512 PROCEEDINGS OF THE AMERICAN ACADEMY. 



complete intersection of a quadric cone and a cubic monoid. The cubic 

 monoid is determined by 15 points, of which 4 must be taken off the 

 quadric to insure the monoid not containing it as a factor. In order to 

 make this monoid contain the sextic that has a quadruple point at the 

 vertex of the monoid, we must make it contain 6*3 — 4-2 + 1 or 11 

 additional points. This is the exact number at our disposal. 



IV. 



On the Composition of Multiple Points. 



1. If a curve of order m has an (m — 2)-tuple point, it can have no 

 other multiple point. The curve is unicursal. We assume 5 < m. The 

 tangents to the curve at this multiple point cannot lie in a plane, but 

 they lie on a quadric cone. As we saw on page 510, this curve has 

 m — 2 apparent double points. It can have no more apparent singular- 

 ities. 



A curve with an (m — 3)-tuple point can have at most one other 

 actual multiple point, viz. a double point. If it had more it would nec- 

 essarily be a plane curve. Thus a twisted quintic cannot have three 

 actual double points. A curve of order m having an (m — 3)-tuple 

 point and a double point lies in general on a cone of order m — 2 having 

 the double point as vertex and the curve as base. This curve cannot 



have more than — double edges. The edge joining the 



vertex to the (m — 3)-tuple point counts for just this number of double 

 edges. The cone can therefore have no more double edges ; that is, 

 when viewed from the double point of the curve, the number of apparent 

 double points of this curve is zero. According to Salmon,* the number 

 of apparent double points of the curve is 2 m — 6 more when viewed 

 from an arbitrary point in space than when viewed from a double point 

 of the curve. A curve of order m with an (m — 3)-tuple point and an 

 actual double point has therefore just 2 m — 6 apparent double points. 

 It can have no other actual multiple point or apparent singularity. It is 

 thus evident that when one of the branches through the double point 

 moves into the (m — 3)-tuple point, the curve loses m — 4 apparent 

 double points, that is an (m — 2)-tuple point may be formed from an 

 (m — 3)-tuple point, a double point and m — 4 apparent double points. 

 A triple point on a quintic curve is thus equivalent to two actual double 



* Salmon's Geometry of Three Dimensions (1882), No. 330, Example 2. 



