476 PROCEEDINGS OP THE AMERICAN ACADEMY. 



and k 1 , respectively, to the two surfaces at that point, i. e. they will he 

 the common generators of the two cones, and no three will in general lie 

 in one plane. Thus, a multiple point at which the tangents lie in one 

 plane will in general be an ordinary point on one of the two surfaces 

 that contain the curve, whereas a multiple point at which the tangents 

 do not lie in one plane must be a multiple point on both surfaces that 

 contain the curve. 



2. If the tangents at a &- tuple point of a curve all lie in one plane, 

 the curve will be met k times at that point by any line that lies in the 

 plane of the tangents and passes through the point ; a line, however, not 

 in this tangent plane can meet the curve only once there. A cone 

 drawn from an arbitrary point as a vertex with the curve as base will 

 have the line joining the vertex to the &-tuple point as a £-tuple edge. 

 The number of actual double points of the curve is therefore affected by 

 a point of this kind, but the number of apparent points is not. For, a 

 line drawn from an arbitrary point in space has two points the same for 

 both surfaces in the case of an apparent double point, whereas in the case 

 of a point of the above kind the line has only one point the same for both 

 surfaces. We assume that the arbitrary point does not lie in the tangent 

 plane nor on a tangent line to the curve at the multiple point. Nor does 

 it lie on the cone having the multiple point as vertex and the curve as 

 base, for the line to the multiple point would then meet the curve again. 

 If, however, the multiple point is such that the tangents do not lie in 

 one plane, the line from an arbitrary point in space will meet the sur- 

 faces in two or more points that are the same for both surfaces, and the 

 number of apparent double points is affected. We shall investigate the 

 number of apparent double points of a curve that has such a multiple 

 point ; we shall use the method of Salmon.* We consider two surfaces 

 U and V of orders /x and v respectively. They intersect in a curve of 

 order yu. v which may break up into a number of component curves. The 

 points on the " lines through two points" that pass through an arbitrary 

 point are given by the intersection of the curve U Fand a certain sur- 

 face, which we shall call S. The equation of this cone S with its vertex 

 at the point (1) is obtained by eliminating A : k between 



km-i Ai £7+ ^V U+ -^f- Ai 3 U+ = 0, 



*"— 2 A k"- 3 A 2 

 ^A, F+^A 1 2 r+1-^A 1 8 F+ = 0; 



* Salmon's Geometry of Three Dimensions (1882), p. 309. 



