1879.] Definite Integrals involving Elliptic Functions. 343 



spectively; we thus find, on substituting for sn 2 ?/ its value in the 

 different cases 



j K log{l-(l-r) m*x}3x =2K{log0O-logG(iK)}, 

 j* i g{l - (1 + V) sn 2 x}dx =2K {log 00-log 0QK + iK') }, 

 J K log{l + h sn 2 x}dx =2K{log GO -log e(JiK')}, 



j*log{l-7csn 2 ;«}c^ =2K{logQ0-logG(K + ^-K')}, 



^ K \og{l-Jc(h-ik') srfx)dx =2K{logG0-logG(JK-^K / )}, 



j K log{l-&(7£ + ^') sn 2 «}^=2K{logQ0-logQ(iK + ^K / )}. 



To obtain the values of the thetas, put x=\~K, y=±K in (36), and 

 this equation gives 



GXiK)=l±f-G30GK, 



whence 



6(JK)=?JV(1 + J') i (53). 



From this, by taking x=^K. in the formula 



Q(x + iK') —i'k l cf l e~^' sn x Qx, 



we deduce that 



Q{±K. + iK')=e4 ^+J^ h\l-k>y . . . (54). 



2M 



The value of G(^K') may be deduced from (53) by putting x—^K. 

 in the formula 



G« = ( — — 1 e 4KK' 0(*aj, 7b ), 



\A;K / cux 



and changing the modulus from h to h' : it is thus found that 



e(^K0 = e^?5?&*(l-&)* • • • • • (55), 



and thence that 



G (K + J*K' )=eml=.tf(l + h)K 



Finally, from (36) by putting x=±K + ±iK', y=±K + ±iK', we find 

 that 



