488 PROCEEDINGS OF THE AMERICAN ACADEMY. 



of two cones of orders p and q not having the same vertex. Every edge 

 of the first cone meets the curve q times and every edge of the second 

 cone meets it p times. As Professor Story has proved, all simple 

 generators of a cone of any order whatsoever meet any curve lying on it 

 in the same number of points. Thus if a generator of a cone K a of 

 order a meets a curve lying on it in j3 points, every generator will meet 

 it in /? points, and the curve, if it does not pass through the vertex, will 

 be of order a/3. The cone K m in this case is the cone K a taken /3 

 times. It is, however, only for special positions of the vertex that this 

 breaking up of the cone K m can occur. In the case of the curve C pq of 

 order pq mentioned above, if we take the point (0, 0, 0, 1) as the ver- 

 tex of the one cone and the point (0, 0, 1, 0) as the vertex of the other 

 cone, and eliminate y between the equations of the two cones, we shall 

 get a homogeneous expression of degree p q in x, z, and s. This is in 

 general the equation of a cone of order pq having the point (0, 1, 0, 0) 

 as its vertex, and the curve C pq as its base. This vertex is an arbitrary 

 point, and so a line from a point off the curve meets the curve in general 

 in only one point. For the eliminant above will not in general break up 

 into two, three, or more equal factors, as would be necessary in order to 

 have every edge of the cone from the point (0, 1, 0, 0) to the curve 

 meet the curve iu two, three, or more points. There may be certain 

 points that, taken as vertices of this cone, will cause it to break up, but 

 for an arbitrary point it will not generally do so. Thus in the case of 

 the quartic of the first kind, if we consider it as the intersection of two 

 quadric cones, there are at most two other quadric cones on which the 

 curve may lie, i.e. there are at most two other points off the curve from 

 which an infinite number of lines can be drawn to meet the curve in two 

 points. Excluding these quadric cones, any other cone whose base is 

 the quartic in question will be a cone of the fourth order, every edge of 

 which meets the curve once. Thus we can always take the vertex of the 

 cone K in such a way that the cone will be of the same order as the 

 curve, an edge meeting the curve in only one point. 



There is, however, from an arbitrary point in space a finite number 

 of lines that meet the curve. C m in two distinct points. These correspond 

 to the apparent double points of the curve and are double edges on the 

 cone K m . As a curve has only a finite number of apparent double 

 points, it will not in general have an apparent multiple point of multi- 

 plicity greater than two. A simple edge of the cone is thus due to an 

 ordinary point on the curve, a double edge to an actual or apparent 

 double point on the curve, and a multiple edge of multiplicity greater 



