THE PARAMETRIC REPRESENTATION OF THE 



NEIGHBORHOOD OF A SINGULAR POINT 



OF AN ANALYTIC SURFACE. 



By C. W. M. Black. 



Presented by W. F. Osgood. Received September 9, 1901. 



INTRODUCTION. 



A. — Outline of Kobe's Treatment of the Problem. 



The problem of the representation, by a finite number of parametric 

 formulae in two variables, of the neighborhood of a singular point of 

 an algebraic surface is considered and alleged to be solved in an article 

 " Sur la theorie des fonctions algebriques de deux variables," * by Gus- 

 tav Kobb. A brief outline of Kobb's method follows : — 



L Treatment of the Original Singular Point. 1) Let the equation 

 of the surface be written in the form 



where F is a function of the three independent variables x, y, z analytic 

 in the point x =^ a, y ^= b, x ^= c. The function F is transformed by 

 means of a change of axes to the form 



* {^, V, = (i, V, 0,n + (.', V, 0,n+l + = (a) 



where the expression (^, rj, ^)„ is a homogeneous polynomial of degree 

 ??, the resulting surface (a) having the singular point considered at the 

 origin, while the function ($, r/, ^),„ is of a form convenient for later 

 treatment. 



2) By the quadratic transformation 



$ a, y,, = C" [(r, <T, !),„ + C (r, cr, l)^^, + ] "^ 



= C'[<f>(r,a)+Cx(r,cr)+ ] V (b) 



* .Journal de mathematiques pures et appliquees, 4th Series, Vol. VIII. (1892), 

 p. 385. 



