ON ELLIPTIC AND HYPEKELLIPTIC' FUNCTIONS. 345 



The equations we have just written down enable us to determine K^ ^K'^ ,, 

 &c. in terms of r^^ j, r^ j, r^ j. Hence also r\ j is known in terms of r/ ,, 

 '"i, 2' ''a, 2» ^^^ '■'i, 2 ' '■'2, 2 "^^^ ^® determined in a precisely similar way. The 

 remainder of the paper is occupied with the discussion of special cases, upon 

 which I shaU not enter, as Konigsberger has gone minutely into details. 

 There are two other papers by Konigsberger on the transformation of hyper- 

 elliptic functions in the seventieth volume of CreUe, which we hope to con- 

 sider in the supplement. 



At the commencement of his paper Konigsberger aUudes to a paper on 

 transformation by M. Hermite, in the ' Comptes Reudus ' for 1855, from 

 which I make the following extracts : — 



Let a^a^a,^a^, hgbjb.^h^, c^c^c^c^, d^c\d^d^ be a system of entire numbers satis- 

 fying the equations 



«o^i + Vi-Co^ -d,a, = 0, 



^A+\<''3-cA -f^«3=o, 

 '^2(^2+^(^3 -<^A -^^'3=0; 



also let 





then, if z- denotes the linear function UiX+b^y, where i is one of the numbers 

 0, 1, 2, 3, and we assume 



e(z^ + G23 -f- Hz„ z^ + HS3 + G'zy^'i^o^^ +^1^2) 



then 



Uix+1, y) = ( -lf^in(a-, y), n(^, 2, + l) = (_l)«in(.r, y), 



n(x + h, y+9')=(-lfm{.v, y)e-^^K^y+9'), 

 n{x+g, y + h) ={-lfm{x, y),-i^K^^-+9)^ 



where g, h, g' are certain ascertained functions of the above quantities, 

 a, h, c, d, G, H, G' and m^, n^, ^j, q^ certain ascertained functions of the 

 quantities a, h, c, d, jx, v, p, q. 



And the method of transformation consists in introducing sixteen func- 

 tions, 0'^^ analogous to Q, but in which G, H, G' are replaced by g, h, g', and 

 then in employing the above relations to express U.(cc, y) hj entire and 

 homogeneous combinations of these sixteen functions. 



I wish to remark that the proofs of Dr. "Weierstrass's theorems, given in the 

 Brighton volume, were obtained by me in the course of the year 1867. I 

 had no assistance, except that derived from the Memoirs themselves. 



