ALGEBRAIC POINT FUNCTIONS IN N-SPACE. 



405 



vectors a^ are the six intersections of four planes, and the vector 

 identity (9) becomes, by putting F(a*) = F(b; X b,) --- i r , 



l p?> 



F(x) = x«(bif 23 b 4 + bif 34 b 2 + bif 24 b 3 + b 2 fi 4 b 3 + b 2 fi 3 b 4 



+ b 3 fi 2 b 4 ) *x 



(17) 



where, if F(x) is an arbitrary quadratic vector function, the six vectors 

 f are arbitrary. 



In both the above illustrations we have K — 2, and note that the 

 polynomials of the standard set are formed by selecting pairs of linear 

 polynomial from a group of K -{- N — 1 possibilities. This pro- 

 cedure is applicable in all cases. For the number of ways in which K 

 factors can be selected from a group of K + A 7 — 1 factors with no 

 repetitions is the same as the number of ways in which A' variables 

 can be selected from N variables allowing repetitions, that is n ways 

 by definition of n. Each linear factor is of the form b;*x where 



b^x = buXi + bi 2 .v» + • ■ ■ + b iN .v N 



(18) 



and there are A T + K — 1 vectors b t so chosen that no N of them are 

 linearly related. We thus arrive at the following: 



Definition II. A standard set of polynomials which has been formed 

 by taking products of K linear polynomials selected from N + K — 1 

 such polynomials without repetition will be called a factored set. 



The vectors ai, a 2 • • • a„ may always be chosen so that Pi(a) = 1. 

 We define the vector [b ; ,b g b r - • • to N — 1 factors] to be the vector 

 whose scalar components are the respective cofactors of the elements 

 of the first row from the determinant 



(19) 



We then define one of the vectors a, say a 4 , as follows ; let mi denote 

 the vector [bib 2 b 3 ■ ■ • b^i] and write 



a 4 = 



mi 



(bjv'mibivr+i'mr • •biv+A'-i'inOiv 



(20) 



The other vectors a are built up in a similar manner: each a is a 

 function of all the b's; the denominator is the Kih root of the product 



