QUADRATIC VECTORS. 387 



it can be written as VpFn-\{p), where F„_i(p) is a vector of degree 



71-1. 



For let the components of Fnifi) be X, Y, and Z. The identity (37) 

 is equivalent to 



xX + yY -\- zZ = 0. (38) 



When y and z vanish together, x does not in general vanish, hence X 

 must vanish. Therefore X, as a polynomial in x, y, and z, can contain 

 no term in a;". We may therefore write X = yw -\- zv where v and w 

 are scalar polynomials of degree n — 1. Similarly, 



Y = zu -\- xw' and Z = xv' -\- yu' • 

 (38) becomes 



yz{u + v!) + zxiio + v') + xy{w + w') = 0. (39) 



When .r = neither y nor z are generally zero, hence u ■\- u' vanishes 

 all over the plane a; = 0. With similar reasoning for y and z we may 

 write 



u-\- u' = px, V -\- v' = qy, w -{- w' = rz, (40) 



where, in the case n = 1, the factors p, q, and r are necessarily zero, 

 since u, u', etc, are constants, but for larger values of n we may have 

 p, q, and r polynomials of degree n — 2. From (39) we now obtain 



p + g + r ^ • (41) 



By eliminating u', v' , w', and p, we have 



X = yw -^ zv, Y = z(u + rx) — xw, Z = — vx — y{u + rx) (42) 



If, therefore, we write 



Fn-iip) = iP + jQ -\-kR = i{u + rx) + j{- v) + kw, (43) 



we find by actual multiplication 



VpFn-iip) = i{yw + zv) + j(zu -\- zrx — xw) — k(vx -{- yu-\- yrx) 

 = Fnifi), by (42). 



