390 HITCHCOCK. 



where 0p is a linear vector. We may thus write (52) in the form 



Vp[F{p)-ct>p-S^p] = (55) 



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

 is of the second degree, giving 



F{p) = <l>p-S^P + pS8p, (56) 



where S8p is, as before, a linear form. It is evident that the three 

 axes of the linear vector 0p are axes of the right member and that any 

 vector in the plane <S/3p = is an axis. It is also clear that this type 

 (b) of reducible quadratic vector contains, as does (a), the more 

 special type (c) as a limiting case, since 0p may itself be reducible, 

 i. e., have an infinite number of axes. By Art. 3, a sufficient condition 

 that a quadratic vector shall be of type (6) is that it shall possess four 

 distinct axes in the same plane. 



Finally, if the left members of (3) vanish identically, theorem III 

 shows that the quadratic vector F{p) is of the form pS8p, and it may 

 then be regarded as a limiting case of either (a), {b), or (c). Collect- 

 ing results, any reducible vector F{p) of the second degree may be 

 written in one of the three type forms 



(a) aSpdp + pS8p 



(6) cl>pS^p + pS8p (57) 



(c) aS/3ipS/32p + pS5p 



10. The following negative theorem is occasionally useful, — 

 Theorem IV. If, for a given quadratic vector F{p), seven distinct 

 axes can be found such that no six lie on a quadric cone, and if VpF(p) 

 does not vanish identically, F{p) is not reducible. 



Proof. If the vector is reducible of type (a), it cannot consist 

 merely of its last term pS8p, since, by hypothesis, VpF(p) does not 

 vanish identically. Its only axes are the vector a, with the cone of 

 axes Spdp = 0. Whence it is not possible to choose seven not having 

 six on this cone. Similar reasoning applies to (c). If the vector is 

 reducible of type (b), we may suppose <^ to possess not more than three 

 distinct axes, for if so it could be written as (c).^° Whence it is im- 



10 For VpF{p) = S0p-Vpt()p. If <t>p has more than three distinct axes, Vp4>p 

 has a linear factor, by reasoning parallel to that of Art. 3. 



