1879.] Definite Integrals involving Elliptic Functions. 341 



Since 0(K + «— /&) = <Z>(2K— K— a + &)=0(K + a + &), &c, and 0( — h) 

 =<fi(h), &c, this 



=&{0(O)+0W • • • + 0(K-a)+0(K-a + fc) . . . +0(K) 



+ 0(0) +00) . . . +0(a)+0(a + fc) . . . +0(K)} 



= 2j K 0(a?)^. 



The lemma is thus evidently true for all real values of y, and from 

 the proof in § 11 it would appear that in general it was probably true 

 when y was imaginary or of the form a-\-bi. It is, however, certainly 

 not true in the case of y—2%K', for 



G (x + 2iK / ) = —eh K '- ix) Qx. 



so that loge(^ + 2t'K / )+log0(a?-2iK , ) = ?^ + 21og0^ 

 and therefore 



I logeO-f 2*K')cfo+ log e(x—2iK!) dx=27r~K ; + 2 1 log Qxdx. 



It is also evident that (44), . . . (47) cannot be universally true., for 

 the left-hand members of these equations remain unaltered when y is 

 increased by 2iK.', which is not the case with the right-hand members, 

 since G is not periodic with respect to 2z'K'. 



To determine the imaginary values of y for which the lemma and 

 the theorems (44), ... (47) deduced from it are true, consider the 

 expansion of logG(^) in a series of cosines, viz. : 



loge^A-^cos^ -^eos^-^cos^- & , (52), 



where A= T V — + i log {-—, ■— j. 



From this we obtain (38), viz. : 



fK 



log Qxdx= AK, 



cos — -dx=0. 

 Jo K 



Jo 



since 



Now log Q(x-\-a) -flog Q(x—a) 



a a x>l %q n f nirfx + a) . mrfx—a) 



=2 A— 2 I — < cos — ^ — — 1-4- cos — - 



nl-q^\ K K 



