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



K2 2 ; WQi'-ik) 



= s 4K ±^ W*(V + ik) (56) . 



77" 



Substituting these values in the integrals and reducing, we have 

 ^l og{ l_ ( l_/ 0sn2 ^^ =i Klog{ T 2 |J} ...... (57), 



J o K ilog{l-(l + /Osn^^}^=-l^ + iKlog{g} . . (58), 



jJlogll + ^sn^^d^-^K' + iKlogj^^l j .... (59), 



^logll-^sn^f^-i^ + iKlogj^LT^j . . . . (60), 



j* log{l-k(k-W) sn 2 x}dx= - JttK' + JK log | ?^i±M J (61), 



j* log {1 - £ (Jb + ik') sn 2 aj} dx = - JttK' + JK log { 2m (V-ik) j (62) 



In (58) as the integral is so written that its value may be real, the 

 term involving i that enters from (54) has been rejected. 



By the addition of (57) and (58) we obtain (17), while (59) and 

 (60) lead in a similar manner to (16), and (61) and (62) to (18). 

 The reason why this happens is easily seen, for since 1 — 2sn 2 a3-f kPsifix 

 is the numerator of cn 2*, we have 



l- 2 s^ + ^n^ = {l-^}{l-_|g_}, 



which = sn 2 x sn 2 ( 1K + iK')} {1 -W sn 2 x sn 2 JK}, 



= {l-(l + &') sn 2 4{l-(l-^) sn 2 ;*}, 



and, similarly 



1-& 2 sn 4 x= {1 — 7^ sn 2 x sn 2 JiK'} {1 -7c 2 sn 2 a sn 2 (K+ JiK')} 

 = {l + &sn 2 «}{l-&sn 2 a?}, 



and 



l—2^ 2 sn 2 £ + k 2 sn 4 aj 



= {1 - & 2 sn 2 x sn 2 (JK - j*K') } {1 - W sn 2 a sn 2 (JK + J*K') } 



= - tfc') sn 2 ^} {l-ft^+tfc') sn 2 »}. 



§ 13. In any integral we may replace sna?, cna?, dnas, by sin a?, cos#, 



(1— Z^sm 2 ^)*, if we also replace dx by f x ■ — -, and make the 



(1 — W sin 2 ^) 4 



