ALGEBRAIC POINT FUNCTIONS IN N-SPACE. 421 



It is interesting to note that q — q' = n[N — 2, K], serving as a 

 check on the correctness of the table. 



8. Special Cases of the General Identity: Method of 



Reduplication. 



A third method of obtaining identities from the fundamental iden- 

 tities (5) or (9) depends on the use of a function of two variable points. 

 I therefore call it the method of reduplication. An example of the 

 use of this method for the case N = 3, K = 2 I have elsewhere given. 4 

 Let F(x, y) be a function (vector or scalar) of two points x and y. 

 Applying (5) or (9) we have 



F(x, y)C + F(a x , y)d + F(aa, y)C 2 + • • • + F(a», y)C n = (59 ) 



and by giving to y in succession the values ai, a 2 , • • • a n we have the n 

 further equations 



F(x, ai )C + F( ai , ai)d + F(a 2 , ai )C 2 + • • • + F(a„, ^)C n = (590 

 F(x, aOCo + F(& u a 2 )d + F(a 2 , a2)C 2 + • • • + F(a„, a 2 )C„ = (59 2 ) 



F(x, a„)C + F(a x , a„)d+ F(a2, a„)C 2 + • • • + F(a„, a„)L7„ =0 (59„) 



We now let x = y in (59o), then multiply these n-\- 1 equations by 

 Co, C\, C 2 , C3, • • • C n , respectively, and add. For simplicity of nota- 

 tion we write ao = x. The result is 



SF(a fj a,-)CiC,- = (60) 



where both i and j run from to n inclusive. From this identity, by 

 giving to the function F various forms, scalar or vector, a great number 

 of relations connecting the C's may be obtained. The method may 

 also be applied, by triplication, to functions of three points, and so on. 



4 The Axes of a Quadratic Vector, Proc. Amer. Acad. Arts and Sci., vol. 56, 

 No. 9. June 1921, p. 333. 



