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 functions algebriques de deux variables," * by Gus- 

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



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

 of the surface be written in the form 



F(x,y,z) = 0, 



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 



* (6 V, = (6 V, Om + (6 V, Om+l + = (a) 



where the expression (£, 17, 'Q n is a homogeneous polynomial of degree 

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

 origin, while the function (£, 77, £),„ is of a form convenient for later 

 treatment. 



2) By the quadratic transformation 



£ = t£ , 7] = a'C, 



$ (£ rj, = tT [(r, a, 1)„, + t (r, <r, 1), )1+1 + ] ) 



= C n i<f>(T,o-) + Z X (T, (r)+ ] (b) 



* Journal de mathe'matiques pures et applique'es, 4th Series, Vol. VIII. (1892), 

 p. 385. 



