490 PROCEEDINGS OF THE AMERICAN ACADEMY. 



curve as the intersection of the cone with a monoid of lower order, but 

 we need never take a monoid of order higher than m — 1. Thus far we 

 have not taken into account the multiple points of the curve. It will be 

 shown later that even when this is done the monoid need not be of an 

 order higher than vi — 1. The hnes of intersection of the inferior and 

 superior cones, ju.(jLt — 1) in number, are commonly known as the lines 

 of the monoid. The cones «n_i and Km, have m (fx — 1) edges m com- 

 mon, all of which meet C,n and therefore M^j, in a point distinct from 0, 

 where is the point (0, 0, 0, 1). They therefore lie on M^, , and with 

 the curve Cm form the complete intersection of K,n and M^ . Since 

 fx ^ m — 1, more lines are common to K,n and M^ than there are lines 

 on J/„ . Some lines of M^ must therefore count two or more times as 

 lines common to K,n and M^ , that is, they must be double lines or lines 

 of higher multiplicity on K,n corresponding to apparent multiple points 

 on Cm ' As the vertex of K,n has been taken in such a way that the 

 curve Cjn has no apparent multiple points of multiplicity greater than 

 two, we see that at least (m — fx) (jx — 1) lines of M^ must be double 

 edges on K^- There will be just this number if all lines of M^ lie on 

 K„i, an additional double edge on K,n being necessary for every line of 

 M^ that does not lie on K,n. The curve Cm of order m that is the par- 

 tial intersection of A'„ and AJ^ thus has in general at least (m — fx){/x — 1) 

 apparent double points. Assuming /tx = m — 1, we see that a curve of 

 order m having no multiple points never has less than m — 2 apparent 

 double points. We wish now to investigate the effect of a multiple point 

 of Cm on this number of apparent double points. If the point is a mul- 

 tiple point on M^ , the line joining it to the vertex will be a line on M^ , 

 for it meets J/^ in enough points distinct from the vertex to make it lie 

 on 7l/„ . We are thus led to the consideration of lines and, in particular, 

 of multiple lines on M^ , and to the investigation of the effect of these 

 on the curve of intersection when they are common lines of Km and M^. . 



2. Ordinary or multiple lines on any surface may be of different 

 kinds. A line is torsal or scrolar on a sheet of the surface containing it 

 according as the tangent plane to that sheet of the surface is or is not the 

 same at every point of the line.* If the line xy is a. i'-tuple line on a 

 surface, the equation of the tangent planes along that line will be given 

 by the terms of the /"th degree in x and y in the equation of the surface. 

 According as the equations of these planes, separately or collectively, do 

 or do not contain z and s will these planes be dependent or independent 



* See Cayley, Collected Papers, VII 334. 



