BLACK. — THE NEKJHBORHOOD OP A SINGULAR POINT. 321 



3. The quadratic transformations to be used are of two types 



1) $^ = $n+iC^, ^,a = (^^i+i + S^+i) ^Mi (40) 



2) 1^ = $^+iv^, i^ = (^,.+1 + ^)^+i)Vn' (41) 



In a succession of transformations of type (14) we see that yi = 0, 

 since the first set of points is taken on the line f = 0. Further, sup- 

 pose after the substitution rj — 8i = rji'm ^ oi (39) the expression 



(f, Vl Om 



contains terms besides the J™ ; then it cannot be composed of m equal 

 linear factors, for that would require a term containing f""^ ; but no 

 such term can arise from the factor X of (37), and, on the other hand, 

 it could not be the product of a term from X by a non-constant term of 

 the ^ factor, for then, on account of the constant term of the -E factor, 

 there would have to be present in <5 a term of degree lower than m. So 

 as soon as the function corresponding to <^ of $ contains more than the 

 »ith power of the $ variable, the function corresponding to (f, rj, t,)„^ is 

 no longer the product of m equal linear factors, and we have one of the 

 cases treated earlier. 



The same considerations apply to the transformations corresponding to 

 type 2), since, when the transformation which deals with the infinite 

 region is introduced, the first one of that order is of form 



Accordingly, the most general succession of transformations here is 

 one in which groups of types 1) and 2) alternate. We shall call them 

 the t, and rj types respectively, and when a change is made from one type 

 to the other, we shall speak of it as a reversal of type. 



We shall treat the subject in two cases, first supposing that there is 

 no reversal of type in the succession of transformations used, and later 

 supposing that reversals of type occur. 



C. — Succession of Quadratic Transformations in which 



THERE IS NO REVERSAL OF TyPE. 



4. After a sufficient number of quadratic transformations the surface 

 can be reduced to the form 



[(C + ?^*^^(^- 0C~'+ • • • + ?;^'''^(^- 0']E{^.. v^, 0, (42) 



VOL. XXXVII. — 21 



