ON ELLIPTIC AND HYPERELLIPTIC FUNCTIONS. 329 



which, if 



/(.r) = S^-llT, 



gives rise to the system of differential equations 



V>i V>. ^A ■ ' V>.. 



a-\(?;rj ar^rf^' a;3<ia,'3 .-y,.c?a,' 



Now let 



V/v, V7-1-, V/r, ■ ■ V7a;„ 



(fee. = . . 



where M, N, L are three rational and entire functions of the nth order. But 

 since 



x^, x.^ . . , x„ may be regarded as the n roots of the equation 



(L + N)2/"- + 2M^+(L-N) = 0, 

 or 



L(l+/)+2M*/ + N(l-/-)=0. 



which may be written 



L=Msin0 + ]Srco8 0, 



where is a new variable. Substituting x^, x.,, x.^ . . for x in this equation, 

 we obtain a system of equations which may be regarded as the complete 

 integral of the above system. 



Part III. On the Transformation of Hyperelliptic Functions. 



In considering the papers of Konigsberger on the transformation of hyper- 

 elliptic functions in the 64th and 65th volumes of CreUe's Journal, it will be 

 convenient in this Eeport to foUow his division as to sections. We commence 

 with the paper in the 64th volume. 



Section 1. — Konigsberger assumes the following connexion between two 

 sets of variables : — 



«^=2K,,i^;, + 2Ki.„t',+ .... +2Ki.pt'p, 



^l,=2K,,lV,+2K2,,.^,+ .... +2K,_^v^, 



a= .... 



«p = 2Kp.ii', + 2K,,.,rv+ .... +2Kp_p^p, 



^1= <Jl, l"i + G2, 2«2 + . • ■ • +Gp,lMpi, 

 V^= Gri,2Wi + G2,2^«;, +.-.-+Gp2Mp, 



a= .... 



''p= ^l.p^'i + G-'.p":; + . . . . +t^p,pl'p; 



