322 REPORT— 1873. 



Rosenhaiu, in the last part of his memoir, proves the remarkable equation 



J^ xfLv C^ dx C^ dx C^ xdx 



i 1 i. J. 



C"-^ ^dx^ p' dx p' dx C''' xdx _(^ 



k^ K- V K^ 



a formula much used hy later writers. 



Section 10.— We now proceed to consider the method of treating the hy- 

 perelliptic functions proposed by Gopel. His justly celebrated paper in the 

 3.5th volume of Crelle's Journal presents very few difficulties, which will 

 make our analysis of it shorter and easier. He commences with the sixteen 

 series of which the analogues have been used by Rosenhain, and writes them 

 thus : — 



e'" P' (M,i6')=2( — 1) (the same expression), 



t'" ^" P"(?(,?«') = 2( — 1) (the same expression), 



e'"' ' " P"'(",w') = ^ • 1 • (th^ same expression), 



ru^+r'u'\^ ^ ,, ^, ^.a+b r{n+r{2a+l)K+2bLf+r'{7t- + {2a + l)K'+2bL'y 

 e IH (m, n) = 2,{ — i) € , 



e Q' (m, m')=2( — 1) (the same expression), 



?'?/ I 7* ilf ft 



e iQ" (m, i£')=2( — 1) (the same expression), 



ru''-\-r'H'- 

 e Q,"'(u, u')=% . 1 . (the same expression), 



r«2+»•'K'^„ . ,, ^, -..a+b r(u+2aK+{2b+l)Ly+r'{u'+2aK'+(2b+l)L'f 



f lH (M, ?t):=2( — i) € , 



ru'- + r'u'^..^. , ,^ , -, i ,,, .V 



e iii (m, m)=S( — 1) (the same expression), 



e B"(i<, t«')=S( — 1) (the same expression), 



e R"'(m, m')=S . 1 . (the same expression), 



^rzc' + 'm'^ g ^^,^ _.^^a+bj{u+{2a-\-l)K+{2b+l)Ly+r'{u'+{2a+l)K'+{2b + l}L') 



ni--\-r'u''^ b 



e iS' (m, «') = 2( — 1) (the same expression), 



e iS (i(, i6)=2( — 1) (the same expression), 



e S"'(tf, w') = 2 . 1 . (the same expression). 



where S applies to («) and (b) and extends from — ck to +qo . 



