ON ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 343 



with two similar expressions for 



<f log, d(u^u,\ , (f log, e(M,M,), 



=— ^j anu ■ j — = . 



mq du^au^ 



Take the equation at the commencement of p. 340, 

 Expanding the members in terms of v, we have 



=(».+ j|^;»,"y8(«,».)'+(^n+ . . . .y8(«...,).- 



+(<|. .,+ .... )V„,».);+(^»,+ ....)V«.".)f,.. 



Hence, equating coefficients of v', we find 



from which the formula we desire to prove immediately follows. This demon- 

 stration will be understood, if we remember that 



|;=0. ^=0, 0=0, 0,,,=O. 



The formula for 



d' logfi(u^ti^\ ^^^ cV \ogS{u,uX 

 du^ du^du^ 



may be proved in a precisely similar manner. 



Combining these three theorems with the last, we find 



(Z' log,e(^, + e^, u^ + g^)s _ rf' log,e(MiM,), 

 dul du\ 



d' log,e(M., + ^, %+eJ, _ d" log,0(M^tg 



5 



•2 



(g'log,0(M,4-gi. ^,+g,)5 _ d^ log,0(«,tt,), 

 where 



du^du^ du^du^ 



e(e„eX=0,e(e^,e,\ = 0,e(e„eX.=0 (B) 



These equations, give by integration. 



