402 



HITCHCOCK. 



2. Geometric Meaning of the Coefficients. 



To take a specific example, let K = 1 and N = 3. The vector 

 determinant of the form (4) becomes 



P(x) , Xi , x 2 , x 3 



P(a0 , an, ai2, ai3 



F(a 2 ) , a 2 i, a 22 , (ha 



F(a 3 ) , o 3 i, 032, a 3 3 



The coefficients Co, Ci, C 2 , and C 3 become determinants of the com- 

 ponents of three vectors. If we denote these determinants by (123), 

 (.r23), (zl3), and (.rl2), the identity (9) becomes 



F(x) (123) - P(ai) (a-23) + F(a 2 ) (a-13) - F(a 3 ) (xl2) - 0, (10) 



a relation very familiar to students of vector analysis, F(x) being now 

 a linear vector function. When K = 1 a similar relation holds for 

 any value of N, whence clearly the vanishing of any C means coplan- 

 arity of the vectors which enter into it. 



Next take K = 2 and N = 3. The vector determinant of the form 

 (4) becomes 



(11) 



F(a 6 ) , 061 2 , «61«62 , «6lfl63 , «62 2 , «62«63 , «63 2 



In this case the vanishing of one of the coefficients C means that the 

 six vectors which enter into it lie on a quadric cone, (which may, how- 

 ever, be a degenerate cone). For the equation d = is homogeneous 

 of the second degree in six vectors, and holds true when any two of 

 them coincide, because two rows of the above determinant become 

 equal. Therefore if, as usual, we regard x as a variable vector, the 

 equation d = 0, for values of the subscript other than zero, is the 

 equation of the cone determined by the five fixed vectors which enter 

 into it. Similar reasoning applies for any values of A T and K. 



3. The Coefficients as Eliminants of K-adics. 



The coefficients C may be interpreted in another way, so as to con- 

 nect them with the theory of matrices and allied operators. We take, 



