BLACK. THE NEIGHBORHOOD OF A SINGULAR POINT. 315 



q x =p + n, r t = 0. 



Next, suppose (77, £)„ does contain terms in 77, but no term in 77", i. e. we 

 can express it in the form 



where (77, £)„_,._, contains terms in both ^" _r_5 and £"- r -* and s > 0. 

 Then if any transformation 



is applied, there can be divided out of (77, £) n the factor £" leaving behind 

 as the term of highest degree one in rf*~*. This cannot be cancelled 

 with any term from another part (77, £)„ +i , for any term from this would 

 have as a factor tf after the £" has been divided out. As long, then, as 

 the ?7 variable does not enter to the highest degree in the expression 

 corresponding to (77, £)„ if n > 0, the degree of the S factor is decreased 

 with each transformation, while the expouent of £ outside may be in- 

 creased. Accordingly, by a finite number of transformations, we re- 

 duce the S factor either to an E function or to an expression in which 

 the 7; variable enters to the highest degree in the collection of terms 

 of lowest order. In the former case we have the form required. In 

 the latter case, suppose for convenience that this condition holds for the 

 function £(77, £). By Weierstrass's Theorem we develop in the form 



S(?h = It + niOv"- 1 + + r n (0]£( v , 



= T( v ,0E( V) 0- . (28) 



Consider the n factors of T(rj, £), 



2 ? (^0 = n[, + fx(0]. (29) 



A=l 



If the factors are not all equal, pair them off, so that in each pair there 

 will be two different factors, leaving a number of equal factors : 



fr! + «ta(0] [*+**«)]} {Lv+st h (.Q]tv+su A (01}bi+s»(Qy- (so) 



Now, for each pair, 



^=[? + ^(0]D» + **(o:i, 



we have the relation 



N k +P k ( V ,i:) 9 ~ k = L k tt)$0, (31) 



at] 



since the two. factors are unequal. Then, by the same reasoning as used 



