On the Transformation^of Elliptic Functions. 425 



^A\«i > u,S' — va! i= I' p • '^^^'^ ^ -^ "^ *' HI 1 1( A 



fxS — V a = Ip, 



/,' ^' being any integers whatever. In fact, to prove this, \^ 

 hrtve only to consider the general form of a factor in tWe nu- 

 merator offpi.r. Omitting a constant factor, this jl<^'' }uWlo(| 



- -- . 1 ,^it((ll:inijl 



1 . ^>i^i-.biii \iu3ryiu.iL^ti& af a^adj ijjcl 



^mco + nv + 2rU ^ ^^^* ^: *. V> 



artdit^fs tci be shown that we can always satisfy the equattfiiirf^ 



or the equations ttG^tmmmm 



^ m + 2 r [ji, = m' a + n' ot'j ' 



pn + 2rv = in'S + n' §' ; 



and also that to each set of values of »z, n, r, there is a unique 

 set of values of m', n', and vice versa. This is done in the 

 paper referred to. Moreover, with the suppositions just made 

 as to the numbers a, €' being odd and a', § even, it is obvious 

 that J7i' is odd or even, according as m is, and n' according as 

 n is, which shows that we can likewise satisfy 



11 11 



m -\- — CO + n + — V + 2r Q = m' + — co' + n' + — o' ; 



and thus the denominator of <pj x is also reducible to the re- 

 quired form. ; ^ ^ _. ,^ j^ 



Now proceeding to the immediate object of this paper, a, €, 

 a', §', and consequently «>', o' are to a certain extent indeter- 

 minate. Let A, B, A', B' be a particular set of values of «, 

 ^, a', &, and O, P the corresponding values of w', y'. We 

 have evidently A, B' odd and A', B even. Also 



AB'-A'B=p, 

 /x B' - V A' = Vp, 

 j«. B — V A = hpf 



O = — ( A CO + B o), 

 P 



U = — (A'w + B'o). 



By eliminating w, y from these equations and the former 

 system, it is easy to obtain '"' '^ 



w' = a O + & U, 

 y' = a' O + 6' U, 

 Phil, Map. S. 3. Vol. 27. No. 182. Dec. 1845. 2 F 



