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



Tlie last two papers relate chiefly to tlie integral 



fK 



log (1 + n sn 2 x) dx, 



Jo 



and one of the methods indicated for obtaining its value depends upon 

 the formula 



2Y amp + 2 Y am q= Y am (p + q) + Y am (j> — g) — log (1 — Jc 2 sn 2 p sn 2 q), 



and is therefore the same in principle as that employed in § 10. 

 Mr. Roberts generally uses the Legendrian notation and the function 

 Y, for example the formula (44), viz. : 



log (1 — & 2 sn 2 x sn 2 y)dx=21Z.log — 

 Jo Qy 



is written 



f^iog^w^) #=E(J)[F(Jj 0)J _ mm{k> e) 



Jo (l — kr sin/ 0) s 



but he remarks (" Liouville," t. xii, p. 453) that some of the results 

 could have been obtained more readily by the use of G instead of Y. 

 Mr. Roberts also gives the values of the integrals 

 Mog (l + cot 2 6>sin 2 0) 



o (1- A; 2 sin 2 0)4 



77 | ■■ — 



^ log {1-(1-Z; ,2 sin 2 0) sin 2 0} , 

 log(l + cot 2 0sin 2 0) 



r 



Jo Jo (l-& 2 sm 2 0)*(l-& /2 sin 2 0y 



Pfcr ffcr log{l-(l-^sm3g) sin 2 0} 



J ojo (l-A; 2 sin 2 0)i(l-^ 2 sin 2 ^)i ^* 



These can be derived from (44) by the substitution of %K! — iy and 

 K + ?'K' — m/ in place of y, for 



k sn(iK' —iy}=i Cn ^ h f ^ 

 sn (y, & ) 



&sn (K + zK' — %) = dn (y, &'), 

 whence we find 



(K' - y y + K log (^) - 2K log 9Q/, A;')— 2Klog sn(y, V) (63), 



— _l 

 ~ 2 K 



