400 



HITCHCOCK. 



which, together with (2), constitutes a set of n + 1 linear homogen- 

 eous equations in the n -f- 1 quantities A\, A»,- • • A n and unity. It is 

 accordingly evident that the determinant 





F(x), xj 



F(ai), <in K , du K ' ] "is, 

 /•'(a-j), a 2 i K , 02i X_1 a22, 



x N 



«2JV 



A" 



/"(a,,), «„i A ', ani^-'a,^, 



a„# 



(4) 



must vanish. Developing by the elements of the first column we 

 may write 



F(x)C + F(& 1 )C 1 + F(a»)G + • • • + F(a„)C„ = (5) 



where Co, Ci, etc. are the cofactors of the elements into which they 

 are multiplied. 



If x be variable, with ai, a 2 , • • ■ a„ constant, the identity (5) will be 

 the expansion of the polynomial F(x) in terms of its values at n fixed 

 points, provided (\ does not vanish. The coefficients (\, C 2 , • • • C„ 

 are polynomials of degree K in x. They are also polynomials of 

 degree K in each of the fixed points with exception of the point having 

 the same subscript: (', is independent of a,. On the other hand C is 

 a constant in the sense of being independent of x. The C's are all 

 independent of the special choice of the polynomial F(x), that is of 

 A\, A-2, • • • A n - 



Let us next take 6i, e 2 , • • • e# a system of linearly independent 

 vectors. Any point x may be regarded as a point vector, 



x = .r,ei + .r 2 e 2 + • • • + .1-^ 

 and in a similar manner 



a,- = aiiO] + 0^262 + • • • + ^.ve.y. 



(6) 



(7) 



Let us furthermore take a system of N polynomials of degree K, 

 namely Fi(x), F 2 (x), • • • F#(x), and write 



F(x) = e^x) + e 2 F 2 (x) + • • • + e^(x). 



(8) 



By definition, therefore, P(x) is a vector function of the point vector x, 

 homogeneous of degree A* in the variables x\, .r 2 , ••• xjj. Vector 

 functions are distinguished by bold-faced type. 



Each of the scalar functions which are thus assigned as components 



