VAN DER VRIES. — MULTIPLE POINTS OF TWISTED CURVES. 477 



1^ 



where ^i = ^i 5^ + ^i o" + «i ^ + «i ^"^ ' ®^^- ^^ ^^^" ^^^® 



S = 



^,U,k^,'V, , Ur, 



Ai U, 



^.U, 



u. 



Ai F, i Ai^ F, 

 Ai F, 



, Fl, 



F 



15 



AiF, 



, ^1 



If the point (0, 0, 0, 1) is a ^-tuple point on Cand a ^'-tuple point on 

 V, the equation £/"= will be of the form 



c/.;. + + <^/,+iS^-^-l + 4>lcS^-^ = 0, 



and F= will be of the form 



•A. + + «Aa.-'+i s"-*'-^ + ^k- s"-^" = ; 



where ^^, , t^t, i/'^, , ^k' are functions of degrees jx, , k^ 



V, , k', respectively, in x, y, and z. The terms of A/ f/^ that are of 



lowest degree in x, y, and z are thus of degree k — r, if r ^ k, and of 

 decree 0, if ^ + 1 ^ r. It is now necessary to find the terms of /S' that are 

 of lowest degree in x, y, and z in order to know the multiplicity of 

 (0, 0, 0, 1) on ^S*. We know that the eliminant of two homogeneous 

 equations is homogeneous in the coefficients of those equations. The 

 terms of *§ that are of lowest degree in x, y, and 2 are thus those that 

 are of highest degree in s. We shall therefore look for the terms of 

 highest decree in s in the following scheme, where we have taken account 

 only of the terms of lowest degree in x, y, and z, and therefore of highest 



degree in s. In this scheme fxifi, are polynomials of degrees 



1,2, , respectively, in x, y, and 2 ; similarly for the y's. We have 



filled in the lacking members of our scheme with terms having zero- 

 coefficients. The scheme then really consists of four parts, the first hav- 

 ing terms of degree /x — ^ in s ; the second, terms of degree v — ^' in s ; 

 the third, terms in descending powers of s from si^—^ down ; and the 



