QUADRATIC VECTORS. 429 



Taking the third factor of (189), if (314) = 0, ^i lies in the tangent 

 plane. The two non-vanishing determinants of the matrix reduce to 



^(124)2(452) (315)2 and ^(124)2(451) (315) (125). 



By hypothesis, g is not zero. We cannot have (124) = since ^2 

 was taken without the tangent plane. We cannot have (315) = 0, 

 since there cannot be two axes in the tangent plane, other than ^1. 

 Similarly, we cannot have (451) = 0. Hence if both determinants 

 vanish, we have simultaneously (452) = and (125) = 0, the four 

 axes |8i, /So, ^i, and 185, coplanar, which is impossible under the hypo- 

 theses. 



Taking the fourth factor of (189), the same reasoning holds. The 

 last factor differs from zero by hypothesis. Hence the solution of 

 (188) is unique, aside from a factor of proportionality. This factor 

 aside, we find for the constant a the value, by an easy computation 



a= - (123) (145) { (314) (315) (245) + g{124.) (125) (145)}, (190) 



the result being independent of gfi. For the constant w we find 



u = (123) (245) [- (314) (315) (345) + ^(145) { (314) (125) 



+ (315) (124)} + .9i(145) (314) (315)]. (191) 



In determining the vector w we shall not fail to remark that neither 

 C'i nor Cf, can be zero. This can be shown from the fact that, if d is 

 zero, (for example), Fp is a limiting case of a reducible vector, Cp 

 defining by its vanishing a quadric having four elements in common 

 with (3) at /3i and passing through 182, so that if it passes through 

 another axis we have six on a quadric. Better, if C4 = 0, it is evident 

 from (186) that jSi lies in the plane of |8i and /So. But Xz vanishes 

 when ^i is put for p. Hence 184 must coincide with ^2 in direction, 

 contrary to hypothesis. Similarly, C5 cannot vanish. 



The most natural way to determine tt, provided (145) does not 

 vanish, is by means of the identity 



7r(145) = i8i(457r) + /34(5l7r) + ^5(14x) (192) 



The component (457r) is given by (189). Multiplying both equations 

 of (187) by jSi and taking scalars we have 



C4(147r) + a(142) (314)2 ^ q, ..q„. 



C,(51t) + a(512) (315)2 = q. ^^^"^^ 



