408 HITCHCOCK. 



x£ m (x)C + ai£ OT (ai)Ci + • • • + a„_ m E„(a n _ m )C„_ m = (30) 



If the number of terms in the general polynomial of degree A in N 

 variables be denoted by n(N, A) we have the identity 



n(N, A) - n(N, A - 1) = n(N - 1, K) (31) 



while m = n(N, K — 1) — 1. The number of terms in (29) is 

 n(X, A) + 1. Hence the number of terms in (30) is n(N — 1, A') + 2. 

 It is clear that we may obtain from (29) as many identities of the form 

 (30) as there are ways of selecting in points from n + 1 points. 



Again, we may take A(x) = L(x)E(x) where L(x) is a linear poly- 

 nomial. Repeating the same steps, with the same meaning of E m (x), 

 we shall obtain, instead of the vector identity (30), a scalar identity 



L(x)E m (x)C + L(a,i)E m (a.i)Ci + • • • + I(a„_ m )A m (a n _ m )C„_ m = 



(32) 



We may now choose L(x) so as to vanish at the X — 1 points whose 

 subscripts are n — m — X + 2, n — m — X -f- 3, etc. up to n — m; 

 that is, L(x) may be taken to be the determinant of the components 

 of these X — 1 point vectors and the variable vector x. The last 

 N — 1 terms of (32) will now vanish, leaving n(X — 1, A') — X + 3 

 terms. We may obtain as many identities of this last form as there 

 are ways of selecting X — 1 points from n(X — 1, A) + 2 points, 

 multiplied by the number of ways (above noted) in which (30) could 

 be formed. 



Instead of factoring F(x) into a linear polynomial and one of degree 

 A — 1 we can evidently obtain other identities in a similar manner by 

 factoring F(x) into any pair of polynomials, or into any number of 

 polynomials such that the sum of their degrees is equal to A. Clearly 

 then there exist a vast number of identities connecting the coefficients 

 C in (5) or (9) with the anologous coefficients in expansions associated 

 with smaller values of A. 



In a former paper l I have made a somewhat detailed study of the 

 case A = 2, X = 3. The " Aconic Function " of Hamilton is a 

 function of six vectors, in the form of a scalar product, which vanishes 

 when these six vectors lie on a quadric cone; equivalent, therefore, to 

 Co for the case in question. In the paper referred to, the properties 

 of the seven C's for this case were derived from Hamilton's function, 



1 An Identical Relation Connecting Seven Vectors. Proc. Royal Soc. Edin. 

 vol. XL, Part II( No. 14), pp. 129-139. 



