QUADRATIC VECTORS. 389 



where a is a constant vector. We may thus write (45) in'the form 



Vp { F{p) -qa}=0 (48) 



By theorem III the vector in braces is a scalar multiple of p, and it is 

 of the second degree, giving 



F(p) = qa + pi (49) 



where i is a linear form in x, y, and z. It is evident that a is an 

 axis of the right member of (49). The cone 9 = is a cone of axes. 

 (49) may be regarded as a normal form for type (a) of reducible quad- 

 ratic vectors. In vectorial language, a scalar quadratic form may 

 always be written Spdp, where dp is a linear vector, and a linear form 

 t may always be written S8p where 5 is a constant vector. (49) then 

 becomes 



F(p) = aSpdp + pS8p. (50) 



No change occurs in the order of reasoning in case the quadratic 

 form s, that is Spdp, is reducible to a product of linear factors. By 

 Art. 3, if a quadratic vector possesses two sets of three coplanar axes, 

 the six axes being distinct, it is a reducible vector. Any vector in 

 either of the two planes containing the sets of three must be an axis 

 of the vector, which may be written, as a normal form (c), 



F{p) = aS/3ip^2p + pS8p (51) 



where |8i and ^2 are constant vectors normal to these two planes, 

 giving S0ip and S^2P two linear forms. 



If the common factor of the left members of (3) is a linear poly- 

 nomial, we shall have, instead of (45), 



VpF(p) = S^p-Gifi) (52) 



where SjSp is the linear factor and G{p) is, consequently, a quadratic 

 vector. Multiplying both sides by p and equating the scalar parts, 



SpG(p) = 0, (53) 



whence by theorem II 



G{p) = Vpcl>p, (54) 



