QUADRATIC VECTORS. 411 



(b) 0-2 = 0, that is, A-22 = Ai3 = 0. The first and second expres- 

 sions in braces vanish of themselves. The third gives 



A32{A32Ai3 — A10A33 + ^21^433) = 0. (132) 



(c) C3 = 0, that is /i32 = ^33 = 0. The second and third expres- 

 sions in braces vanish of themselves. The first gives 



^23(^23^12 — A13A22 + ^31^22) = 0. (133) 



There are no other possible cases for irreducible quadratic vectors, 

 these conditions are, then, necessary and sufficient for a triple axis /3i. 

 In subcases (b) and (c), the meaning of the condition is, geometrically, 

 as follows: let VpFp = define three cubic cones by separation into 

 components along Vl3iP2, V^2l33, and VPsjSi. The second of these 

 always has a double line at j8i. If C2 = 0, the first also has |3i for a 

 double line. The condition (132) then requires than the tangent 

 plane to the third cone shall touch one sheet of the first. It is easy to 

 show that it also touches one sheet of the second, whence a triple axis. 

 We have a similar meaning for (133). 



20. We may note in passing that the rule developed in Art. 18 

 for detecting double and triple axes is applicable to vectors of any 

 degree, — with, however, one important modification. In the cjuad- 

 ratic case, S^aVVpFp can be obtained from the polar vector. When 

 Fp is of higher degree the rule may read : 



General rule for Double and Triple Axes of Vectors. 



If j8 is a double axis of a vector Fp, (homogeneous in p), the derived 

 vector Sp'V-VpFp takes the form Syp'V^ir when j8 is written for p 

 after the differentiation ; and the second derived vector S-^yV • VpFp 

 takes the form V^t when j3 is written for p after the differentiation. 

 The axis /S is of multiplicity higher than two if (and only if) S/Sttt = 0. 



These two formal conditions may be combined in one vector equa- 

 tion as follows. Let dp and 5p be two independent differentials of p. 

 The vector 



V{8VpFp) (dWpFp) (134) 



must vanish if, after the differentiation, /3 is written for p, and dp is 



