480 PROCEEDINGS OF THE AMERICAN ACADEMY. 



first group, and d is p columns to the right of a, the degree of b in s is p 

 less than that of a, and c is of a degree the same as that of d. Therefore 

 when we substitute b and c for a and d respectively we do not increase 

 the degree of this term in s. Similarly, if both constituents come from 

 the second or third group. If b and c are in the first and second groups 

 respectively, a is of the same degree in s as the degree of b, whereas d is 

 of the same degree in s as the degree of c, if c — d < k', but of a degree 

 less by c — d — k 1 if Tf < c — d. Thus the substitution of a and d for 

 b and c respectively does not increase the degree in s of our term. If a 

 and d are in the first and third groups, respectively, the substitution of b 

 for a decreases the degree in s by b — a, whereas the substitution of c for 

 d increases the degree by b — a only if b — a < v — k', and by less than 

 b — a if v — k' < b — a. These interchanges of pairs of corresponding 

 constituents, however often performed, do not increase the degree of s in 

 our term. A zero-coefficient, whenever it occurs, causes the term to 

 vanish. We can therefore in no way get a term of higher degree in s 

 than the term first selected. In this term, s appears to the degree 



- 1) Ql - k) + (k - 1) - F), if k' < k ; 

 and to the degree 



(ji - 1) (v - k') + (If -1) Qi- k), if h < V. 



These are both, however, equivalent to 



v fx — h k' — fx. — v + & 4" &'> 



which is thus the highest degree to which s can occur in £. As the 

 eliminant is homogeneous of degree (/a — 1) (v — 1) in the variables, 

 the lowest degree to which x, y, and z together occur is 



f> — 1) (v - 1) - (vfi — klf — /i — v + k + k') = (k - 1) (V - 1). 



The point (0, 0, 0, 1) is thus a [(& — 1) (k' — l)]-tuple point on the 

 surface S. 



The points of intersection of U, V, and S are the "points" of the 

 "lines through two points;" they are in general p (fx — 1) (y — 1) in 

 number. The number h of "lines through two points" or of apparent 

 double points being one-half of this number, we have in general 



2h= fJLV (ll-1) (v- 1).* 



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



