340 REPORT — 1873. 



Section 8. — This section is very short, and contains some formulse for 

 transformation when the moduli are doubled. 

 From the equation 



e(u^ + v^ Up + Vp, ri,i. . . .rp,p)0(M, — I', ■Wp — V T,,i rp,p) = 



2(2h,. . ..2up, |. . . .'-|, 2r:.,. . . .2rp,p)6{2v^. . • •, | • • ■ • 2r,.,. . . . ) 



is deduced by means similar to those used ia the last section, 

 fl(2w, .... 2u^, 2r,, , . . . . 2rp_ ^0(2., .... 2.^, 2r,^ ,. . . . 2r^_ ^) 



and from this equation one or two other expressions are derived. 



In section 9 the application of these principles is made on a more extended 

 scale to h vperelliptic functions of the first order ; as, however, this is pre- 

 sented in a more developed state in the sixty-fifth volume of CreUe's Journal, 

 we proceed at once to the second memoir, and shall follow, as before, Kooigs- 

 berger's division as to sections. 



Section 1. — We now recur to the equations at the beginning of Konigs- 

 berger's first paper. Patting p = 2, we have 



M,=2K,_jV-F2Kj^,v„ v, = Gj_ ,«, + Ct,_,?i„ 



whence 

 whence 



with similar values for 7',^ 2' ^'n, i' '"'2, a- 



The following notation is adopted in Konigsberger's second paper : 



-R(x)=x(l-x)(l-c'.r)(i-l\v)(l-m'x), 

 E/2/)=.y(l-.v)(l-P.7)(l-XV)(l-/"l'/). 



— p= + /=^- =:clu,, — ,^ — --■■ + /..^ — - =a>i^, 



Vll,2/, VR.?/3 " VR,2/, VE,?/, 



3 



a-^(li\ xj.v^ 



.^R^; +;7l^=^^«'3=W-,+K 



These equations are plainly connected together : and, the usual notation of 

 Dr. Weierstrass being used, we have 



