[harkness] study of ELLIPTIC FUNCTIONS 87 



Suppose that Uj is a singular point of the kind just considered at 

 which X (u) becomes indeterminate and that it bars the hitherto un- 

 interrupted process of continuation along a line 

 that started from u^. As we approach Uj the radii 

 of convergence tend to zero. Accordingly we can 

 find points on L such that circles with as small 

 radii as we please shall contain u ^ in their interior. 

 Such a circle includes therefore a point at which 

 X (u) is indeterminate. This conflicts with Picard's theorem that we can 

 always describe, from an arbitrary point u^ as centre, a circle with a fixed 

 radius, such that within this circle the integral u shall be one-valued. 



II. Method based on conformai representatio)i. 



When aj, 0,2, ag, a^, or a, b, c, d as we shall now call them, are real, 

 there is no difficulty about effecting the conformai representation of T^ on 

 a rectangle. The two branch-cuts must lie along straight lines connecting 

 branch-points if the sides of this rectangle are to be sti aight lines. The 

 case where a, b, c, d lie on a circle is easily reducible to the above case. 



In the case where a, b, c, d do not lie on a circle it is easily 

 seen that the contour of T^, composed of the two banks of A and 

 the two banks of B, along which the values of u differ by 2oj^ 2(ii^ 

 respectively, must map on to a figure composed of four curves. The 

 curves that arise from the banks of A are congruent with regard to 2w; 

 i.e. the one is derived from the other by a translation 2o). A complica- 

 tion that arises is the possibility that a curve may be cut by another 

 that is congruent to it. In such a case no region would be afforded by 

 the curvilinear parallelogram. This difficulty would be avoided by 

 choosing the cross-cuts A, B so that they shall map into straight lines. 

 That this can be done was shown by Schwarz in his memoir Conforme 

 Abbildung der Oberflache eines Tetraeders auf die Oberflache einer 

 Kugel (Ges. Werke IL p. 84) This memoir is concerned primarily 

 with a problem which does not concern us here and deals with an integral 

 u which contains the elliptic integral of the first kind as a special case. 

 We shall select from it those parts which provide the material for 

 effecting the inversion and add the necessary complements to complete 

 the solution of the inversion problem. 



Let u-u,= /;(,_,)«-! (,_,/-! (x-<.)^"'(x-d)'-'dx 



where a, b, c, d are distinct complex numbers, and a, /3, 7, ô are real 

 numbers lying between and 1 and such that a + /3 + 7 + 5=2. 



From a, b, c draw lines L (ad), L (bd) L (cd) which shall not cut 

 themselves or one another. Let the plane of x be cut along these lines, 



