LIBRARY 

 NEW YORK 

 BOTANICAL 



CiAKl>€N 



IDENTITIES SATISFIED BY ALGEBRAIC POINT 

 FUNCTIONS IN N-SPACE. 



By Frank L. Hitchcock. 



TABLE OF CONTENTS. 



Page. 



1. The fundamental identity 399 



2. Geometric meaning of the coefficients 402 



3. The coefficients as eliminants of K-adics 402 



4. The method of standard sets 403 



5. The method of factoring . 407 



6. Application to determinants whose elements are linear polynomials 412 



7. Rule for constructing these determinants 417 



8. The method of reduplication 421 



1. The Fundamental Identity. 



Let F(x) denote a homogeneous polynomial of degree K in N vari- 

 ables X\,x<l, ■ ■ • x N . Let the number of terms in this polynomial be n, 

 that is 



n = (K + 1) (K + 2) (K + 3) • • • (K + N - l)/l.2.3. • • • (N - 1). (1) 



Let jF(a,) be the value of the polynomial when a set of values an, 

 «f2, ••• UiN is assigned to the variables X\, x-i,---x-$. We say that 

 F(a,i) is the value of the polynomial at the point a t . 



As a temporary notation, let the coefficients of the various terms of 

 the polynomial be A\, A*,- ■ • A n , with the terms written in some 

 determined order. We may suppose this order to be descending 

 order of Xi, .r 2 , etc., meaning that the term containing Xi K is the leading 

 term, followed by all terms containing x\\ then" all terms containing 

 Xi K ~ 2 and so on ; and that in each of these groups we follow descending 

 order of a*2 by a similar rule, and so on. Thus 



F(x) = A lXl K + A***- 1 * + A 3 .n K -\v 3 + • • • + A n x N K . (2) 



Now let there be a set of n points ai, a-2, • • • B>r • • a„. If we expand 

 the value of the polynomial at each of these points in the form (2) 

 we shall have n equations of the form 



Ffa) = Aia a K + A 2 a il K - ) a i 2 + ^au*- 1 ^ H + A n a iN K , (3) 



