1879.J Definite Integrals involving Elliptic Functions. 349 



§ 15. It was shown in § 5 that 



k fK' < y 



log cn (x + iy) dxdy — KK/ log — ■ ; 

 o J -k' h 



the value of this integral when the limits with regard to y are K' and 

 may be found as follows. 

 In the formula 



Jc'i T.-«e( w + K+?K'), 

 cn u— — a 4 e2K — - 



put u—x + iy—'K—iK.', 



and, taking logarithms, we find 



+ log0(aj + i'7/)-loge(a3 + %-K-?:K / ) . . . (72) 



(p(x-\-iy— K— {K')dxdy — \ (£>(x + iy)dxdy 

 o Jo Jo 



if be an even function, so that (72) gives 



K/ log cn (x + iy)dxdy=±KK f log — -±7rK' 2 

 o Jo h 



+ J ^ (iK 2 K' + i^KK^ - K2K ; - lKK'3) 



=iKKMog-^-^VKK / (73). 



§ 16. I conclude with the determination of the values of the in- 

 tegrals 



when w is a positive integer. 



e^2Ka^ _ 1 _ ^ cQg 2a , + 2 ^ 4 cog ^ _ 2 ^ 9 cQg 6aj + &c ^ 



civ 



whence 



e-^Q^^^j-e-x^ _ 2ge-*- 2 cos 2ra + 2^e-* a cos 4aas- &c. 



Now f e"* 2 cos 2&a3^=i^e- 55 , 



so that we have 



J V* 2 e(^-)cfe^ + &c. 



