BLACK. — THE NEIGHBORHOOD OF A SINGULAR POINT. 307 

 Let the result be written as follows : — 



C2'"i{[x3"'i + ri(y2)a:3"'i-' + + r,„,(9/r,)]E(i/^)-{-Z2Fs(x3,i/o,Z2)} =0.(k) 



From this last equation we deduce the following theorem : — 

 AH points of the surface O = m the neighborhood of the curve 



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 ^ (x, y, z) = of the lemma, the values of m 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^ being 

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

 the point z.2 = of the function 



^a(2/2)> A. = 1, 2, mi, 



all points of the surface $0 = i» the neighborhood of the curve 



4>{^,y) = ^, 2 = 0, 



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

 xz'^ + ri (2/0) Xg" -1 + + r„,^ (yo) = , Zj = , 



on the surface $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 



xr^ + n (6 - a) X,"'-' + + r,„^ (b-a) = (/x) 



are ai, 02, a„, then points of the surface $2 =^ for which 



^2! < 8, l^ol < 3, !/ = b, 



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



x^ = z„{x^ + a^), y„ = b — a, o- =1,2, m^. 



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

 for which qo(y.,) vanishes, we have an upper limit for a^- as a root of 

 the equation (fx), and thus by taking ^2 and x^ small enough we can make 

 X2 as small as we please. Then the transformations (8) and (Q still 



