1879.J Definite Integrals involving Elliptic Functions. 331 



the bodies examined by them, and obtaining only negative results, and 

 always a certain result with a crystal of the salt, they have insisted 

 that this is the only nucleus. Others, again, have sought for an 

 explanation in some catalytic or other mysterious force ; while a third 

 set of observers have declared it to be a matter of uncertainty or 

 hazard whether a foreign body acts as a nucleus or not. In reviewing 

 the subject and repeating my experiments in various ways, I see no 

 reason for withdrawing from the theory which I had the honour of 

 submitting to the Society eleven years ago, namely, that the action of 

 nuclei is simply mechanical, and is capable of being expressed by the 

 familiar word adhesion. 



YIII. " On Definite Integrals involving Elliptic Functions." By 

 J.W.L.Glaisher, M.A., F.R.S., Fellow of Trinity College, 

 Cambridge. Received July 31, 1879. 



§ 1. The chief object of this paper is to apply to definite integrals 

 involving elliptic functions certain special methods which have been 

 employed for the evaluation of integrals of a similar kind involving 

 circular functions. 



§ 2. One of the most elegant and direct investigations of the value 

 of the integral 



log sin x dx — \tt log (J) 



Jo 



is afforded by the product 



. 7T . 2lT . (n — Y)7T / n-ifn- 11 



n n In 



for, taking the logarithm of both sides of this equation, and writing 



v 7 



n 



Klogsin/i + logsin2/i. . . +logsin^)= 3 rlim., =ao lo ^^ n ' ^~ i(n ' l) } 



n 



= _l 7r lo g2 . 



The same principle also gives the value of the integral 



log (1— 2acos£ + a 2 )cfo (1), 



Jo 



which =0 or 27rlog a according as a < or > 1, and it is easy to see in 

 general that if 



then [°^\o S ^x)dx=l lblLij!S±SL =bti^L. 



J a n 00 ) 



2 a 2 



