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



so that the lemma will necessarily hold good so long as the series 



^, 1 4<q n riirx nira 

 2 - - — ±— cos — — cos — — 

 nl — (f n K K 



is convergent. 



Let a=miK! 1 then 



cos— — =i(2 Wtt -Hg[ »*)=i_L-3 — , 



and the series 



= 2 — q a(] - m) cos , 



which is convergent when m< 1. 



It follows therefore that the lemma and the theorems (44), . . . (47) 

 are true when y is of the form a + hi, where b lies between K' and 

 — K', and a is unrestricted. 



In the case of b = K.' the series becomes neutral, but it is easy to see 

 that the lemma and theorems are still true ; for, transform (44) by 

 putting y=iK' + z, so that 



7 1 



snz 



then log Qy=log Q(iK'+z) 



= J log Jc + 1~ + log sn z + log Qz + log i - -g. 

 Thus we have 



f K log (sn 2 z - sn2^) clx = - WK' + K log (^^) - 2K log Qz + wr (* - K) . 



JO \ JvTT J 



The imaginary term %tt{z — K) is due to the fact that sn 2 z— sn 2 # 

 changes sign when x=z, so that we might expect the term 

 (K— 2) log (— 1) to appear in the equation. Writing, therefore, the 

 integral in the real form 



fK 



Jo 



and throwing away the imaginary term, we have (45). It may be ob- 

 served that (46) and (47) are also connected by the same substitution 

 of y=iK! +z. 



§ 12. In the formula (44), viz. : 



j K log (1 -k 2 sn 2 x sn 2 y)dx=2K{log 60-log Qy}, 



put y equal to JK, JK + iK', JiK', K + J&K', JE— ^'K', ^K + ^K' re- 



