BLACK. — THE NEIGHBORHOOD OF A SINGULAR POINT. 295 

 C. — The Number of the Neighborhoods, ti, t.^, t^, 



REQUIRED TO REPRESENT T IS FiNITE. 



6. In the foregoing paragraph it has been shown that the neighbor- 

 hood of each tangent line to the surface $ = 0, at the singular point 

 can be mapped on the neighborhood of a (regular or singular) point 

 of the surface y = 0. We now proceed to show that the whole 

 neijihborhood 



o 



T: Kl < 8, h|<S, |^|<S 



can be covered by the neighborhoods of a finite number of such lines. 

 We distinguish two cases : — 



Case I. — The polynomial </> (f, rj) has no multiple factors. 

 Case II. — This polynomial has multiple factors. 



Theorem: T/ie neighborhood T can be completely covered by a finite 



number of regions 7\, T^, T^, which overlap each other and which 



are mapped respectively on the following regions ty, to, t^: 



In Case I: 1) the region t^, i = 1, 2, k, consists of the neigh- 

 borhood of a singular point of the surface y*' = ; 



2) the extent of each of the neighborhoods ti, t^, t^ having been 



arbitrarily determined, the regions tj, J =: k + 1, v, then consist 



of regidar regions of surfaces g^^ = 0. 



In Case II : 1) the region ^;, V = 1, 2, k, consists of the neigh- 

 borhood of a singidar point of the surface ^''' = ; 



2) the extent of each of the neighborhoods <i, <2 5 '« having been 



arbitrarily determined, the regions tj,j=^K-\-\, v, then consist 



of regions of surfaces g'-'^ = defined as follows : omitting the index j 

 throughout, we write 



9 (^^ Vv = Li + Pi (Vv 0C + +Pr (Vv 0] ^ (^„. Vv 0, 



where p^ (171, ^) is analytic throughout a region 



\rix\<h, l^l<8- 



Here r,for a given value ofj, is a positive integer satisfying the relation 

 1 < r <m. 



Case I. — The polynomial <^ (f, ^ contains no multiple factors. 

 Here, the equation 



