BLACK. — THE NEIGHBORHOOD OF A SINGULAR POINT. 327 



For if a transformation of type (40) or (41) in which the S or c is not 

 zero were used, we should have in all the coefficients of X^ in (42), out- 

 side of the £J factor, only powers of one variable. Suppose it to be ^ ; 

 then, by means of a succession of transformations of type (46), we can 

 reduce some term to a form in which the exponent of C is less than r, 

 and thus secure a reduction of singularity. 



10. A sufficiently long succession of transformations of types (49) and 

 (50), applied to surface of type (42), unless it first secures reduction 

 of singularity, will secure the condition that, for some term (the rth), 



s = 2, 3, m. 



r s 



Consider the two terms 



Any transformation of type (49) leaves the p^ and pt unchanged, and 

 increases the 



qs hy Ps — s, 



qt " Pt — t. 



Any transformation of type (50) leaves the q, and q^ unchanged, but 

 increases the 



Ps by q, — s, 



Pt " qt — t- 



Represent 



q, — r 



r = 2, 3, m, (51) 



So, for each transformation of type (49) the K^ is increased by the IT,., 

 and for each transformation of type (50) the 11^ is increased by the K,.. 

 We shall show that finally we must have one of two conditions 



«) n, > n„ K. > K„ 



b) n, < n,, K, < K,. 



Suppose, at any stage, neither of these conditions holds, and we have, 

 for example, 



n, > n„ K, < K,. (52) 



