BLACK. — THE NEIGHBORHOOD OP A SINGULAR POINT. 311 



1) neighborhoods of singular points of transformed surfaces ; 



2) regular pieces of transformed surfaces. 



The pieces of class 2) lead at ouce to representation by means of para- 

 metric formulae of type (A). The singular points of class 1) are all 

 of lower order than the original singularity except in one case, and it 

 is this case that it remains to consider in §§ 3, 4. The case can pre- 

 sent itself at the outset only if the polynomial (i, rj, ^),„ is the product 

 of m linear factors in ^, rj, ^, all vanishing for a single set of values of 

 the arguments $, tj, ^ not all zero. Geometrically, the tangent cone, 

 (t) v^ Om = 0? of the surface <J> (^, 77, ^) = at 'the point (0, 0, 0) con- 

 sists of m planes having a common line of intersection. It is found 

 necessary to distinguish two sub-cases according to whether the planes 

 themselves are not all coincident, or are all coincident. 



To sum up, then, we already have reduction in all cases except when 

 ■we are led to singular points in class 1) of the particular type just 

 described. 



§3. 



A. — The Singular Points of Special Type. 



1. In the special case in which the function (^, 77, Om is composed of m 

 linear factors, each vanishing for all points on a common line, it is possi- 

 ble to reduce the singularity by means of a finite succession of quadratic 

 transformations together with certain additional transformations. 



We consider two cases : — 



Case A. — The m linear factors of ($, -q, ^)„, are not all equal. 



Case B. — The m linear factors of (f, r/, ^)„j are all equal. 



2. Case A. — (|, -q, ^),„ is composed of m linear factors not all equal. 

 The surface can be expressed in the form 



4>(|, rj, = (i, V\> + (f, V, 0.«+i + = (13) 



where ($, rf)^ contains terms in both f" and rf". 

 If the surface were in a form 



f(u, V, w) = (u, V, iv)^ + (u, V, w)„+i + = 



with the condition that the m linear factors of (m, v, w)^ all vanish for 

 the line 



u = ato, V = ftw, 



we could make the transformation 



