QUADRATIC VECTORS. 391 



possible to choose seven distinct axis not having four in the plane 

 SjSp = 0. This plane, with the plane of two other axes, constitutes 

 a quadric cone. 



There are no other possibilities; that is, it is never possible to 

 choose, for a reducible quadratic vector, seven distinct axes without 

 six on a quadratic cone, (excluding the quadratic vector pSdp) . This 

 is the theorem. 



When the axes are not already known, we may test F{p) for re- 

 ducibility by resolving VpF{p) into scalar components in any con- 

 venient manner, and examining these scalars for common factors 

 according to any of the well-known geometric or algebraic processes 

 for detecting reducible polynomials. 



When, by any method, a scalar factor has been found for VpF{p), 

 we throw F{p) into the proper type form, by the processes of Art. 9. 



It is of value, in theoretical investigations, to have tests for re- 

 ducibility not requiring resolution into components. These are 

 always possible. For example, if F(p) is of type (a) or type (c), 

 VpF{p) is always in the plane at right angles to a. Hence if pi, P2, 

 and p3 are any three values of p we must have 



S(Fpii^Pi) (Fp,Fp2) {Vp,Fps) = (58) 



The further study of these tests leads naturally to the use of differ- 

 ential operators, and lies beyond the purpose of this paper. 



