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 + pt (49) 



where t 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 g = 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 S5p where 6 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) = aSl3ip^2P + pS8p (51) 



where jSi and 182 are constant vectors normal to these two planes, 

 giving Sl3ip and SjSop 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-G(p) (52) 



where S^p 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) = Vpct>p, (54) 



