BLACK. — THE NEIGHBORHOOD OF A SINGULAR POINT. 315 



qi=P •{■ n, ri = 0. 



Next, suppose (t], ^)„ does contain terms in rj, but no term in rj", i. e. we 

 can express it in the form 



where (tj, ^)„_r_s contains terms in both 7]"~''~' and ^-^' and 5 > 0. 

 Then if any transformation 



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

 as the term of highest degree one in f]"''. This cannot be cancelled 

 with any term from another part (rj, C)n+i> foi" ^^J term from this would 

 have as a factor ^' after the C" has been divided out. As long, tlien, as 

 the rj variable does not enter to the highest degree in the expression 

 corresponding to (?;, ^),j if n > 0, the degree of the S factor is decreased 

 with each transformation, while the exponent of ^ outside may be in- 

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

 duce the S factor either to an £ 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 S (rj, C)- By Weierstrass's Theorem we develop in the form 



S(v, - Iv" + niOv"-' + + r„(0:\E(r], 



=^T(r],OE(r,,0- (28) 



Consider the n factors of T(r], Q, 



T(r,, Q = Ulrj + S,(02. (29) 



A X 



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 : 



Now, for each pair, 

 we have the relation 



^. + P.iv, oi^' = L^O $ 0, (31) 



drj 

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



