ALGEBRAIC POINT FUNCTIONS IN N -SPACE. 411 



Of the polynomials of a standard set, not fewer than n — m — t can 

 be connected by a relation of the form 



2c<L<(x)P<(x) = (37 c ) 



unless the linear polynomials Lj(x) all vanish at more than t of the points 

 ai, • • -a„; it is understood that Li(x) is of the form (37b), and that c i is 

 independent of x. 



Proof. Suppose a relation of the form 



cM*)Pi(x) + • • • + c w L w (x)P w (x) = (37 D ) 



where w = n — m — t — 1 and where the L's all vanish at the / points 

 9>w+ir ' ■ » &w+t, hut at no others of subscript less than n-\- 1 . 



The polynomial Lj(x) is the determinant of the coefficients of the N 

 vectors x, y, a,, and the N — 3 adjoined vectors, any or all of which 

 might coincide with an equal number of the original set ai---a n , 

 according to the value of t. Thus Li(x) is a linear function of a,, and 

 Ly(x) is the same linear function of a, except perhaps in sign. If we 

 write L;(x) = Z/(a<) we may write (37/)) in the form 



c 1 L(a 1 )P 1 (x) + • • • + c u X(a u .)P«,(x) = (37,) 



It suffices to show that Z(a»_ m ) = contrary to hypothesis. 



Now Ci must be a polynomial of degree K — 1 in a*. For all the 

 terms of the supposed identity (37,) are of degree K in a; with excep- 

 tion of the iih term; is independent of a;; and L(s,i) is linear in a*. 



Furthermore Ci is of degree K — 1 in each of the a's from a„_ OT+ i to 

 a„ inclusive. For (37,), being an identity, will subsist if we make 

 a» coincide with a„_ m+3 ; Pi will be unaltered since it is independent 

 of a ; ; all other P's vanish; L(a„_ m+ y) by hypothesis does not vanish; 

 hence c 4 must vanish when a t coincides with a„_ m+J ; that is, Cj is a 

 polynomial of degree K — 1 in aj vanishing at the m = n(N, K — 1) 

 — 1 points a n -m+i" • • a„, and is therefore a polynomial of degree K — 1 

 in each of these points. We may accordingly identify the c's of (37,) 

 with the polynomials E m (a,i) of (32) except, in each case, for a constant 

 multiplier </j. 



If, finally, we make a, coincide with a„_ m , r, will not vanish, for a 

 polynomial of degree K — 1 cannot be made to vanish at more than 

 m arbitrary points; Pi is independent of a»; and so L(a„_ TO ) must 

 vanish, contrary to hypothesis. 



Hence the identity (37/)) is impossible unless the L's all vanish at 

 more than / of the points ar • -a,,, as was to be proved. 



