QUADRATIC VECTORS. 439 



The onlv terms other than those in 3' are 



— kay' + khiy' 



Hence h\ = a gives 2 = as the tangent plane to (3) at i. 



The reasoning of the preceding article may now be extended to 

 cover the case of a quadratic vector with one triple axis and either one 

 or two double axes. We proceed as in Art. 34. If the axes originally 

 selected to be zeros along with jSi remain single axes, we still obtain 

 the form of (201). The resulting vectors a cannot be coplanar and 

 we may factor as before. If, on the other hand, the axes selected 

 to be zeros approach each other, we obtain the form of (203). The 

 reasoning already employed shows that the plane of the resulting 

 vector cannot be constant, and we may factor as in Art. 34. 



36. If the given quadratic vector has two triple axes and one 

 single axis, the preceding reasoning fails. A normal form for this 

 case is (159). It will be simplest to consider two subcases, according 

 as the three distinct axes are, or are not, diplanar. 



Case 1. The three axes /So, ^-o, and ^^ are diplanar. Let these 

 axes be made zeros by adding to the normal form (159) a proper 

 term of the form pSbp. The vector then takes the form of the right 

 member of (201). The constant vectors a will not be in the same 

 plane, and we factor into V4>pdp as in Art. 33. 



Proof. Suppose the vectors a to be coplanar. Let one of them 

 be expressed in terms of the other two. The scalar coefficients of 

 these two will then define, by their vanishing, two quadrics. The 

 four intersections of these quadrics can only be at 182, jSs and /S?, since 

 there are no other axes. Hence one of the axes jSa or ^b must be a 

 double intersection for the two quadrics. Suppose the axes to be 

 numbered so that /Ss is a double intersection for the two quadrics. 

 The four intersections of these quadrics count as four of the seven 

 axes of the quadratic vector. The plane of the vectors a must contain 

 the other three, which, however, can only be at 182 and ^5 since there 

 are no other axes in that plane. It was proved in Art. 35 that if a 

 quadratic vector of constant plane has one axis in that plane a zero, 

 and only one other axis in that plane, the plane itself is the tangent 

 plane to the cones (3) at the zero. That is, the tangent plane to 

 (3) at ^2 passes through /Ss. 



Consider the normal form (159). The tangent plane to the cones- 

 (3) at ^2 is the plane (23p) = 0. Hence we must have (235) = 0. 



