ALGEBRAIC POINT FUNCTIONS IN N-SPACE. 417 



The two-row determinant does not vanish identically since it does not 

 vanish when x = a 2 + a 4 . Nor does the three-row determinant vanish 

 identically, for if x = a 4 + ai it becomes (45671) times the two-row 

 determinant 



(14672); (14673) 

 (14572); (14573) 



which, in turn, does not vanish if we let ai = a 2 + a 5 . Hence our 

 original determinant does not vanish identically. 



To complete the factorization, consider the minor of the leading 

 element in (53), namely 



(*4672) (o;4573) - (.r4572) (.r4673); (54) 



and let x be expanded in terms of the five vectors a 2 , &s, a 4 , a 5 , and a;, — 



x(23457) - a 2 (z3457) + a 3 (.r2457) - a 4 (a-2357) + a 5 (.r2347) - 



a 7 (.r2345) = (55) 

 and multiplying by [.r467] 



- (z4672) (a-3457) + (.r4673) (a!2457) + (.r4675) (.r2347) = (56) 



whence it is evident that (54) contains the factor (.r4567). In a 

 similar manner it may be shown that the other minors of the elements 

 of the first row in (53) contain the same factor. Thus the determinant 

 contains this factor. In the same way we may show that the minors 

 of any other row contain a common factor, hence the determinant is 

 the product of three linear factors. Again, the same process shows 

 that (52) is a product of two linear factors. The factorization is 

 therefore complete, and the adventitious factor of our original determi- 

 nant is here, as in the former case, a mere product of determinants 

 linear in the points which enter into them. 



7. Rule for Constructing these Determinants. 



It remains to indicate how, in general, the b's may be chosen so 

 that the determinant contemplated in theorem V shall not vanish 

 identically. We have seen that the order q of this determinant is 

 n[(N — 1), K] — N + 2. Let q' be the number analogous to q but 

 associated with polynomials of degree less by one, that is 



q ' = n [N - 1, K - 1] -N + 2 (57) 



The elements of the determinant are of the form b •a,jE(a,j). A neces- 

 sary condition that the determinant shall not vanish identically is 



