318 PROCEEDINGS OP THE AMERICAN ACADEMY. 



For suppose such a term to be aljrjj'l'', where /+ g < m. Then, by 

 a succession of p transformations such as defined, we have 



^y = ^i'$v+p-, V''='Q'-nv+p, 



(derived from form of (19) when all y's and 8's are 0). Substituting 

 this in the expression above we get 



But we must divide out of this C^'', so that we have left the term 



" t v+p Vv+P *= 



This term could not combine with any other derived in a similar way, 

 for if we had another term 6 J-^t;^^*", we should get 



7 >/ 9 c-k+plf+g—m) 



0$y+pVu+pC 



and this would not combine with the other unless k = h. Now, if the 

 degree of the singular point is not reduced, we must have for the sum 

 of the exponents 



f+ 9 + h + p(f+ g — m) ^m 



or (p + l)(m -f—g)S h, 

 and SiS m > f -{- g 

 h 



+ 1^ 



m —/— g 



thus securing an upper limit for p, the number of transformations which 

 leave the term and the singular point of the ?«th order. 



So it is only in the case in which all terms of Vl (tV, y]v, ^^^ ^^ 

 degree not less than m in I" and 7]^ together that we do not have a re- 

 duction of singularity by the succession of transformations of type (14). 

 But, in this exceptional case, we have the conditions of the Lemma of 

 § 2, where in equation (6) we take 



the singular line being 



^. = 0, 1 = 0. 



There is in Q (IJ,, rj^, ^) a term in H,*", and so the expression go (y^ 

 does not vanish when ^2 = 0. Accordingly, within a neighborhood 

 about this point, we can break up the singularity by the methods of 



