i 



Theory of the Jacobian Elliptic Integrals. 271 



This process is general, and we can theoretically find a 



general expression for 1 kK'Kf'dk, which, however, is of 

 no importance. * ° 



Results will now be obtained involving the complete 

 elliptic integral 



E(£) = f 2 Vl-^sin^^ ... (28) 

 Jo 



The formulae connecting E and K are 



r/E 

 *^ = E-K, (29 a) 



Integrating (10) by parts, 



P.WK(i)4 



r K(A) ,, ,.dP,-] 1 1 f 1 „ ,,dP„ dK(k) , 



= l-^i^-^-dfL, + i^+T)}J 1 ~' i) 'dJ--dk dli 



"0* + l)Jo ^ 



= -JL C Y n (2/c--l){E-k'- 2 K}kdk. 



.-. f^-{E-^K}PJ(2P-l)^=g^j 2 . . (30) 

 Therefore changing the variables Is, U, 



C k{E'-t?K>}J>n'(?*?-l)dk = (-)"- 1 $£$ • (31) 



An integration of (30) by parts give- 



rap = [(■-«fr-M] 1 _ I -j>«4ra { E " Ki ' 2 K 



(2 



The integrated term is zero, for Iu =( , £' 2 logy, = 0, 

 and, again, £_ J £ _ ^ "> K by (29) 



