298 PROCEEDINGS OF THE AMERICAN ACADEMY. 



regarded as an equation in £/r], has equal roots. Thus — is the largest 

 value of t) for which the equation 



has a critical point. So the functions are analytic and give all points 

 of the original neighborhood for which 



I- 



< hi — «n \v\ < 8 2> 



or for which 



/ > jt^— = t + rrr 1 — \ = ** + «*» f*> = r)» 



thus securing the limits 



\v\ < Sai Ul < S 3 , U| > (^ + e 2 )|4l, 



where A 2 is the distance to the furthest point in the r/-plane for which the 

 equation 



has a critical point, and if e 2 is first chosen arbitrarily small, 8 3 can be 

 determined not zero. 



Now consider the neighborhoods of the critical points of the curve 



*(?, v) = 0. 



In these, however small we take the 8, all the remainder of a circle in 

 the 77-plane including all the values for which the curve cf> = has 

 critical points can be covered with circles such as were determined for 

 the domains of the regular points above, these circles overlapping the 

 circles about the singular points and not reaching out to these points in 

 any case. Let the radius of the large circle be G where 



G > 1 , G > h, + e 2 . 



Then, if we take for 8 4 the smallest value of any ^ or S", the develop- 

 ments within these circles together with the neighborhoods of the set 

 of new singular poiuts will represent all points of the original neigh- 

 borhood for which 



Finally, taking for 8 the smallest of the three quantities $ 2 , 8 3 , o 4 , the 



