ALGEBRAIC POINT FUNCTIONS IN N-SPACE. 401 



of the vector F(x) will satisfy an identity of the form (5). We have 

 then a set of # identities 



Fi(x)C + Fi(ai)C, + Fi(a2)C 2 + • • • + F 2 (a B )C„ = 

 F 2 (x)C + F 2 (ai)d + F 2 (a2)C 2 + • • • + F 2 (a„)C„ = 



F N (x)C + F N (^)C\ + F N (av)C 2 + • • • + F N (& n )C n = 



wherein we assume that the fixed points a b a 2 , • • • a„, while wholly 

 arbitrary, are constant for all the identities of the set. Hence the 

 coefficients Co, C\, C 2 , ■ • • C„ are unaltered as we pass from one 

 identity to another. If, then, we multiply these identities in turn 

 by 6i, e>, • • • e#, and add the results, we shall have the vector identity 



F(x)C + F(a 1 )6' 1 + F(a,)C 2 + • • • + F(a„)C„ = 0. (9) 



It is obvious that the left member of this identity may, if we wish, 

 be written in the form of the determinant (4), the elements of the 

 first column being now vectors. 



Just as with the scalar identity (5), we may regard the vector iden- 

 tity (9) in two ways, — either as the statement of the linear relation 

 connecting the values of the vector polynomial F(x) at n + 1 arbi- 

 trary points; or as the expansion of this polynomial in terms of its 

 values at n points arbitrary except that C does not vanish. The 

 vector function F(x) depends on A T scalar polynomials, each contain- 

 ing n arbitrary numerical coefficients. Whenever these coefficients 

 need to be separately considered they may be designated according 

 to the matrix 



A\n , Ao N , A 3 ff , •••, A n ff 



Thus the set of numbers An, like the set a^, contains Nn elements. 



We may now state the following simple but fundamental theorem: 



Theorem I. The values taken on by an arbitrary vector polynomial 



at n + i arbitrary points arc linearly related. The multipliers Co, Ci, 



C 2 , • • • C n are functions of the arbitrary points, but are independent of 



the coefficients An of the terms of the polynomial. 



