480 PROCEEDINGS OF THE AMERICAN ACADEMY. 



first group, and d is p columns to the right of a, the degree of 6 in 5 is jO 

 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 — 0? ^ k', but of a degree 

 less by c — c? — ^•' if k' <C_ 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 hy b — a, whereas the substitution of c for 

 d increases the degree hy b — a only if 6 — a < v — k', and by less than 

 b — a ii 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-coefhcient, whenever it occurs, causes the terra 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 



(v - 1) (jji - k) + (k - 1) (u - k'), \ik' <k; 

 and to the degree 



(/x - 1) (1/ - k') + {k> -1) (fji- k), if k < k'. 

 These are both, however, equivalent to 



V fx — k k' — fx — V -\- k -\- k', 



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

 eliminant is homogeneous of degree (/i, — 1) (i^ — 1) in the variables, 

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



{fx. - \) (y - \) - {v IX ~ kk' - ^JL - V + k + k') = {k - 1) (k' - 1). 



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

 surface »S'. 



The points of intersection of U, V, and aS* are the "points" of the 

 "lines through two points;" they are in general fxv {fx — 1) (v — 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 



2A=/xv(/x-l) (v-1).* 



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



