318 PROCEEDINGS OP THE AMERICAN ACADEMY. 



For suppose such a term to be aljrjjt, h , where /+ g < m. Then, by 

 a succession of p transformations such as defined, we have 



L = £, p $ v+ p, V" = £ p yv+pj 



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

 this in the expression above we get 



a? v+P V,,+p£ 

 But we must divide out of this £ mp , so that we have left the term 



n t } J yh+piZ-hg-m) 

 "^v+pVv+p^ 



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

 for if we had another term b$* rj g £\ we should get 



7 >/ 9 yk+p{f+g—m) 



o? v+p v v+P £ 



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+g + h + p(f+g — nij^rn 



or (p + 1) (m — /— g) ^ h, 

 and as m > f + g 

 h 



+ 1^ 



»» — /— g 



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

 leave the term and the singular point of the mth order. 



So it is only in the case in which all terms of Q (£„, r/ v , Q are °f 

 degree not less than m in £" and rj v 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 (0) we take 



lv = a? a , Vv = z-i, t = y-i, 

 the singular line being 



^ = 0, |„ = 0. 



There is in D, (£„, rj v , £) a term in £ v m , and so the expression q (y 2 ) 

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

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



