120 Mr. Lubbock on some Elementary 



In this example s^ = + I, g^ = + ^ ^ -^i + ^ *^2 = ^ ^ "~ ^' i . 



By the known properties of the function F, (see Theorie 

 des Fonctions Elliptiques, vol. ii. p. 417,) 



J ^ ^ rip+q) 



^^ - 3 r| - 6 ^ r(^i-) 



log TT^ = 0-2485749 logr(— ) = 9*9734262 



logr(|-)* = 9-9508435 log 6 = 0*7781513 



log 5 = 0-6989700 0-755775 



0-8983884 

 0-7515775 



log\pi = 0-1468109 

 ^i = 1-402202 



^'l^c'^*^^^*^^^-^^^^^^- 



It will now be seen that the numerical values of the inte* 

 grals <p 0^1, <p ^2 verify the equation 



4/a:i + \t/^2 = v(/, — 4^'l» 







for -508264 + -893917—^1/', = ^I/l-^^^ 



"0" "B" 



•508264 + -89391 7 = ^i 



•334901 + 1-06728 = rt/i- 

 The preceding results may easily be shown to be in ac- 

 cordance with the known theory of elliptic integrals by put- 

 ting for X 



1 — _^_?_ . (See Legendre, vol. ii. p. 381.) 

 tan"* i (p 



d<p 



>/3 8in2^ 



J ^l-A-3 ^ ^ J ^X- 



The equation 



3 = (1— ^i) (1— ^2) giv^s 



• -1 = 1-33333 and ^ — 1-83333, &c. (See theTable of the value* 

 3 o 



of log Ta, Thiorie des Fortciiom EUiptiqties, vol. ii. p. 493.) 



