426 On the Transformaiion of Elliptic Functions. 

 where ,r/,,K)ff.,"| ^,tf.fj 



a = l.{uB'- SA'), b= _±(«B-gA), 



p p ^ ' 



The coefficients «, i, «', 6' are integers, as is obvious from 

 the equation ju, (a B' — § A') = jo (L' « — / A'), and the others 

 analogous to it ; moreover, «, V are odd and a\ h are even, 

 and 



a i' - a' ^ = 4- ( A B' - A' B) (a g' - «' g) ; 



i.e. aV — cdh ■=■ \. 



Hence the theorem, — " The general values aj', o' of the 

 complete functions are linearly connected with the particular 

 system of values O, U by the equations, w' = « O -f 6 U, 

 y' = a' O + i' U, in which a, V are odd integers and «', h 

 even ones, satisfying the condition ah' — a' b ^= 1." 



With this relation between O, U and w', u', it is easy to 

 show that the function (p^ x is precisely the same, whether O, 

 U or w', y' be taken for the complete functions. In fact, sta- 

 ting the proposition relatively to (p.r, we have, — " The inverse 

 function ^ x is not altered by the change of w, y into c«', y', 

 where co' = a a> + ^ y, y' = «' «> + S' u, and «, &, «', S' satisfy the 

 conditions that a, §' are odd, «', § even, and aS' — a' § = I." 

 This is immediately shown by writing 



m CO + n V = m' oo' -{- 7i' v', 

 ox m = Tu' a + w'a', 



n =■ m' S -{• n' S'. 



It is obvious that to each set of values of m, n there is a unique 

 set of values of m', n\ and vice versa: also that odd or even 

 values of w, m' or n, w' always correspond to each. It is, in 

 fact, the preceding reasoning applied to the case of p = 1. 



Hence finally the theorem, — " The only conditions for de- 

 termining w', v' are the equations 



oo' = — (a w + S v). v' = — («'«; + & u). 



where a, §' are odd and a', § even, and 



u^' — u' ^ =: p^ |U< §' — V «' = y Pj ju, § — V a = Ip, 

 I and V arbitrary integers: and it is absolutely indifferent 

 what system of values is adopted (or J, y', the value ot (pyX 

 is precisely the same." 



We derive from the above the somewhat singular conclu- 

 sion, that the complete functions are not absolutely deterrai- 



