Theory of the Jacobian Elliptic Integrals, 283 



From (07) and (7), 



i ' (4ir-2 cos- 1 k + IJ . K' . <M = 4( 1- I, + _V . .) 

 Jo V o D / 



■■ by (21), 8 ' 



(77) 



f 1 K'.(I 1 -2c03- 1 *).^=^7T 3 . . . . 



Jo *> 



By (68) and (7), 



\(2cos- l k-7r + I 2 ).K.dlc=lo- z . . . . (78) 



Jo ^ 



Integrating all the above by parts, a large number of 

 integrals involving E and K, E' and K' may be deduced. 



Bv combinations o£ (67) and (68) with previous series of 

 Legendre functions, Ave may deduce a large number of definite 

 integrals of functions of k from to 1. The following are 

 examples : 



(1) 7,-PQ J cos- 1 ** ~(l u I,)} where P=K, K', E, E' 



(2) *PQ { rin-^+i^, T 2 )} „ Q = K, K', E, E' 



and these integrals, when found, will on integration by parts 

 give rise to great numbers of elliptic integrals, integrated 

 with respect to their modnlns. 



It may here be noted, in connexion with the result (62) 



and similar results, that by a similar investigation with series 



of the type 



iry . 1 . , 1 . - 



f = «my- £2 sm 3y + _, sin 5y. . . 



77- 

 when y is between ±'5*, we may in all cases expand, in a 



series of Legendre function-, the integral 



p|F(X)(tan- 1 ^rsinx)j cos\d\ 

 J S«+i?"sfi?X 



