406 HITCHCOCK. 



of K determinants of the form b*m, that is of the form (19); any 

 selection of A 7 — 1 of the b's determines one of the n vectors mi, m 2 

 • • • m„ which forms the numerator of the corresponding a. All the 

 remaining b's occur explicitly in the denominator. 



The polynomials Pi, Pi,- ■ -P n may now be taken of the form 



P h = (b p 'x) (bg«x) (b r «x)- • -to K factors. (21) 



The a of corresponding subscript is that a which contains precisely 

 the same choice of b's in its denominator; for example, if ai be defined 

 by (20) then Pi must be taken as 



Pi = b N -xbjv+i -x • • • b N+K ^ -x (22) 



By inspection of (20) and (22) it is evident that Pi(aO = 1, and 

 similarly for any other distribution of b's between numerator and, 

 denominator. That is P^(a^) = 1 for all values of h, as was to be 

 shown. 



It is clear that a set of polynomials defined by (21) will fulfill the 

 conditions for a standard set provided no group of A 7 vectors chosen 

 from the b's are linearly related. For the a's may be taken as in (20) 

 and no denominator can vanish. At least one of the b's which occur 

 in m,- must occur in Pi when i is different from j. Hence Pi(a,-) = 0. 



Definition III. A factored set of polynomials, together with a set 

 of points ai, a>, • • • a ?l such that P;(a/) vanishes when i is different from j 

 but equals unity when i equals j, will be called a normal reference 

 system. 



Any polynomial Pi of a normal reference system agrees with the 

 coefficient d of (5) except for a constant factor, since Pj and C,- both 

 vanish at every a except a,. By putting a^ for x in (5) we have 



P(ai)C + P(a i )C i (a i ) = 0; (23) 



but F is an arbitrary polynomial, hence P(a t ) is in general not zero. 

 Therefore we have identically 



Co + Ci(&i) = (24) 



and since P;(aj) = 1 it follows that 



Pi=-% (25) 



By substituting d = — CoPi in the vector identity (9) we arrive at 

 the following special case of theorem I, — 



