ALGEBRAIC POINT FUNCTIONS IN N-SPACE. 407 



Theorem III. An arbitrary vector function homogeneous of degree 

 K in N variables may be written as the sum ofh terms 



F(x) = Z[f A (b,-x) (b g 'x)- • -to A' factors] (26) 



h 



where the vectors i u f 2 , • • ■ f n are arbitrary; in each term occur K linear 



-polynomials selected from N+ K — 1 linear -polynomials; these linear 



polynomials are under the sole restriction that no N of them are linearly 



related. 



It follows from the form of (26) that, if a& be of the form (20), 



F(a A ) = U (27) 



It is apparent that expansions like (26) will be of advantage in 

 investigations where symmetry of form is to be sought. As a simple 

 illustration suppose A = 2. We have N + A — 1 = N + 1 and 

 may take as our linear polynomials the N variables Xi, x 2 , • • • x n 

 together with their sum Xi + x* + • • • + .r#. We see at once that 

 any quadratic vector function can be written in the form 



TXiiX&j + <p(x)2x h ; [i different from j; h = 1,2, • • • N] (28) 



where <p(x) is a linear vector function in N-space. 



5. Special Forms of the General Identity: Method of 



Factoring. 



I propose next to consider a large class of identities which we may 

 obtain from (5) and from (9) if, instead of making some special choice 

 of the vectors ai, ao, • • • a„, we take particular polynomials A(x) and 

 F(x). 



As a first example, let A(x) be a polynomial of degree A — 1 and 

 take F(x) = xA(x); that is, the vector function F(x) is chosen to be 

 the point vector x multiplied by an arbitrary scalar polynomial. The 

 identity (9) becomes 



x£(x)C'o + a 1 A(a 1 )C 1 + a 2 £(a 2 )C 2 -| 1- a„£(a n )C' n = (29) 



We may now choose the polynomial E(x) so that it shall vanish at a 

 number of points, say at m points. Let these points be a n - m +i, 

 a n - m+2 , etc. up to a„. In other words let A(x) be a function of the 

 same type as the C's but associated with the case A — 1. This 

 particular form of E(x) may be distinguished as E m (x). The last in 

 terms of (29) will now vanish and we have 



