[harkness] study of ELLIPTIC FUNCTIONS 85 



absolute value of f (x, u) for this range of values in the x and u planes, 

 then the circle of convergence for the power series for x (u) in the 

 theoiem is at least r {1-e-^^-^'). 



This theorem of Cauchy's shows readily that there is only one 

 holomorphic integral that reduces to x,, when u=Uo. This theorem 

 has been supplemented by the further theorem that there are no solutions 

 x= F (u) that satisfy the condition F (Uq) =Xo and are of the following 



type:— 



(a) at all points u^q within a sufficiently small neighbourhood of Ug, 



F(u) - F (u^) = (u - u^) P (u - u^) 



(b) lim F (u^o) = F (Uq) , when u,, tends to u^^ as limit along a path 

 1, which may be finite or infinite in length. [This path may e.g. tend 

 towards Uq along a spiral. The path 1 must, however, be such that e 

 having its usual meaning, there shall be a point on 1 beyond which 

 every point of 1 is at a distance less than e from u^ .] 



(c) F (u) — F (Uq) is not expressible as a power series in u — u^. 

 Another supplement to the theorem takes account of the case 



where f (Xq, uJ is infinite, whilst 1/f (x, u) is holomorphic about x = Xg, 

 u = Uq. In this case the differential equation has an integral which 

 reduces to x^ when u = u,,, and this integral is of the form 



X — Xq = Ci(u — U(,)^^^+ C2 (u — Uo)"/^+ . . .Cj 4= O; h a positive 

 integer. This integral has an algebraic critical point at u = Ug. 

 Furthermore this is the only integral which tends to Xq when u 

 tends to Uq. 



With these theorems in mind let us return to the elliptic differential 

 equation. Starting at the ordinary place (x,,, yj on T, which gives us 

 the lower limit for the elliptic integral of the first kind, apply Cauchy's 

 theorem to the differential 



dx/du = R (x) = y, 



for which the initial condition is x = Xq and y = Jq. When u = o, we 

 see that the ec^uation admits a holomorphic integral x (u) which reduces 

 to Xq when u = o, and whose derivate then takes the value y^. The 

 region of definition of this integral is, at first, only the domain of con- 

 vergence of the power series for x — Xg in terms of u, but this power 

 series can be continued, bit by bit, by the aid of Weirestrass's process of 

 anal3'tic continuation, and these continuations satisfy the diffeiential 

 equation. If every point Uj ( 4= c») of the u-plane can be reached by 

 these continuations, so that Uj lies inside the domain of one of the 

 power series in the aggregate of power series formed by the primary 

 one and those derived from it, then x (u) is one-valued in the finite 

 part of the plane. 



