[harkness] study of ELLIPTIC FUNCTIONS 91 



rities other than poles, except at u = oo . There is a certain neighbour- 

 hood of u=o, of radius Rq, to each value u of which corresponds one 

 and only one place (x, y) ; to u = o corresponds (a;, o). For the 

 branch place (a;, o) we use the parameter t, where 



t^ 

 """^i^ R' (aO' 



then y = x^R[x) = -J R (ai + 7Û7^) = v't^ + terms in t*, t\ etc. 



= t [1 + t^PCt^)] 



By insertion of these values for x, y in the integral, we find that 



u = Cit+C2t^+ , where Ci4:o. 



Reversion of the series for u in terms of t, gives a series for t in 

 terms of u in which the first power of u is present. When this is sub- 

 stituted in the equation connecting x, y with t, we have for x, y two 

 power series in u, say 



x = $ (u) ,y = ^ (u), 



which give the place (a;, o) when u=o, and which satisfy the dif- 



dx 



ferential equation ^ = y identically when | u | does not exceed Rg . 



These one-valued representations of x, y in terms of u have only 

 a limited range of validity in the u-plane. But they can be replaced 

 by others with an arbitrarily great range by the use of Abel's theorem 

 (or, what comes to the same thing, Euler's theorem of addition). 

 For let R be an arbitrarily great positive number and let the positive 

 integer n be chosen so that R <. n R,,. Then 



^-(t)--"(t) 



are power series in u which converge when | u | < R, since they con- 

 verge when |u| =nRo. They satisfy the algebraic equation 2y- = R(^); 

 they also satisfy the differential equation 



and reduce to the place (aj, o) when u=o. 



r^^V tlx , r^^r] dx rx,Y ax -r. r 1 



Hence u=n / ' — and n / ^ ' — = / — • It fol- 



J a„o y J a„o y J a,,o ^ 



lows from the addition theroem that x, y are rational functions of ^, t], 

 and therefore are expressible as quotients of power series, which con- 

 verge when lui = R. From the mode of construction the numerators 



