quadratic vectors. 377 



Vectors of the First or General Type. 



5. Theorem I. A quadratic vector of type I is completely deter- 

 mined by its axes, aside from a constant non-vanishing multiplier h 

 and an additive term tp. 



The truth of the theorem would, on the proviso that the axes are 

 independent, appear from a count of the scalar constants involved. 

 For the three quadratic forms X, Y, Z involve eighteen scalars. The 

 multiplier h, and the three scalars which determine t, leave fourteen. 

 The seven axes, if independent, involve fourteen. 



That the axes may, in fact, be assigned arbitrarily, 1 shall show by 

 expressing F{p) in terms of its axes by means of a vector equation. 



The theorem further implies that, no matter what special relations 

 exist among the axes of F{p), provided they are distinct and of finite 

 number, anv other vector having the same axes mav be written 

 hF{p) + tp/ 



To prove the theorem, let any seven axes be chosen, distinct, with 

 no six on any quadric cone. Let vectors in these seven directions be 

 denoted by |3i, /S-i, . . 187. We may, without loss of generality, suppose 

 that jSi, ^2, and ^3 are not coplanar, and, at the same time, that 04, 0b> 

 and jSe are not coplanar. For the seven axes cannot all lie in the same 

 plane, because such a plane, with any other plane, would constitute 

 a degenerate quadric cone, contrary to hypothesis. Accordingly, let 

 any three non-coplanar axes be called j3i, ^2, and 0s- The remaining 

 four axes cannot all lie in the same plane, for, with the plane of /3- and 

 02, it would constitute a quadric cone. Let any three which are not 

 coplanar be taken from the four and called 184, jSs, and 0(,- 



I shall now determine the constants p, q, and r, of the linear form t, 

 so that 01, 02, and 03 shall be zeros of the vector F{p) -\- tp; that is, 

 so that we have, simultaneously, 



i^(i8i) + t0, = 0, F(P2) + t02 = 0, ^(^3) + t03 = 0. (5) 



Let the components of /3i, 82, 0z be given by 



01 = ibn + jbn + A-6i3, ^2 = 1621 + i&22 + ^'^23, 



03 = ibzi + jb32 + ^^33- 



Let the determinant of the components be denoted by (123), i. e. 



(123) = 



(G> 



