Theory of the Jacobian Elliptic Integrals. 269 



and 



\\-K>Vk = ]<T, (17) 



Jo ^ 



and we note that if n is even, 



C k(K-K')Fn(2k"-l)dk = . . . (18) 



Jo 

 and also that 



r*(K 2 -K /3 )^=0 (19) 



Jo 



Combining (7) and (8) 



(-)" 



= 42 



(2re + l)- 



.-. f , Kdt=2(l-i + p-...) . . (20 ) 

 while 



J "2 o(2w+l) 2 



LK'tfts^ (21) 



Again, 



f> K '*--Iro.C p - wK '(\/ [ f i )' i '' 



-{>-*+*•■•} 



t ) o o\ ^ tne use o£ Ruler's numbers 



7T 3 



... CkKK'dk^ (22) 



Jo 16 



Expressions may be fouud for integrals containing higher 

 powers or* K, K', but cannot be concisely exhibited. 



A theorem will be used, which will be stated as follows, 

 and is easy to prove. 



