VAN DER VRIES. — MULTIPLE POINTS OF TWISTED CURVES. 489 



than two to an actual multiple point of the same multiplicity. As before, 

 it is not necessary to consider the case where the line from the vertex to 

 a multiple point crosses the curve again. Nor shall we consider the case 

 where the vertex lies on a tangent to the curve, these tangents gener- 

 ating a definite developable surface on which the arbitrary vertex may 

 also be assumed not to lie. 



The fact that we can always find a point such that an edge of the cone 

 having this point as vertex and the curve as base generally meets the 

 curve in only one point euables us to obtain any curve C m as the partial 

 intersection of a coue K m and a monoid M u of order (x having the same 

 vertex as K m . For, every line through the vertex of M a , and therefore 

 every edge of K m , meets M u in one point distinct from the vertex ; viz., 

 the point where C m crosses the edge of K m . An ordinary edge of K m is 

 not necessarily a line on J/ w ; a certain number of them, however, gener- 

 ally lie on the monoid. A double edge of the cone due to an apparent 

 double point of the curve always lies on the monoid. For it meets C m , 

 and therefore the monoid M u on which C m lies, in two points distinct 

 from the vertex. It thus meets M„. in fx + 1 points and therefore lies on 

 it. A line to an actual double point or to a multiple point that has its 

 tangents lying in oue plane meets the curve in general only once at that 

 point, which is not enough to make it lie on M u . The line from the 

 vertex to a multiple point that does not have its tangents all lying in oue 

 plane is a line on J/ M . This will be shown more fully later. 



The general equation of M^ is of the form : — 



u^i s + u„. = ; 



where « M _ i and « M are homogeneous functions of x, y, and z of degrees 

 fx — 1 and //. respectively. These functions thus represent cones, known 

 as the inferior and superior cones, respectively, of the monoid. The 

 number of independent constants in the above equation is fj. (/x. -f 2). In 

 order to make 3I U contain C m , we must make it contain mix. + 1 points 

 of the curve. We can do this if 



w/i f 1 ^ ]i{jt + 2), 



m 2 



i. e. if — — + \ V»i 2 — 4 m + 8 ^ n, 



i. e. if m — 1 < /x; 



that is, we can always obtain the curve as the intersection of a cone of 

 order m and a monoid of order m — 1. We may be able to obtain the 



