404 



HITCHCOCK. 



identities may be obtained from the scalar formula (5) as well as from 

 the. vector formula (9). 



An elementary method is, evidently, to assign numerical values to 

 the elements an, with the sole restriction that Co shall not vanish. 

 The coefficients C\, C 2 , • • • C n are polynomials of degree K. By (5), 

 an arbitrary polynomial F(x) can be expanded in terms of these n 

 polynomials, the coefficients F(&i) being found by direct substitution. 



It is evident, therefore, that &, Co, ■ • ■ C n are linearly independent. 

 It is equally evident that not every linearly independent set of n 

 polynomials homogeneous of degree K in N variables can be taken as 

 these nC's; for the totality of all vectors where two or more C"s 

 vanish is comprised by the set of vectors &i, a 2 , • • • a„. We may 

 embody this distinction in the following definition: 



Definition I. A set of polynomials Pi, P 2 , • • -P n , homogeneous of 

 degree K in N variables, such that Pf(ay) vanishes when i is different from, 

 j but not when i equals j for each of the points ai, a 2 , • • • a n , will be called 

 a standard set. 



By use of standard sets of polynomials, many of the identities of 

 elementary algebra may be made to appear as special cases of (5). 

 To take the simplest of illustrations, let N = K = 2 so that n = 3, 

 with variables x, y. Let the matrix an be 



By easy calculation C = 1,C X = xy — x 2 , C 2 = — xy, C 3 =xy —y 2 . 

 Choosing F(x, y) = (.r + y) (x - y) yields F(l, 0) = 1, F(l, 1) = 0, 

 F(Q, 1) = — 1. The identity (5) appears as 



(x+y) (x-y) + (xy - .r) + + (-1) (xy - if) = 



To take a less familiar illustration, let F(x) be a quadratic vector 

 function in ordinary space, A r = 3, A' = 2. A simple way to build a 

 standard set of polynomials is to choose four vectors bi, b 2 , b 3 , b 4 of 

 which no three are coplanar, and form six products of linear factors 

 X'bi-byx which are the required polynomials. The six fixed points or 



