLE. E= E'= cnu. 1.55073 19509 84013 1.06105 93337 53857. dnu.

ELLIPTIC FUNCTIONS q. = 0.00324. 54674 43524 990. q'= 0.17865 56628 04652 r. 6. = 13' D(90) = 1.01306. TABLE. 6. 66847 71240 1/D(90) =0.98710 18512 72317.

ELLIPTIC FUNCTIONS. K = 1.59125 K'= 2.90256 r. TABLE. 43820 13687 49406 70027. u=(r/90)K=F(<^,k). = 13' E = 1.55073 d. E'=. 19509 84013 1.06105 93337 53857. dnu. snu. 30. 0.71168 0.72383 0.73575 0.74744 0.75890 0.77013. 15730 02993 33084 72872 90057 53167. 0.98710 0.98665 0.98620 0.98576 0.98532 0.98487. 18512 47354 83665 32870 00372 91545. 44. 0.98444 0.98400 0.98357 0.98314 0.98272. 11726 66210 60245 99021 87672. 95690 30186 98708 46489 19013. ZZ 30 29 74 87. 0.98231 0.98190 0.98150 0.98110 0.98071. 31261 34782 03147 41188 53645. Z6 59 Z6. 45165 20295 83475 39038 91201. 88 49 97 92 23. 00 64 67 50. 0.70250 0.68997 0.67724 0.66432 0.65119 0.63788. 21982 80415 96357 11217 66837 05482. 74 34 90. 35 67 13 10 70. 0.62437 0.61069 0.59682 0.58278 0.56857. 69815 02880 48083 49180 50251. 63. 41. 60 28. 0.55419 0.53966 0.52496 0.51012 0.49513. 72. 0.79562 0.81330 0.83098 0.84866 0.86634 0.88403. 71910 77952 83994 90037 96079 02122. 52 53 54 55. 0.90171 0.91939 0.93707 0.95475 0.97243. 08164 14207 20249 26292 32334. 74 19 64 08 53. 0.78112 0.79186 0.80237 0.81262 0.82263. 31560 95423 15773 64451 14124. 56 57 58 59 60. 0.99011 1.00779 1.02547 1.04315 1.06083. 38376 44419 50461 56504 62546. 97 42 87. 0.83238 0.84188 0.85112 0.86010 0.86881. 38283 11236 08108 04839 78176. 61 62 63. 68589 74631 80674 86716 92758. 20. 65. 1.07851 1.09619 1.11387 1.13155 1.14923. 0.87727 0.88545 0.89337 0.90102 0.90839. 05671 65680 37351 00628 36239. 84 14 56 28 50. 0.47999 0.46472 0.44931 0.43377 0.41811. 61999 21386 43324 74157 60412. 12 70. 0.98033 0.97996 0.97959 0.97924 0.97889. 66 67 68 69 70. 1.16691 1.18460 1.20228 1.21996 1.23764. 98801 04843 10886 16928 22971. 43 88. 0.91549 0.92231 0.92885 0.93512 0.94110. 25696 51287 96074 43882 79303. 56 94 16 76 17. 0.40233 0.38643 0.37043 0.35431 0.33810. 48790 55 86150 46 19501 76 95993 05 62902 33. 0.97856 0.97824 0.97792 0.97762 0.97733. 44060 01590 67636 45911 39992. 46 30 26 49 77. 71. 1.25532 1.27300 1.29068 1.30836 1.32604. 29013 35056 41098 47141 53183. 66. 72 7Z 74 75. 87681 55116 68452 15276 83912. 65 30 09 03 49. 0.32179 0.30539 0.28890 0.27233 0.25569. 67627 57678 80665 84296 16364. 65 18 75 75 50. 0.97705 0.97678 0.97653 0.97629 0.97606. 53316 89175 50715 40929 62658. 63 69. 45. 0.94680 0.95222 0.95735 0.96220 0.96675. 76 77 78 79 80. 1.34372 1.36140 1.37908 1.39676 1.41444. 59225 65268 71310 77ZS2 83395. 89 34 79 23 68. 0.97102 0.97500 0.97869 0.98208 0.98518. 63418 43579 14905 68625 96685. 54 64 Z7 40 74. 0.23897 0.22218 0.20533 0.18842 0.17146. 24742 57375 62275 87515 81222. 00 04 65 93. 0.97585 0.97565 0.97546 0.97529 0.97513. 18586 11236 42970 15983 32305. 49 63 20 45 43. 81. 1.43212 1.44980 1.46749 1.48517 1.50285. 89438 95480 01523 07565 13607. 12. 69 90 20. 0.15445 0.13740 0.12031 0.10319 0.08603. 91569 66775 55096 04823 64273. 20 24 69. 91. 91745 47171 57040 16129 19919. 84. 0.97498 0.97486 0.97474 0.97464 0.97456. 93795 02143 58862 65295 22606. 87. 57 02 46. 0.98799 0.99051 0.99273 0.99466 0.99629. 16. 82 83 84 85 86 87 88 89 90. 1.52053 1.53821 1.55589 1.57357 1.59125. 19650 25692 31735 37777 43820. 35 80 24 69 14. 0.99762 0.99866 0.99940 0.99985 1.00000. 64587 47010 64760 16103 00000. 36 46 34 45 00. 0.06885 0.05166 0.03444 0.01722 0.00000. 81791 05737 84493 66447. 04 92 30 84 00000 00. 0.97449 0.97443 0.97440 0.97437 0.97437. 31783 93634 08792 77706 00647. 45 46. 47 48 49 50 51. 64. 07 51. 96 41. 85. 31. 76. 65 10 54 99. 2>2,. 77 22. 11. 56. 00. 50 11. 29 85. 91. 52. 19. 80 95. 11. 99 51. 15. 2>6. 81. 40 37 60 44. 40 72 20 99 57 79 01. 39 25 63. 11. 27 58. 19. 84 66 52. 11. 94 50 58 85.

ELLIPTIC FUNCTIONS q. = 0.00324. 54674 43524 990. q'= 0.17865 56628 04652. 66847 71240 1/D(90) =0.98710 18512 72317. 6. E(0,k). 56. 57 58 59 60. = 13" D(90) = 1.01306. TABLE. A(r). D(r). 0.79188 0.80933 0.82677 0.84421 0.86163 0.87905. 90071 76109 83153 11382 61072 32594. 26 69 36 32 34 49. 0.78818 0.80540 0.82260 0.83979 0.85696 0.87412. 09430 05910 46509 31652 61954 38215. 99 86 59 80 35 06. 0.70709 0.71932 0.73133 0.74313 0.75469 0.76603. 18853 51756 93697 08068 58942 11079. 49 52 10 60. 0.89646 0.91386 0.93125 0.94864 0.96602. 26414 43092 83280 47724 37261. 52. 61421 32743 53532 25321 49819. 26 08. 44 56 07. 0.89126 0.90839 0.92550 0.94260 0.95968. 81. 0.77713 0.78799 0.79862 0.80900 0.81914. 0.98339 52816 63 1.00075 95406 85 1.01811 66134 91 1.03546 66190 12 1.05280 96846 45. 0.97675 0.99380 1.01084 1.02787 1.04488. 28911 64652 59270 15157 34869. 30 95 93 85 45. 1.06188 1.07886 1.09584 1.11280 1.12974. 21121 76784 04884 08593 91227. 1.14668 1.16361 1.18052 1.19742 1.21432. 18. 33419 13523 90848 62621 26074 78449. 39. 39. 1.00653 1.00676 1.00698 1.00721 1.00744 1.00766. 29939 81693 33233 52181 06900. 61 95 90 81 76. 1.00789 1.00811 1.00833 1.00855 1.00876. 17002 39005 41751 22558 78766. 20 40 93 09 90. 0.82902 0.83866 0.84803 0.85715 0.86601. 66504 00865 80627 77209 62819. 22 56 17 52 84. 1.00898 1.00919 1.00939 1.00960 1.00980. 07751 06917 73708 05604 00132. 36 63. 23 90 60 10 68. 0.87461 0.88293 0.89099 0.89878 0.90630. 10460 67. 93938 87869 67692 09670. 1.00999 1.01018 1.01037 1.01055 1.01073. 54860 67407 35443 56693 28936. 54 77 96 20 59. 56246 07241 47940 82195 13979. 04 94 78 06 65. 0.91353 0.92049 0.92717 0.93357 0.93968. 90901 89324 83727 53750 79895. 76 94 23. 50014 17831 30353 85617 81728. 91. 26. 1.01090 1.01107 1.01123 1.01138 1.01153. 47387 86622 35998 99930 82932. 04 38 53 92 34. 0.94551 0.95105 0.95630 0.96125 0.96592. 43531 26899 13118 86189 31001. 56 78 54 37 62. 1.01168 1.01181 1.01194 1.01207 1.01219. 16862 89272 97285 39308 13827. 61. 72. 51. 01. 68 83 67 52. 19. 97 43. 62 63 64 65. 1.07014 1.08747 1.10479 1.12211 1.13942. 59460 55471 86396 53830 59443. 66 67 68 69 70. 1.15673 1.17402 1.19132 1.20860 1.22589. 04982 92260 23164 99643 23716. 31. 71. 72 73 74 75. 1.24316 1.26044 1.27771 1.29497 1.31223. 97459 23011 02566 38372 32731. 84 57 56 00. 1.23120 1.24807 1.26494 1.28179 1.29864. 76 77 78 79 80. 1.32948 1.34674 1.36398 1.38123 1.39847. 87990 06547 90838 43343 66578. 76 09 45 57 55. 1.31548 1.33232 1.34914 1.36596 1.38278. 89607 24648 92827 98994 48066. 69 58 90 32 68. 0.97029 0.97436 0.97814 0.98162 0.98480. 33337 79874 58192 56776 65018. 07 34 90 93 84. 1.01230 1.01240 1.01250 1.01259 1.01267. 19412 54715 18475 09518 26759. 81. 93466 35471 86321 18278 34001. 50 35 19. 1.39959 1.41639 1.43320 1.44999 1.46679. 45028 38 94921 67 02841 90 73931 76. 0.98768 0.99026 0.99254 0.99452 0.99619. 73222 72606 55305 14375 43792. 51. 1.01274 1.01281 1.01287 1.01292 1.01296. 69201 35941 26165 39155 74286. 92 33. 83 84 85. 1.41561 1.43295 1.45018 1.46742 1.48465. 86 87 88 89 90. 1.50188 1.51911 1.53634 1.55356 1.57079. 36166 27466 10605 88298 63267. 1.48358 1.50037 1.51715 1.53394 1.55073. 26392 65. 0.99756 0.99862 0.99939 0.99984 1.00000. 38457 94195 07757 76823 00000. 47. 1.01300 1.01303 1.01305 1.01306 1.01306. 31027 08944 07698 27048 66847. 24 33. 61. 82. 85 53 19. 06 99. 72 04 93 50. 21. 81. 79. 72 11. 19 75 95. 13375 41. 18233 94169 59494 19509. 01. 86 41. 84 53. 11. 79 71 71. 01. 26 58 50 71. 12. 35 26 00. 30 77 73 31 73 51. 58 32 52. 20 38 71. 99 62. 55 46 07. 75 35 71.

ELLIPTIC FUNCTIONS. K = 1.59456 K'= 2.83267 r. 83409 31825 25829 18100. TABLE. = 14° E = 1.54755. 45758 69993. £'=1.06860 95329 78401.

ELLIPTIC FUNCTIONS q. = 0.00376. 92262 86977 846. q'= 0.17059 45383 49476 r. 6. = 14° D(90) = 1.01519. TABLE. d. 14249 78510 1/D(90) =0.98503 59010 08687.

ELLIPTIC FUNCTIONS. K= 1.59456 K'= 2.83267 r. 83409 31825 25829 18100. TABLE. = 14" E = 1.54755. 6. E'= 1.06860. 45758 69993 95329 78401.

ELLIPTIC FUNCTIONS q. = 0.00376. 92262 86977 846. q'= 0.17059 45383 49476 r. 6. = 14» D(90) = 1.01519. TABLE. 14249 78510. 1/D(90)= 0.98503 59010 08687.

ELLIPTIC FUNCTIONS. K= 1.59814 K'= 2.76806 r. 20021 12540 31453 68768. u=(r/90)K=F(.^,k). TABLE. = 15° E = 1.54415 ^. E'= u. 04969 14673 1.07640 51130 76403. dnu.

ELLIPTIC FUNCTIONS q. = 0.00433. 34205 09983 127. q'= 0.16303 35348 21580 r. 5. = 15° D(90) = 1.01748. TABLE. 6. 52236 81446 1/D(90) =0.98281 52554 21419.

ELLIPTIC FUNCTIONS. K = 1.59814. 20021 12540. K'=2.76806 31453 68768 r. TABLE. = 15' E = 1.54415. 6. E'=. 04969 14673 1.07640 51130 76403.

ELLIPTIC FUNCTIONS q. = 0.00433. ^. = 15". D(90)= 1.01748 52236 81446 1/D(90) =0.98281 52554 21419. 34205 09983 127. q'= 0.16303 35348 21580. TABLE. 5. D(r). A(r). E(</.,k). 1.00874 1.00904 1.00935 1.00965 1.00995 1.01026. 26104 772Z\ 24641 64622 93468 07490. 53. 1.01056 1.01085 1.01115 1.01144 1.01173. 03016 89 76397 30 24009 37 42261 88 27599 39. 34 83 02 86 03. 1.01201 1.01229 1.01257 1.01284 1.01311. 76506 85512 51194 70183 39166. 57 45 70 75 94. 0.87460 0.88293 0.89099 0.89878 0.90629. 42661 47 29755 64 27302 62 10727 85 56283 76. 1.01337 1.01363 1.01388 1.01412 1.01436. 54892 14173 13892 51002 22535. 52 62. 82. 0.91353 0.92049 0.92717 0.93357 0.93968. 41056 42975 40815 14206 43641. 58. 1.01459 1.01481 1.01503 1.01523 1.01543. 25602 57395 15198 96379 98404. 17 99 10 54 72. 11005 48220 66812 72632 71652. 74 13 90 32 17. 0.94551 0.95104 0.95629 0.96125 0.96592. 10478 96947 86157 62102 09661. 22 92 48 38. 1.01563 1.01581 1.01599 1.01615 1.01631. 18834 55329 05650 67667 39354. 47 04 94 68 35. 1.31188 1.32851 1.34513 1.36175 1.37835. 69964 73774 89389 23217 81752. 90. 0.97029 0.97436 0.97814 0.98162 0.98480. 44247 69 44989 83 55224 96. 1.01646 1.01660 1.01672 1.01684 1.01695. 18796 04190 93849 86201 79795. 07 39 40 87 07. 1.39495 1.41154 1.42813 1.44471 1.46129. 71574 99339 71768 95646 77809. 60. 0.98768 0.99026 0.99254 0.99452 0.99619. 65250 87 66280 31 50443 72 10791 68 41296 97. 1.01705 1.01714 1.01722 1.01729 1.01735. 73296 65496 55306 41765 24037. 65 18 63 75 20. 1.47787 1.49444 1.51101 1.52758 1.54415. 25140 44560 43019 27493 04969. 0.99756 0.99862 0.99939 0.99984 1.00000. 36856 93292 07355 76722 00000. 1.01740 1.01743 1.01746 1.01747 1.01748. 01411 73307 39271 98979 52236. 56 23. 23353 13 40362 Z2> 50209 15 53521 84 51178 16 44303 32. 0.70708 0.71931 0.73132 0.74311 0.75468 0.76602. 02248 37273 81506 98330 51808 06691. 29 90. 48. 0.78911 0.80625 0.82337 0.84047 0.85755 0.87461. 00 69 63 58 56. 0.89165 0.90867 0.92567 0.94265 0.95960. 34267 22684 11407 02526 98364. 84 93 77 40 43. 0.77712 0.78798 0.79861 0.80899 0.81913. 28429 83184 37836 59997 18019. 54 24 27. 0.98540 1.00274 1.02006 1.03738 1.05468. 69187 29 11064 57161 09077 68522. 46 94 78 12. 0.97655 0.99347 1.01037 1.02725 1.04412. 01475 46 14639 29 40857 89 45552 22. 0.82901 0.83865 0.84803 0.85715 0.86600. 81005 18816 02085 02218 91414. 1.07198 1.08927 1.10655 1.12382 1.14108. 37315 17384 10761 19583 46084. 01. 62 63 64 65. 1.06097 1.07780 1.09461 1.11141 1.12819. 31103 43850 87837 67302 86673. 25 03. 66 67 68 69 70. 1.15833 1.17558 1.19282 1.21005 1.22728. 92596 90 61546 73 55451 71 76917 53 28634 83. 1.14496 1.16171 1.17845 1.19517 1.21188. 50556 63738 31175 57987 49453. 74 87 57. 71. 72 73 74 75. 1.24450 1.26171 1.27891 1.29611 1.31331. 13375 33991 93407 94620 40695. 94 56 32 30 43. 1.22858 1.24526 1.26193 1.27859 1.29524. 76 77 78 79 80. 1.33050 1.34768 1.36486 1.38204 1.39921. 34761 80009 79684 37086 55563. 89 37 33 18. 81. 82 83 84 85. 1.41638 1.43354 1.45071 1.46787 1.48502. 38509 89359 11585 08692 84214. 64 69. 86 87 88 89 90. 1.50218 1.51933 1.53649 1.55364 1.57079. 41711 84762 16964 41927 63267. 0.79406 0.81151 0.82894 0.84637 0.86378 0.88119. 47874 14908 76486 32864 84427 31684. 52 53 54 55. 0.89858 0.91597 0.93334 0.95070 0.96806. 75274 15957 54621 92274 30046. 56 57 58 59 60. 61. 51. 20 31. 35 60 01. 02 84 70 67. 41. 59 48 59. 13 13. 35. 07 95. 83351 11. 19. 85 16. 51. 61. 96 06 21. 01. 36 27 58. 80 53. 92 02 15. 61. 35 32 20 08. 31. 98 44 15 71. 11. 14607 95 6.'^612. 75. 54 99 89 82 00. 18 73 98 14 46 75. 16 61. 73. 11. 17 81.

ELLIPTIC FUNCTIONS. TABLE. 6. = 16'.

ELLIPTIC FUNCTIONS q. = 0.00493. 84132 64213 395. q'= 0.15591 66592 65792 r. 2. = 16= D(90) = 1.01995. TABLE. 6. 07026 62439 1/D (90) =0.98043 95422 14776.

ELLIPTIC FUNCTIONS. K = 1.60197 K'= 2.70806 r. 45. 85300 86952 76145 90486. u=(r/90)K=F(^,k). TABLE. = 16' E = 1.54052 6. E'=. 15741 27631 1.08442 52193 72543. dnu.

ELLIPTIC FUNCTIONS q. = 0.00493. 84132 64213 395. q'= 0.15591 66592 65792 2 r. — 16° D(90) = 1.01995. TABLE. 6. 07026 62439 1/D (90) =0.98043 95422 14776.

ELLIPTIC FUNCTIONS. K = 1.60608 K'= 2.65213 r. 13494 10364 80046 30204. u=(r/90)K=F(^,k). TABLED = 17° E = 1.53666 E'= 1.09265 u. 97975 68556 03455 37715. dnu.

ELLIPTIC FUNCTIONS q. = 0.00558. 45970 58516 804. q'= 0.14919 73690 67429. <?. 07081 95194. 1/D(90)= 0.97790 83576 50672. 2. D(r). A(r). E(0,k). 41. = 17' D(90) = 1.02259. TABLE. 72 78141 1(> 49529 18. 1.00000 1.00000 1.00002 1.00006 1.00010 1.00017. 00000 68808 75148 18770 99255 16016. 00 20 96 89 Zl. 00000 02299 51483 94453. 61. 0.00000 0.01745 0.03489 0.05232 0.06974 0.08714. 72002 20844 02448 07474 26689. 23 10 07 60 52. 0.10451 0.12185 0.13915 0.15641 0.17362. 55661 43663 60758 54278 71687. 28 64 08 68 69. 1.00024 1.00033 1.00043 1.00055 1.00068. 68304 55200 75625 28336 11927. 18 60 63 03 43. 0.19608 0.21386 0.23163 0.24939 0.26714. 50976 71349 78965 65133 21332. 55. 0.19078 0.20788 0.22492 0.24189 0.25878. 60591 68756 44123 34828 89210. 41. 69 29 94 02. 1.00082 1.00097 1.00114 1.00132 1.00151. 24835 65340 31563 21475 32895. 95 18 28 19 21. 63 12 02 72 21. 0.28487 0.30259 0.32029 0.33797 0.35564. 39213 10618 27588 82372 67438. 00 42 53 59. 55833 83499 21265 18456 24683. 1.00171 1.00193 1.00215 1.00239 1.00264. 63494 10799 72193 44921 26092. 55 25 16. 81. 0.27560 0.29233 0.30898 0.32553 0.34198. 85. 0.37329 0.39092 0.40854 0.42613 0.44370. 75483 99442 32494 68077 99888. 92 35 99 59 63. 0.35832 0.37456 0.39068 0.40669 0.42257. 89856 64201 98275 42979 49575. 1.00290 1.00317 1.00344 1.00373 1.00403. 12683 01544 89397 72847 48381. 86 38 90 88 28. 0.46126 0.47879 0.49630 0.51378 0.53125. 21896 92 28348 61 84 72992 91 01121 97. 0.43832 0.45394 0.46942 0.48476 0.49995. 69701 55384 59055 33565 32198. 1.00434 1.00465 1.00497 1.00530 1.00564. 12372 61089 90695 97255 76740. 86 64 40 35 98. 50421 23 91362 64 05087 12. 0.54868 0.56610 0.58349 0.60086 0.61820. 93578 46085 54676 15698 25818. 35 25 18 74. 0.51499 0.52987 0.54459 0.55914 0.57352. 08685 17218 12467 49588 84242. 34 65 02. 1.00599 1.00634 1.00670 1.00706 1.00743. 25034 37935 11164 40365 21119. 90 90 06 95 96. 0.63895 0.65652 0.67407 0.69161 0.70914. 89972 44557 67541 57788 14326. 0.63551 0.65280 0.67007 0.68731 0.70452. 82019 81612 22229 01830 18704. 75 35 36. 0.58773 0.60176 0.61561 0.62927 0.64274. 72608 71391 37843 29769 05547. 16. 1.00780 1.00818 1.00856 1.00894 1.00933. 48941 19289 27569 69141 39326. 65 25 18 64 26. 0.72665 0.74415 0.76163 0.77910 0.79656. 36348 35 23211 84. 0.72170 0.73886 0.75599 0.77310 0.79018. 71466 59060 80760 36167 25208. 0.65601 0.66908 0.68195 0.69461 0.70706. 24132 45077 28540 35300 26764. 1.00972 1.01011 1.01050 1.01090 1.01129. 33407 83 46642 00. 00000 51864 94015 16756 10413 65354. 00 09 80. 18. 0.00000 0.01784 0.03568 0.05353 0.07137 0.08920. 46425 97558 15049 94219 30435. 35 34 84 22 57. 0.10703 0.12486 0.14268 0.16049 0.17829. 0.19619 0.21400 0.23181 0.24961 0.26741. 19123 55769 35928 55224 09365. 69 94 07 90 90. 0.28519 0.30298 0.32075 0.33851 0.35627. 94140 05428 39202 91535 58607. 0.37402 0.39176 0.40949 0.42721 0.44491. 36703 22227 11696 01754 89170. 0.46261 0.48030 0.49798 0.51564 0.53329. 70844 43809 05239 52445 82887. 0.55093 0.56856 0.58618 0.60378 0.62138. 94168 26 84043 82. 0.00000 0.01784 0.03569 0.05353 0.07137 0.08921. 00000 52673 00491 38602 62169 66375. 00 68 06. 0.10705 0.12488 0.14272 0.16054 0.17837. 19 71. 00 42 53 50. 16. 69 13 l(i. 19. 54 24 49 78 81. 74441 80 89729 07 68930 12. 18 10. \Zm. 06. 81. 40. 15. 64 83 46 03 67. 00 65 97. 15. 92 32 14. 10 41. 27 ZZ 09 14 38 14 18 61. 71. 60 85 98 11. 14 16. 55. 29 92 10. 30 37 91. 48 59. 91. 12. 32. 74261 11 11479 95 53501 63.

ELLIPTIC FUNCTIONS. K = 1.60608 K'= 2.65213 r. 13494 10364 80046 30204. TABLE. = 17* E = 1.53666 <?. E'=. 97975 68556 1.09265 03455 37715.