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 ^nd ^(124)^(451) (315) (125). 



By hypothesis, g is not zero. We cannot have (124) = since 182 

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

 since there cannot be two axes in the tangent plane, other than /3i. 

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

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

 axes /81, ^2, iSi, and jSs, coplaiiar, 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{l24:) (125) (145)}, (190) 



the result being independent of ^1. For the constant u we find 



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



+ (315) (124) } + srx(145) (314) (315)]. (191) 



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

 Ci nor C5 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 |Si and passing through (80, 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 j8i and /Sa. But .T3 vanishes 

 when 184 is put for p. Hence ^4 must coincide with ^2 in direction, 

 contrary to hypothesis. Similarly, C5 cannot vanish. 



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

 vanish, is by means of the identity 



7r(145) = /3i(457r) + ^4(5l7r) + jS^iUw) (192) 



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

 of (187) by |8i and taking scalars we have 



C,{Ut) + a(142) (314)2 ^ q, ,.__^ 



C,(51t) + a(512) (315)2 = q. ^^^'^^ 



