ALGEBRAIC POINT FUNCTIONS IN N-SPACE. 409 



instead of from the determinant (4). It was shown that in this case 

 any three C's are connected by a relation of the form 



L P C P + L q C q + L r C r = (33) 



where the A's are linear polynomials. It is easy to see from (30) 

 that a similar relation holds for all values of N and K. For, writing 

 for the sake of brevity 



n - vi - N + 1 = n(N - 1, K) - N + 2 = q; m = n(N, K - 1)- 1 



(340 

 and with a notation like that of the preceding article, 



we multiply b into (30), taking the dot product. The first term 

 vanishes because b«x = 0. The last A T — 2 terms also vanish. The 

 products b«ai etc. up to b»a g+ i are obviously linear polynomials in x, 

 while E m (a,i) etc. are constants. We now have 



b*ai£ m (ai)Ci 4- • • • + b«a,+iE m (a, + i)C g+ i = (35) 



A standard set of polynomials, as already pointed out, becomes a 

 set of C's when each is multiplied by a proper constant. We have 

 therefore by (35) the following theorem: 



Theorem IV. Any q -\- 1 polynomials selected from a standard set 

 are connected by relations of the form 2LjP; = 0, where Li- • -L g+ i are 

 linear polynomials of the form b *a;. 



If S[p, q] be the number of ways in which p points may be selected 

 from q points, (i.e. S is a binomial coefficient), the number of such 

 relations connecting the polynomials of a given standard set is 

 S[q + 1, n]S[N - 2, n - q - 1]. 



The factor last written may be denoted by Q, that is 



Q = S[N -2,n-q-l] (36) 



This is the number of relations of the form XLiPi = which may be 

 written down connecting a given group of q + 1 polynomials selected 

 from the n polynomials of a standard set. (We assume N > 2, K > 1). 

 If N = 3, A' = 2, we have n = 6, q = 2, Q = 3. If either A T or K be 

 larger, we have Q > q -\- 1 . 



We may now prove that, regarding a relations of form (35) as 

 equations satisfied by a particular selection of g 4- 1 of the C's, not 

 more than q of these equations can be independent. For consider 

 the m + A T — 2 points a g+2 , a g+3) - • -a„. These are the a's which 



