BLACK, — THE NEIGHBORHOOD OF A SINGULAR POINT. 



309 



type (^) is needed, and after collecting all terms of the mi-ih order we 

 make at once the substitution 



and proceed in the same manner as before if the degree is not reduced. 

 For convenience, we suppose the succession of multiple curves of the 

 same order to begin with that on the surface $ = 0, and use a nota- 

 tion independent of that hitherto employed. 

 Our successive transformations are of the type 



^ —Pi (y) = ^1 



(-) 



whence 



x^Piiy) +P2(y)z + ps(y)z'+ +pAy)^''~^ + ^y- (p) 



Develop the function $ in (y) by Weierstrass's Theorem : 



$ (x, y, z) = Ix^ + q, (y, «) x—^ + + q^ (y, «)] J^{^, V, «) 



z=F(x,y,z)E(x,y,z). 



From (p) we derive the relation : 



9F _9F_9x^_'^9F_ 

 9x 9xt,dx z^dxv 



The succession of transformations (tt) so long as it does not reduce the 

 deofree in x and z, takes out of the F factor at each step the factor s", 

 since, on account of the constant term in the E factor, no power of z 

 could come out of it. So, after the v transformations (tt), we have 



F{x.y,z)=z^-FXx^,y,z) = 



z^[xj" + q^^(y, z)xj"~' + + q^^iy, «)], (r) 



and by (<r) 



9_F 

 9x 



^vim—l) 



9F, 



IXv 



Now we may consider F as having no multiple factors vanishing at the 

 point (0, 0, 0). So we have the relation 



9F 



L{x, y,z)F+ M{x, y,^)g^ = R{y^ 2) = «*^i(y, ^) 



(y) 



