410 HITCHCOCK. 



correspond to the terms absent from (35). Let E\, Ei, ••■E q+ \hz 

 polynomials of degree K — 1 which vanish, respectively, at q + 1 

 different selected groups of m points chosen from the above m + A T — 2 

 points. Let bi, b2, • • -b g+ i denote those linear vector functions of x 

 of form (34) into which, in each case, enter the a's not occurring in 

 the corresponding E. Corresponding to each choice of E and corre- 

 sponding b we may write an identity of the form (35). Thus 



bi*aiEi(ai)Ci + • • • + bi'a,+i£i(a g+ i)C 9+ i = 

 b 2 -a 1 E 2 (a 1 )C7 1 + \- b 2 «a 9+1 E 2 (a a+ i)C g+1 = (37) 



bg + i , ai/ig + i(ai)Ci-r • • • -f- b g+ i , ag + i£ g+ i(a g+ i)(y g+ i — 



We may regard these as q + 1 linear equations satisfied by the q + 1 

 C's which enter. We know that these C's are actual polynomials of 

 degree K in x. Hence the equations cannot be independent, as was 

 to be demonstrated. 



On the other hand we may in general select q such equations which 

 shall be independent. For consider the coefficients of Cy in the suc- 

 cessive equations: they are of the form b^a/F^ay), and may be 

 thought of as polynomials of degree K in a ; . They all vanish if ay 

 be made to coincide with any one of the points a g+2 - • a„ or x, by 

 hypothesis, that is at m + N — 1 points. In general n polynomials 

 may be linearly independent. Hence of these polynomials n — 

 (m + A T — 1) may be linearly independent. But this number is q, 

 as was to be shown. A specific rule for selecting the q equations will 

 be given below. 



Corollary to Theorem IV. The following corollary to theorem 

 IV will be essential in the applications made below. We suppose 

 given a standard set of polynomials P](x),- • -P„(x) based on a set 

 of vectors ai, • • -a n as already defined. We now adjoin A T — 3 other 

 vectors which need not be distinct from the others, and which we may 

 call a n+ i • • -etc.; and write as in (34 2 ) 



bi = [xa/,.a ; - • •] (37^) 



where the A' — 2 vectors a*, a ; , • • • which occur in b; are precisely 

 those a's (out of the total number of n -f- A T — 3), which do not occur 

 in P^ It is evident that bi«y will be a linear polynomial in x. Write 



Li(x) = bi-fr (37 B ) 



where y is an arbitrary vector. The corollary may now be stated: 



