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 — or g r = 0. Let G be an arbitrarily 

 chosen (large) positive quantity; then any point of T for which 



\i\< G\t\, \-n\<G\Z\, \C\<8, 



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

 sidered on the surfaces g = 0. 



If '(, — 0, but £ 3 77 do not both vanish, then the point (£, 77, s) is car- 

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



9 T = Q- 



§ 2. 



A. — A Lemma. 



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

 following 



Lemma. — Given an analytic surface 



*(*» y, z ) = £(*> y) + »*(*i y> z ) — o, (a) 



<£(*, y) = 



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

 point x = 0, y = 0, 



<f>(x,y) = [x+p(y)]'»JE(x,y), (/?) 



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

 ty (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 



b) 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,tf, z) = 



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



x =■ xz, 



