QUADRATIC VECTORS. 387 



it can be MTitten as FpF„_i(p), where Fn-i{p) is a vector of degree 

 n-1. 



For let the components of Fn(p) 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 x". 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 -\- u') + zx{v + v') + xy{w + w') — 0. (39) 



When a: = neither y nor z are generally zero, hence u -{• u' vanishes 

 all over^the plane x = 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, v! , 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+q+r^O (41) 



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



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

 If, therefore, we write 



i^n-i(p) = iP + jQ + kR = i{u + rx) + j(- «) + kw, (43) 



we find by actual multiplication 



VpFn-iip) = i{yw + 2v) + J(2w + zrx — xw) — k{vx -\- yu-\- yrx) 

 = Fn{p), by (42). 



