BLACK. — THE NEIGHBORHOOD OP A SINGULAR POINT. 317 



of 4 shows that the left side becomes divisible by £ (m-1) M. So we have 

 either 



(m—l)/x<q l or (m — 1)^ < y 2 



and thus an upper limit for /x, the number of transformations which 

 leave the order of the singular point unchanged. 



Now, to consider the transformation (33), we see that it is a one- 

 to-one transformation by which the surface remains analytic near the 

 origin. (Dx (£) and w 2 (0 contain no constant term, for otherwise the 

 factor 



rjv + wj (£) or £, + w 2 (£) 



could be combined with the E factor. Then the transformation (33) 

 leaves the E factors still E factors, and the factors vanishing at the 

 origin still vanishing there. Also, it is easily seen that this transfor- 

 mation leaves the terms of type (£, rf) m still in the part (£„, tj v , £) m . 

 Further, if the function «!>„(£„, -q v , £) goes over into X(£ v , rj v , £), we have 



9® v _9X _9X9l v _9X 

 9 £ v 9i„ 9 £„ 9£ v 9 $ v 



and similar conditions hold for the partial derivative with reference to 

 7/ v . Accordingly, if by the transformation (33) <&„(£„, Vv, £) goes over 

 into £2(£„, rj v , £) we replace equations (25), (26), and (27) by 



1 v 

 L v (i V} £, t)X(l, Vv , + M v Q vi y v , 0— - = frvSiEfa 0, (34) 



9t v 



P v (l, vv> QBQ„ £,, £) + &<&, Vv, 0-J?= ^~^E($ V , 0, (35) 



9rj v 

 Q(|*» Vv> = XQv* Vi"> CAlC?** V"> — &(€vj t]v, O-^aClfj Vv, 0- ( 36 ) 



Now, in a further succession of transformations of type (14) on the 

 surface f2 (£„, Vi>> £) — 0, if there enters either a y or a 8 not 0, then on 

 the right side of equation (34) or (35) the only factor remaining outside 

 of the E factor is a power of £, and we must finally have a reduction as 

 shown above. So it is only in the case in which all the y'a and S's of 

 the later transformations are that we are not already sure of reducing 

 the singularity. Now if in £2 (f„, r),,, £) there is any term of degree less 

 than m in £„ and -q v combined, such a succession of transformations must 

 reduce this term to a degree less than m and thus reduce the singularity. 



