388 HITCHCOCK. 



The vector Fn-i{p) is not uniquely determined, since we may add to it 

 an arbitrary vector term of the form pt, where t is a scalar polynomial 

 of degree n — 2. 



Theorem III. If a vector Fnifi), whose components are homo- 

 geneous polynomials in x, y, and z of degree n, satisfies the identity 



VpFn{p) = (44) 



it can be \\Titten in the form pt, where i is a scalar polynomial of degree 

 11 - 1. 



Proof. Identity (44) impHes that all directions of p are axes of the 

 vector, or that equations (3) become identities for the vector in ques- 

 tion. It follows that X vanishes all over the plane ar = 0, and we 

 may write X = tx where / is a polynomial of degree n — 1. Simi- 

 larly, Y = ty and Z = tz, the factor t being the same in all three cases, 

 by (3), This proves the theorem.^ 



9. Returning now to the case of a reducible quadratic vector F{p), 

 if the common factor of the left members of (3) is a quadratic poly- 

 nomial which is irreducible, we have 



VpFip) = qct>p (45) 



where q is the quadratic scalar and 0p is a vector of the first degree 

 in p. If we multiply both sides of (45) by p and take scalars we have 



Spct^P = 0, (46) 



because S-pVpF(p) = S-p^F(fi) = 0. Therefore by theorem^II 



# = Vap (47) 



9 Theorems similar to II and III may be proved by Euler's theorem for any 

 vectors whose components are homogeneous of the same degree. In general, 

 if F{p) is of degree n, we have the identity 



F(p) s VpFn-i(p) + Vs, 

 where s is a scalar function of degree n + 1 and V is the differential operator 



From this, theorem II follows at once, s being zero. The vector i^n_i(p) may 

 always be taken parallel to VvF{p). See Phil. Mag., 29 (May 1915), p. 

 704. 



