ALGEBRAIC POINT FUNCTIONS EN N-SPACE. 413 



Hence we may take as the required cubic the determinant 



(a:17)£ 1 (a 7 ) , (arl8)E 1 (a 8 ) , (.rl9)£ 1 (a 9 ) 

 (x27)EM , (.r28)/>(a 8 ), (z29)£ 2 (a 9 ) 

 (.r37)£ 3 (a 7 ) , (.r38)£ 3 (a 8 ), (.r39)E 3 (a 9 ) 



(38) 



That this determinant vanishes when x coincides with ai is evident, 

 for the elements of the first row all vanish; similarly the second and 

 third rows vanish when x coincides with a 2 and a 3 respectively. The 

 columns vanish when x coincides with a;, a 8 , a 9 . Since as has been 

 shown the determinant must also vanish when x coincides with a-i, aj, 

 or a6, we are presented with three new identities of the form 



(417) J B 1 (a 7 ), (418)£ 1 (a 8 ), (419)£ 1 (a 9 ) 

 (427)£ 2 (a 7 ), (42S)£ 2 (a 8 ), (429)£ 2 (a 9 ) 

 (437)£ 3 (a 7 ), (438)£ 3 (a 8 ), (439)£ 3 (a 9 ) 



- (39) 



the other two having 5 and 6 in place of 4. In fact if we multiply the 

 elements of the first row of (39) by the three-row determinant (561), 

 those of the second row by (562), of the third by (563), and add, the 

 sum of the elements of each column is zero ; for 



(561) (41.T)£i(x) + (562) (42a-)£ 2 (x) + (563) (43.r)£ 3 (x) = (40) 



is an identity of the type (33), a special case of (35). It holds there- 

 fore when x is replaced by a-, a 8 , or ag. 



Determinants of similar form to (38) may evidently be written down 

 at once for any value of K when A T = 3. The E's will in all cases 

 denote polynomials of degree K — 1; vanishing at n(N, K — 1) — 1 

 points. 



When N is greater than 3 we have always q > K; hence if we select 

 from the equations of form (37) a set of q equations which are inde- 

 pendent we may write the C's proportional to a set of determinants 

 having linear elements; these determinants must accordingly be reduci- 

 ble polynomials and must possess a common factor of degree q — K, 

 the other factor being one of the C's. 



Not every set of q equations of the type of (37) will be independent. 

 For example let A T = 4, K = 2, so that n — 10, m = 3, q = 4, and 

 Q = S[2, 5] = 10. If we choose our four vectors br'-b^ to be 

 [xa;a 5 ] where i = 1, 2, 3, 4, the coefficients of C/ in our four equations 

 will be (xibj) (pqrj) where p, q, and r are different from i and from 

 each other and are less than five ; while j has any value from six to ten 

 inclusive. The four polynomials in a,- are linearly related; for we 

 may write as a special case of (29) 



