QUADRATIC VECTORS. 437 



constant plane, is in the tangent plane at /Ss, in contradiction with the 

 result just obtained. Hence tt, t, and f are not coplanar. 



I shall now show that the right member of (203) can always be 

 thrown into the form V(f)pdp when f , tt, and r are not coplanar. Write 



<f)p = XiVtt + XiV^it + xzij., (204) 



where ii is to be determined. Also 



dp = giXiV-KT + giXiV^TT + {gifx + giVirT)xz, (205) 



these simultaneous forms of 4> and 6 corresponding to coincidence of 

 two axes of the linear vector <t)~^dp. If we now take the vector product 

 V(i>pdp and equate to (203) we find, comparing vector coefficients of 

 XiXi, x^, and 0:2X3, 



(f/a — gi)S^irT = 1, gsS-rrn = 1, St/j, = 0, 



t= {gi- g2)VV^TT-n - gsirS^TTT. 



From these equations we obtain the solution, for g^ and n, 



gsS^TTT = -1, fjL= V^T-\- Vtt (206) 



whence 4> and 6 are known, gi and g2 being arbitrary provided their 

 difference be constant. 



35. Taking next the case of a quadratic vector having a triple axis, 

 but no axis of higher order, a number of situations may arise. Sup- 

 pose first that we have one triple and four single axes. Let the triple 

 axis be jSi. It is always possible so to choose ^2 and /Ss that of the two 

 other axes neither shall lie in the tangent plane to (3) at jSi, (because 

 only one single axis can lie in that plane), and, at the same time, so 

 that these two remaining axes shall not be coplanar with 81, (because 

 four axes cannot be coplanar). Let /3i, ^2, and jSs be rendered zeros. 

 We then have the form of the right member of (201). The three 

 vectors ai, a2, and az must be diplanar, and we can factor into V(t>pdp 

 as before. 



Proof. Suppose (aia2a3) = 0. These three vectors will not all be 

 parallel, for Fp is not reducible. Let one of them be expressed in 

 terms of the other two, (non-parallel). The scalar coefficients of these 

 two define, by their vanishing, two quadrics. Two cases may arise. 



Case 1. These quadrics are tangent at ^i. Neither of the single 

 axes not first chosen to be zeros can be now a zero. But all non-zero 



