340 Mr. J. W. L. Glaisher on [Nov. 20, 



f K 



log (1 — & 2 sn 2 x — W sn 2 y + h 2 sn 2 a? sn 2 y)dx 



Jo 



=Klog(^-5)-2Klo g e?/ . (47). 

 The last two integrals may also be written 



fK 



2 l°g ( cn2 83 ~~ ^ n2 93 sn2 2/) 2< ^> 



log (dn 2 x—W cn 2 a? sn 2 y)dx, 



and i£ these be transformed by the substitution of K— x for s#, they 

 become respectively 



I -Jlog (sn 2 a3 — sn 2 ^) 2 ^ — 2 j log dn xdx + 2K log h\ 

 Jo Jo 



| log (1 — & 2 sn 2 a3sn 2 ?/)cfo: — 2 | log dn a3(fe -f 2K log h\ 

 Jo * Jo 



so that of the four integrals (44), (45), (46), (47), the pair (44), 

 (47) are convertible one with another, and also the pair (45), (46), by 

 the substitution of K— x for x. 



Integrating (44), . . . (47) with regard to y between the limits K 

 and 0, and using (38) we obtain the following evaluations : — 



f * j* log (1 - A; 2 sn 2 x sn 2 y) dxdy = - ^KK' + f K 2 log (^p) . . (48) , 



j* j* J log (sn 2 a-*n« y)Uxdy= - §7rKK' + fK 2 log'(-^) ■ . (49), 



| K | K ilog(l ~sn 2 a3 -sn 2 ?/ + W srfx $tfy)Hxdy= — firKK' + fK 2 %(^) 



. . (50), 



I log (1 — W 1 sn 2 x —h 2 sn 2 y-\-~kr sn 2 03 sn 2 y)dxdy 

 Jo Jo 



= _i 7rK K'+fK 2 log( 2 ^) . . (51), 



§ 11. The lemma at the beginning of the last section may be de- 

 duced directly from the definition of integration when y is real ; for, 

 in virtue of the equation 0(a? + 2K) =0(#), (£>(x + y) + <fi(x— y) can 

 always be reduced to the form <f)(x + a) +(fi(x — a) where &<K; and 



<f)(x-\-ct)dx-\-\ (/)(x—a)dx 

 Jo Jo 



=M0(a) + 0(af^)+0(ft + 2^) . . . + 0(a + K) 

 + 0(-<O+0(~a + fc) + 0(-&)+0(O)+0(fc) . . . +0(K-a)}. 



