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 f is a scalar polynomial 

 of degree n — 2. 



Theorem III. If a vector F„(p), whose components are homo- 

 geneous polynomials in x, y, and z of degree 7i, satisfies the identity 



VpFnip) ^ 



(44) 



it can be written in the form pt, where tis a, scalar polynomial of degree 

 n - 1. 



Proof. Identity (44) implies 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 x = 0, and we 

 may write X = tx where f 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 



VpF{p) = q4>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 



Spcl>p = 0, (46) 



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



<f)p = Vap ■ (47) 



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

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

 if F(p) is of degree r?, we have the identity 



F(p) = VpFn-iip) + Vs, 

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



8x dy dz 

 From this, theorem 11 follows at once, s being zero. The vector Fn-i(p) may 

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

 704. 



