ON ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 341 



and differentiate, we shall have 

 ..0,.,vV...)., ,_jL,H..,;i^-i^!...)..^»K..,iK;....).: 



a V a a 



apply the formula (A), this becomes 



du da du ^ ' '" du ^ ^ ' '" 



a V V a. 



But 



— aX(M ) ^'2a-l^'2.-l «A"i----)2.-l 



^'^ ^ > • • • • - «2._l-«2.-l ■ «^'(«1 • • • • )2a-l" 



Also, using formulae (1) and (2) of section 1, 



d 2v-l 



^aXK-K,. ...).= ±^'2a-l«'2.-l«^'K- • • •)2.-l«^"(Wl- • • •)2a-l 



VR.V 1 



But by a formula (Crelle 47, page 292) proved by Weiorstrass in his 

 second paper, page 322, we have 



d . (^li^l^l^^^£ha-l _ ^\.-\ J. . J . 



du„ V ±{^20.-1 ~" 2,^-1) 



whence 

 and 



d.l0g.«?KM,....)2.-l , „, . ^1 -Jji- 







du„ 



a 



whence we see that 



du 2,|-i fnog ^aX(» , t6,.... ),,„_! (nog^aX(«,»,....),^_i 



and we have 



d' log, «?(» ,.. -Oa^-^ ^ c?log^a\(K,M,....)2,_i ^ c/log,aX(M,it,....)2^_ i 



' ■ ^'2a-l g%.-l/^^'K----)2.-l _ 



«2v-l-«2a-l'«^'K----)2a-l' 



This proposition has been proved by Brioschi, ' Annali di Matematica,' torn, i., 

 in a paper entitled "Sopra alcune proprieta delle funzioni Abeliani," Section 4. 

 Brioschi, however, uses the notation in Wcierstrass's second paper. 



