313 KEPORT — 1873. 



Consequently 



log ^log J— 4A^ 



But 



^ ^ to logoff -2At; _ jk' log^p 



log^p log^ 5-4A'' ~ log^^>log^^-4A- ' 

 Hence 



_ / "" S 2?zw'+9i^log j' '^ »j=logy 2?K2y'-f4«»«A' 



geP-CO -QO 



«^ log 5''4-4«TOA'+we- log jj'+2;B2'y'-)-2«M)' 





'V ~kF^^3, 30'^'» ^"''i^'' 2' ^')- 



3gei5 



Rosenhain gives two other theorems of a precisely analogous nature (p. 397) 

 for transforming 



and also 



v^log j-|-tt^log p—4Avw 

 e HpH?-4A'^ ^^^^ ^ (^,, ,,, ^,, ^, A) into ^,^ 3 (iv\, iw\,p\, q\, A' J, 



where the new variables and constants emanate from the former according 

 to a certain law. 



Section 4. — Bosenhain next enters upon investigations relative to the 

 multiplication of functions 6, commencing with elliptic functions, and thence 

 advancing to hypereUiptic functions. He proves without difficulty that, by 

 directly multiplying the functions 6^ together, 



n-l 

 



nA(«' + aA.5) = 2„P„e2««'03(,m+s+alog^^,2"), (1) 



«P„.2«t^e3(«M' + S + a log, q, f) = \e - n,d,iw + cc,+ ^, q), . . (2) 



01 »i 



and P^ is a certain constant, — where, as is obvious, the product 11 extends 

 • to the quantities a^, a^ .... a,„ s = aj + a^ + Kj +....+ a,,, and a is an integer 

 less than («). 



