BLACK. THE NEIGHBORHOOD OP A SINGULAR POINT. 287 



present article, supply the deficiency, and treat at once the more general 

 case of an analytic surface, i. e., the case that the function F (x, y, z) is 

 not merely a polynomial, but is any analytic function which vanishes 

 at the point (a, b, c.) 



§ 1 

 A. — The Fundamental Theorem. 



1. The theorem, the proof of which forms the subject of this article, 

 is the following. 



Theorem: Let F (x, y, z) be a function such that 



1) F (x, //, z) is analytic in the three independent variables in the 

 neighborhood of the point x = a, y = b, z = c ; 



2) F(a, b, c) = 0; 



3) (— \ =( 9 ~) =( — ) =0- 



\dzj[a.b,c) \5y/(a.6,c) \dzj[a,b,c) 



then we can represent all values of (x, y, z) satisfying the equation 



F(x,y, *)=0 



and lying in the neighborhood of the point (a, b, c) : 



\x — a | < S, \y — b\<8, \ z — c \<$ > 



by a finite number of parametric formulae of the following type : 

 x = <f> p (u, v) 1 



y = <A P ( M > «0 y p = 1, 2, p, (A) 



z = Xp( ?/ ' v ) J 



where t/> p , if/ p , \p are analytic in the arguments (u, v) throughout a cer- 

 tain region ; further for each set of values of (x, y, z), the values (0, 0, 0) 

 excepted, there corresponds for at least one value of p a pair of values 

 (u, v) lying within the region in which the functions <f) p , \p p , x P are con- 

 sidered, and for any value of p for which this is the case, there corresponds 

 no second pair of values. To the set of values (0, 0, 0) corresponds at 

 least one, and in general an infinite number of pairs of values (u, v) for 

 every value of p. 



2. Explanation of Symbols. The symbol (x, y, z, )„ indicates, 



in the expression in which it appears, the total collection of terms 

 of degree n in the arguments taken together, which belong to that 

 expression. 



