BLACK. — THE NEIGHBORHOOD OF A SINGULAR POINT. 307 



Let the result be written as follows : — 



<£ 2 (x 2 , y 2 , 2 2 ) = z 2 '"x<P 8 (x 3 , y 2 , z 2 ) = 



z 2 m i{[x 3 "> l + r l (i/ 2 )x z m i- 1 + + r m {y 2 )]E(y 2 ) + z 2 F & (xz,y 2 ,z 2 )} =0.(k) 



From this last equation we deduce the following theorem : — 

 All points of the surface <!> = in the neighborhood of the curve 



<f>(x,y) = 0, 2i = 0, 



are mapped upon a finite number of new neighborhoods which are 



1) neighborhoods of singular points of degree < m, which neighbor- 

 hoods may be taken arbitrarily small ; 



2) neighborhoods of new multiple curves on surfaces constituted like 

 the surface <I> (a:, y, z) = of the lemma, the values of in thus arising 

 never exceeding the original m of the lemma. 



By the same kind of reasoning as in § 1, 5, we show namely that for 

 any one of the above values of a, the corresponding value of y 2 being 

 in or on the circle of convergence of the Taylor's development about 

 the point z 2 = of the function 



r x(V2)> A=l, 2, 



r/h 



i\y2j> k — i, ±, /«!, 



all points of the surface <£ 2 = in the neighborhood of the curve 



<f>(x,y) = 0, sz = 0, 



are represented by points in the neighborhoods of points of the curve 

 * 3 m ' + r x (yjx^- 1 + + r, % (y 2 ) = , z 2 = 0, 



on the surface 4> 3 = 0, i. e. if such a value of y is b, so that the corre- 

 sponding value of y» is (b — a), and if the roots of the equation 



a^K + n (b - a ) x 3 m ~ l + + r mi (b - a) = (p) 



are u x , a 2 , a,„ then points of the surface 4> 2 — for which 



|*«| < 8, K| < 8, y — b, 



are connected with the points of the surface (k) by the relation 



x 2 = z, (x o + a v ) , y, — b — a, a = 1, 2, m x . 



Further, if we limit y. 2 to a circle not reaching out to the nearest point 

 for which qo(y 2 ) vanishes, we have an upper limit for a„ as a root of 

 the equation (u), and thus by taking z 2 and x a small enough we can make 

 x 2 as small as we please. Then the transformations (8) and (£) still 



