ON ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 315 



Now from the definition of 0^ 3 {v, iv) it is easy to see that 







?±^,. / 'kin Utt a 



X y 



-QO -00 



= ''2i^2) 2^ 2 e('*'+^)'l%^+^('*'+^)(^+>')-A+(v+y)Mog^j 



-co -00 



= (using the reasoning of section 4, and so putting nx for a?, nt/ for y) 



2n^ .p'5%2/5(t'+2yA)^r=,2,(.+2/3A)|^ | 



-00 -00 

 n^~^ log^^+4n^^yA+nV log^ 2+2>w(/3 \og^'p-k-'^yK-\-v)-ir%iy{y log^ y+2/3A+«;) 



=nye2/3(t-+5yA)^y^^2y(«,+5/3A) 



03.3('<^+/51ogeP+2yA), n(M;+ylog^2+2/3A), p«', ^w', An«), 

 whence 



= B€^'^'+^^''..^3,3 0^^+^log^i' + 27A, nM; + ylog^g+2^A, p^, (t,k.n), . (2) 







which agrees with Eosenhain, p. 404. 



Hence, combining (1) and (2) together, we obtain 



« 



1 



(Eosenhain, p. 405). 



In this way formulae are found for the multiplication of hyperelliptic 

 functions. Two others of a similar nature are given by Eosei^hain, toge- 

 ther with the expression just written down ; and they are presented in a 

 somewhat modified form on page 406. The quantities B^ ^ are expressed 

 by means of functions (p^^ 3 with constant arguments, by a method analogous 

 to that by which the constants Q^ were determined previously. 



Section 6. — Having thus discussed some of the properties of f^ ^ (y, w), 

 Eosenhain proceeds to develop a number of similar functions defined as 

 follows, p. 499 : — 



CO ^8 2??2 V 



f., ,• (^'. "')=2„,p e 0^(w + 2mA, (7), 



-co 



