BLACK. — THE NEIGHBORHOOD OF A SINGULAR POINT. 809 



type (£) is needed, and after collecting all terms of the m r th 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 4> = 0, and use a nota- 

 tion independent of that hitherto employed. 

 Our successive transformations are of the type 



x —pi (y) = *i 



(»■) 



x v - x -p v (y) = x 



whence 



x - p x (y) +p*(y)z + pz(y)z' x + + p v (y)z v ~ 1 + x/. Q>) 



Develop the function <J> in (y) by Weierstrass's Theorem : 



<P(x, y, z) = [x m + q x (y, z)x m ~ 1 + + q m (y, z)-]E(x, y, z) 



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



From (p) we derive the relation : 



9F _9F9x v __ 1 9F 

 9x 



9x v 9x 



z v 9x v 



(-) 



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

 degree in x and z, takes out of the F factor at each step the factor z v , 

 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 (ir), we have 



F(x,y,z) = z Vm F(x v ,y,z) = 



*"'"[<" + q»(2f, z)K"- 1 + + 9 mv bt, *)]. to 



and by (o-) 



9F , 9F„ 



Qx 9x v 



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

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



L(x,y, z)F+ M(x,y, *)^= Rfa z)=z*R l (y, z) (?) 



