BLACK. — THE NEIGHBORHOOD OF A SINGULAR POINT. 299 



whole set of functions thus determined will represent all points of the 

 orififiual neighborhood for which 



hl<3, Kl<s. 



The new set of singular points may or may not be all of degrees lower 

 than m, but if they are we have simplified the problem ; we have reduc- 

 tion, as we shall say, borrowing a term frequently used in the theory 

 of algebraic invariants of a linear transformation ; and if not, the further 

 treatment will be considered later. 



D. — An Example. 



Before taking up Case II, however, we consider an example in which 

 the degree is reduced by one quadratic transformation, and the para- 

 metric representation {A) is at once secured. 



Let the surface be 



The transformation 



secures for the equation corresponding to (3) 



Here 



<^(f,^) = ? + ^^-l 



and the critical points are 



i==0, ^ = -1. 



Let 



and we have 



Hence 



Also let 



and we have 



il = $, r]i = Tj — I , 



Vx=-l + V$ia-L) + 1. (a) 



ii = h Vi = V+^) 



rj, = l-^/i,{^-i,) + 1. (b) 



In (a) and (b), only that branch of the radical is taken which becomes 

 + 1 for zero values of the arguments. 



