BLACK. — THE NEIGHBORHOOD OF A SINGULAR POINT. 295 

 C. — The Number of the Neighborhoods, t u U, t, 



REQUIRED TO REPRESENT T IS FlNITE. 



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

 hood of each tangent line to the surface 4> = 0, at the singular point 

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

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

 neighborhood 



'b' 



T: |*| <*, hl<S, \£\<& 



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

 We distinguish two cases : — 



Case I. — The polynomial </> (£, 77) has no multiple factors. 

 Case II. — This polynomial has multiple factors. 



Theorem: The neighborhood T can be completely covered by a finite 



number of regions T u T. 2 , T v , which overlap each other and which 



are mapped respectively on the following regions t x , f 2 , t v : 



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

 borhood of a singular point of the surface g w = ; 



2) the extent of each of the neighborhoods t x , t. 2 , t K having been 



arbitrarily determined^ the regions t fi j — k + 1, v, then consist 



of regular regions of surfaces g <J> = 0. 



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

 borhood of a singular point of the surface g {i) = ; 



2) the extent of each of the neighborhoods t x , t. 2 , t K having been 



arbitrarily determined, the regions fj,j = K + 1, v, then consist 



of regions of surfaces g' j) = defined as follows ■' omitting the index j 

 throughout, we write 



9 (£., Vv = [£ + ft (Vv C + + p r (Vv Ol^d,, Vv 0, 



where p e (r^, £) is analytic throughout a region 



M<h, \£\<8- 



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

 1 5; r < m. 



Case I. — The polynomial </> (f, 77) contains no multiple factors. 

 Here, the equation 



