304 PROCEEDINGS OP THE AMERICAN ACADEMY. 



Any point in T'can be carried by a suitable transformation into a 

 point on one of the surfaces g^z= or g^ — 0. Let G be an arbitrarily 

 chosen (large) positive quantity; then any point of 7' fur which 



\^\<G\^\, \v\<G\i\, |C|<8, 



is carried by the transformation (4) into one of the neighborhoods con- 

 sidered on the surfaces ^ ^ 0. 



If ^ = 0, but ^, r] do not both vanish, then the point (^, i], s) is car- 

 ried by (5) into one of the neighborhoods considered on the surfaces 

 ^. = 0. 



§2. 



A. — A Lemma. 



1. The treatment of the multiple curves of Case II depends on the 

 following 



Lemma. — Given an analytic surface 



^{x, y, s) = <^(a;, y) + «*(a:, y, «) = o, (a) 



where 



is a multiple curve ; let <f>{x, y) have the form in the neighborhood of the 

 point X = 0, y = 0, 



^{x,y) = [x+p{y)YE{x,y), (/?) 



where p (y) is analytic at the point y = 0, and p (0) — 0. The function 

 ^ (x, y, z) shall be analytic at the point (0, 0, 0), but shall not be divisi- 

 ble by X + p(y) at that point. Consider a region for which \y\ <, h, 

 and let h be chosen 



a) less than the radius of convergence of the Taylor's series which 

 represents the function p (y) developed about the point y = 0, and 



h) sufficiently small, so that the points (x = p (y), y) will lie in the 

 region in which E(x,y) is analytic and different from zero. Then the 

 part of the neighborhood of the curve 



x + p(y)--=0, z = 0, 



which lies on the surface 



^{x,y, s) = 



can be transformed, by means of quadratic transformations of the type 



