THE VALUES OF -^-('e^'dt. 2.77 



VTTJ 



For t = 2-8, e ~ < 2 = 0-000 393 669 040 655 078 210 9805 ; 

 A G = _ re - <2 {l-2-8r + 4-893r 2 -5-9173r 3 + 5-159 413r 4 - 3-237 4968> + 1-361 565 765 0793r 6 

 -0-259 346 702r7- -103 377 617 382 716 05r 8 +-103 997 546 129 383r 9 

 -•036 027 867 912 33r 10 + -001 055 801r 11 + -004 625 1919r 12 --001 989 645r 13 , etc}. 



and -|-e" (3 = 0-000 444 207 944 205 666 629 3623 = E ; 



J-rr 



AH = Er- 0-001 243 782 243 775 866 56r 2 + -002 173 657 540 313 0620r 3 



- -002 628 526 475 179 665r*+ -002 291 852 390 107 31r 5 - -001 438 121 837 3856r 6 

 + -000 604 818 329 407r 7 - -000 115 203 865 431^ - -000 045 921 158 89r 9 



+ •000 046 196 536 17r 10 - -000 016 003 8651r n + -000 000 468 995r 12 + -000 002 054 511r 13 

 -•000 000 088 38r 14 +etc. 



Lastly, for t = 3, e'< 2 = 0-000 123409 804 086 679 549 4976 ; 



AG=-r€ _(3 {l-3r + 5-6r 2 -7-5r 3 + 7-3r 4 -5-3r 5 + 2-8d4 7619r 6 -0-967 857jr 7 + -099 867 724> 



+ 112}r 9 --077 510 822r 10 + -021 764 0692r n + -000 886 0583r 12 

 -•003 249 626 464 0122r 13 + etc.}. 



and -4-e- 12 = 0-000 139 253 051 946 747 853 89 = E ; H = 0-999 977 909 503 001 4145 ; 

 Jtt 



AH = Er- -000 417 759 155 840 243 561 67?* 2 + -000 789 100 627 698 237 8386r 3 



- -001 044 397 889 600 608 904r 4 + -001 016 547 279 211 259 33r 5 



- -000 738 041 175 317 7636r 6 + -000 390 571 655 222 069r 7 - -000 134 777 060 991 32r 8 

 + •000 013 906 885 4788r 9 + -000 015 616 235 lllr 10 --000 010 793 618 59r n 



+ -000 003 030 713 07r 12 + -000 000 123 386r 13 - -000 000 452 53r 14 + etc. 



These data will enable anyone to verify the table, and also to recompute to the like 

 degree of accuracy Kramp's first Table of the values of G. Any value of G may also be 

 found for verification by multiplying 1 - H by ^ s/ 7r - 



TJie constant p and its derivatives. 



20. The value of t = /j , in the solution of the equation — 



t* 1 t 5 1 t 1 J * 2 / t s tf t 7 \ 1 



'-3+L2-5 +LT3T - ^ = 4 ; ° r 7^ ('-3 + 275-3!7 + J = 2 (37) 



is of importance, as it enters into the coefficients of various formula?. Bessel 

 employed the value 0'476 9364, Encke, followed by De Morgan, uses 0*476 9360, 

 and Airy gives 0'476 948. # To obtain this value with extreme accuracy, we may 



proceed thus: Since 0*475 =\ (I-2V) an d 0'475 2 = 5+^8 + 40^0' the computation of the 

 series for this value is comparatively easy, and gives — for t= *475 — 



* It seems strange that the late Astronomer Royal, so late as 1861, should have adopted a value differing from 

 that so generally recognised as correct at least to six decimal figures ; he gives its reciprocal also as 2'096 665 {Theory 

 of Errors, pp. 23, 24). Laplace (Throne Anal, des Probabilites, 2 e ed., p. 238), in one of the very few examples he 

 gives, makes « 2 = -210 2497, which would give p = -45853. M. Poisson, also (Connaissance des Temps, 1832, Add. p. 20), 

 gives -47414 for the value of p, and "67336 for that of p\fv, and again (Bech. sur la Prob. des Jugements, p. 208), he has 

 •4765 and -6739 for the same quantities. Gauss (Werke, Bd. iv, S. 110) gave the value as *476 9363, which is correct 

 to the nearest figure in the seventh place. Lastly, O. Byrne (Dual Arithmetic, p. 200) finds 0-476 936 2744, which 

 errs only in the last two decimal figures. 



