[harkness] study of ELLIPTIC FUNCTIONS 89 



lie wholly in the negative half-plane, and let X, for the present, refer 

 to the dissected plane with these cuts L (o, z), etc. 



For all values of x interior to X the functions log du/dx, logx, 

 etc., that enter into the equation 



, du 

 ^°S d^x = 



i [logx + log (x - 1) + log (x - z)J 



are one-valued and continuous functions of x. 



Schwarz uses the theorem to prove that the positive half-plane 

 of X is thereby mapped on a simply connected region, U (o, 1, oo). or Uj, 

 which forms a part of the complete map of X; and the boundary of 

 U (o, 1, 00 ) is everywhere convex to the outside. 



The proof is simple. Let x proceed along the real axis from - oo 

 to -h 00 , then u traces out the contour of U i . To an element dx 

 corresponds an element du and the angle 6 between dx, du is the angle 

 d of slope of U 1 at the point at which the element du is situated. From 

 the equation 



du 

 dx 



du 



dx 



du 

 it follows that ^ =the coefficient of i in log -r-. We can therefore follow 



dx 



the variation in the angle of slope by following the variations in the 

 coefficient of i in - ^ [log x + log (x - 1) + log (x - z)]. These 

 variations are entirely due to the third^term, except at x=o and x = l 

 where there is an abrupt increase in each case of 7t/2. The vaiiation 

 due to - ^ log (x -z) is a continuous increase. When x describes the 

 whole of the real axis beginning at - oo , passing successively through 

 0, 1 , + 00 , and ending at - oo , the point u describes a curvilinear 

 triangle Ui whose angles =7r/ 2 and whose sides are everywhere convex 

 to the outside ; and the total increase in the angle of slope = 27r. Hence 

 the contour of Ui cannot be intersected by a straight line in more than 

 two points. 



Returning to the integral in the form which it had before it was 

 subjected to the bilinear transformation, the positive half-plane must 

 be replaced by the region interior to, or the region exterior to the circle 

 through a, b, c; the interior region if d lies without the circle, otherwise 

 the exterior region. Call this region of the x-plane (a, b, c) and 

 let its u-map be called U (a, b, c). In U (a, b, c,) we have a 

 curvilinear triangle which is everywhere convex to the outside; its 

 angles are right angles. 



Similarly if we had used a, b, d instead of a, b, c, we should have 

 obtained a region (abd) in the x-plane, which is bounded internally 

 or externally by the circle through a, b, d, according as c lies outside or 



Sec. Ill, 1913—6 



