BLACK. — THE NEIGHBORHOOD OP A SINGULAR POINT. 313 



in which the plane ^ = 1 cuts the planes corresponding to these factors 

 is at a finite distance. We have now the conditions 2) and 3) of § 1, 3, 

 and are ready to apply the transformations 



giving 



^i (iu Vi^ = C' [(li, Vu 1)- + C(li. Vu l).+i + ] = 0. (17) 



Now if f] = y2, 7/1 = 8oj is the common point for which the m factors of 

 (li> Vi) ^)m vanish, then the substitution 



^2 = ?1 — 72 J V2 = Vl ~ ^2i 



gives a group of mth degree terms in ^^ and r]^ exactly corresponding to 

 the terms of ($, t])„. So in the successive collection of terms of the 

 mth degree, the terms of (i, r])„ are always carried over with merely 

 a change of subscript, and thus we never introduce the condition of m 

 equal linear factors. Accordingly so long as the degree of the singular 

 point is not reduced, the intermediate transformations are of the type 



thus securing the succession of transformations (14). 



4, The succession of transformations in 3 will lead to the relation 



e^L (^„ r,^ ^^ + ^{U V^^O ^'"-'"' ^ = Ji(yj,^)$0 (18) 



where 



Combining transformations (14) we have the relations 



^ = 71^+72^^ + + 7.^' + ^"$.) 



r) = 8U+ 82^' + + 8,C + ^"7?. \ ^^^^ 



As $ contains both ^^ and rj"' terms, we can develop by Weierstrass's 

 Theorem. 



a>(^, rj, = [e"+p,(v, oe'-' + +Pn.{v,o:iM^, v, o] 



^(i, V, = [r + qid. Ov""' + + qm(i, 0]^2(l, V, 



As the function E-, (f, -q, ^) contains a constant term, when the first trans- 

 formation of (14; is made, the factor C" must come out of the <P, and a 



(20) 



