ON ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 313 



To reduce this he makes use of the following theorem : — 



«2«%2«e.0^(riM;+«nlog,<?,<?»^) = 2j6-^'03(tc; + ^,^). ... (3) 



As Eosenhain has not demonstrated this formula, I give the proof here. 

 Let 



rm 

 In — , where (s) is a prime number, aU the remainders are diiferent as 



m increases from to s — 1. Hence we easily see, forming n linear equa- 

 tions, by putting A;=0, 1. . . .n — 1, 



2rHn 



But 



\ '*■ / -oo 



2.€ » eAw+-~,q)= 2„,S,?^ 



A ** / -00 



r,m 2mw — - 







2miTr 



i — e " 

 This expression vanishes except when m=nfx, ju being an integer, or 



'\ TO ^/ ^ _^^ 



-00 



the formula required. 



This formula may be written 



cfi »— 1 _ 2ariTT i j.j^ 1 1 



^2 « e^««'03(«it; + a log^ g, j**) = 2^e "^63 { ^f' + — , ?» | , 

 so that equation (1) becomes 



1 \ « % ^ / 



Rosenhain then shows how, by giving w the n values 



w, w+ - log^ J, w+ - log^ 2, w+ !illi log 5, 



we may obtain n equations to determine the constants Q^ in terms of 

 functions with constant arguments. 



