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 (&, *, = C" [(& vi, 1). + C(fi. vu i) m+ i + ] = 0. (17) 



Now if f] = y 2 , r;x = S 2 , is the common point for which the m factors of 

 (li> t)u l)m vanish, then the substitution 



& = fi — y2> % = vi ~ ^2> 



gives a group of with degree terms in £ 2 an d 770 exactly corresponding to 

 the terms of (£, rj) m . So in the successive collection of terms of the 

 wth degree, the terms of (£, rf) m 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 



vh = s ~~ vi-i ' vr ~ \ — <V+i * 



thus securing the succession of transformations (14). 



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



e-Z (&, * <*>, + M(U Vv ,£) tf-*» 9 ^ = B( V ,£)$0 (18) 

 where 



**(£,, >to = &*(£», Vv, l)E(Jsv, yv, 0- 

 Combining transformations (14) we have the relations 



£ = yi£ + y 2 £ 2 + + y,P + P&\ 



v = s 1 z + 8 2 f+ + 8 v z» + e> Vv ] 



*(6 v, = C"**(t» n» 



As <I> contains both £ m and rj m terms, we can develop by Weierstrass's 

 Theorem. 



*(*, * = [£" + j»i(* OF'' 1 + +Pm(v,Q]Ei(e, v, 



*(6 * = Df + ?i(& 0v m ~ l + + ?„(*, 0]^(£, v, 



= * (^ 17, ^ 2 (6 V, 0- ' 



As the function i£, (£, 17, £) contains a constant term, when the first trans- 

 formation of (14 y i is made, the factor £"• must come out of the <i>, and a 



(20) 



