BLACK. — THE NEIGHBORHOOD OP A SINGULAR POINT. 323 



and arrive at the form (42) required. Now any further transformation 

 of type (40) in which the 8 is not zero will leave the rj variable present 

 only in the E factors, so that the general term (after the first) of the 

 function Xv is of type 



r^^(^".,OC~*' ^--=2,3, m. 



Suppose, after this, there are p transformations of type (40). Then the 

 corresponding term after the factor ^™'* has been divided out is 



and if this is of degree not less than m, as it must be if we are not to 

 secure reduction, we have 



m — s -\- q^ — ps > m 



or p S J 



~ s 



thus securing an upper limit for the number of transformations of type 

 (40) which do not give reduction of singularity. Accordingly, after the 

 form (42) is reached, it is only when all later 8's are zero that we are 

 not sure of reduction.* 



5. A sufficient number of transformations of type (43) applied to (42) 

 secures either 



1) reduction of singularity, or 



2) the condition that for some term (the rth) of the X factor 



s = 2, 3, m. 



If, for any term 



a transformation of type (43), after the factor C™ has been divided out, 

 yields 



decreasing the exponent of ^ by r — p^. This decrease takes place at 

 every such transformation, and thus the exponent of ^ must finally be 



* We do not lieed to consider the possibility of liaving all the coeflBcients of the 

 powers of |^ lower than the m-th vanish, for then the function X^ would have 

 m equal factors 1^ and this case has been excluded. 



