84 THE ROYAL SOCIETY OF CANADA 



term. Then u — u^ in each case takes the form of a power series 

 beg;inning with the first power of t. By reversion of series, we see that 



t = Ci (u — u') + Cj (u — u')- + (Ci+o) . 



It follows at once that x — x^ = a power series in u — u\ no matter 

 whether x^ belongs to an ordinary place, or a branch-place, oi is situ- 

 ated at 00. There is a similar conclusion for y — y'. 



The fact that in all cases u — u' = P (t) shows that u is every- 

 where finite on the Riemann surface T, and also is a holomorphic function 

 of X in the neighbourhood of the place (x\ }•'), in the sense that it a 

 holomorphic function of t in the neighborhood of t = o, where t is 

 the parameter attached to (x\ y^). The fact that in all cases x — x^ = 

 P (u — u*) shows that x (u) is also a holomorphic function of u in the 

 neighbourhood of u — u^ 



What is established is that the surface which arises from T^ by 

 means of u must cover only a finite part of the plane, and that this 

 surface has no branch-points. We have not proved that there is no 

 multiple covering of the u-plane. By Cauchy's theory of integration, 

 as applied to the surface T\ u can take only one value u^ at the place 

 (x'j y^) ; but there is no reason a priori why it should not take the 

 same value u^ at another place x", y". In this case x — x^ and x — x", 

 by what precedes, are power series (u — u*) P (u — u^), (u — u") P (u — u"), 

 and the places (u^, x'), (u\ x") on the surface that gives the u-region 

 are covertical (the u-plane being regarded as horizontal). 



I. Method Employing the properties of Differential Equations. 



In this method x (u) is considered in connection with the differ- 

 ential equation / iv\-' 



which is supposed subject to the condition that x=Xo when u=o, and 



that y, or VR (x), = y^ when u=o. Cauchy proved the following 



theorem by means of his calculus of limits: — 



If the differential equation 



dx 



d-^ = f(x,u), 



in which f (x, u) is a holomorphic function of x, u in the neighbourhood 

 of x=X(„ u=Uo, is subjected to the initial condition that x = x„ when 

 u = Uo, it is satisfied by a function x (u) which is holomorphic in 

 the neighbourhood of u^ and takes the form x — x,, = (u — u^) P (u — u^) . 

 If f (x, u) is holomorphic within and upon the circumferences of 

 circles of centres x^ and u^ and of radii R and r, and M = the maximum 



