BLACK. — THE NEIGHBORHOOD OF A SINGULAR POINT. 299 



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

 original neighborhood for which 



h| < 8, |C|< 5. 



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

 than ?n, 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) 



*(6^0 = P + ?-l-?C = O. 

 Here 



(£$ = ? + ?-l 



and the critical points are 



1=0, 5=1, 



?=0, , = -l. 



Let 



and we have 



Hence 



Also let 



and we have 



d = €> Vi = V ~ l t 



^ 2 + ^ 2 +2t 7i -^C=0. 



m = -i + Vti(t-&) + i- ( a ) 



£2 = l> f]i = V + ! > 



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

 + 1 for zero values of the arguments. 



