Se 



V 



336 REPORT — 1873. 



(remembering that Se'^^*""+''^ vanishes, except when /i=0, when it be- 



n 



comes unity) 



V 



which is what we want to prove. 



Taking now the general case for elliptic functions, we have 



0(w) = e(m'M+a' : s<iO0K''«+a'" = s^'O • • • .B{'nu+u : s") 



=^At + !!_i(m(»a<'> + wi'=W='+ . . . .mV)-^(m<'>s^'>+ . . . .mV)\. 



It is easy to develop this expression by means of the principles already laid 

 down ; and we have, finally, 



a -1 



./2w(i)«+^-^-^^(»w'i>«(i'. .wV) -2'?^l^l(;„(i ),(!)+ . .w(^)s^) +2a'+2siri, , + v'"r,, i)^/ 



e 



Se 



X 



Putting in this expression t\=m^^\' + n'^> , .'f^'=m'^V + /i<'>, r''> = m'3V+/x<", 

 where /x'^' is less than m^^K . . . , we see that the expression vanishes, except 

 when /x<^' = 0, ju'-'=0. . . ., and that consequently the expression takes the 

 form Cd(ru, tt^ ^). Another theorem for <p{u^v,^. . . . m ) is given by Konigs- 

 berger in this section. 



Section 5. — Konigsberger here gives two series of hyperelliptic functions, 

 and proposes to determine the coefficients of the second series in such a way 

 that they may be expressed rationally by means of the first. It follows as a 

 consequence that the periods of one set of these functions can be expressed 

 linearly in terms of the periods of the other, the coefiicients in these linear 

 relations, however, being subject to the condition 



Section 6. — Konigsberger then proceeds more immediately to the transfor- 

 mation of functions 6, the expression of 



0(n!tj nwpn,^ j. . . , nrp.p) by d{u^ t/.p, pr^^ ,. . . . rp, p)a. 



