334 Mr. J. W. L. Glaislier on [Nov. 20, 



but if, in virtue of the formulae 



. sn(s?, k') 

 snix=i — )-'— 

 cn (%, k ) 



cn 



cn (x, &')' 



cn {x, k ) 



these equations be written 'in a real form, they only reproduce (5), 

 (6), and (7), as might have been anticipated a priori. 

 § 5. The double products (" Eundamenta Nova," p. 66) 



n 2 2mK + 2m'iK' _ (-)^-i ^ 



Ilcn^ : 



Hdn 2 2mK + 2m '*' K ' -frW-i) 



give rise to the remarkable integrals 



*K f*K' 



logsn (x-\-iy)dxdy= — KK'log& . . . (13), 

 ^^logcnix + ii^dxdy^KK'log^j . . . (14), 

 K J logdn(x + iy)dxdy=KK'logk f . . . . (15). 



These may of course also be written 



m 



§ 6. Using the values of the three integrals in (5), (6), (7), and 



log {sn (x + iy) sn (x—iy)}dxdy = — KK'log& ; &c. 



observing that 



[*K (*2K 



log sn 2xdx = J log sn xdx, 

 Jo Jo 



(*K p2K 



= J log sn o?^£c + i\ log sn xdx, 

 Jo Jk 



f K f K 

 =-J logsn$(fe + 7? logsn (2K— x)dx, 



Jo Jo 



f K 



= log sn xdx, 

 Jo 



