QUADRATIC VECTORS. 393 



shall vanish. /3i is thus a multiple axis of F(p) when, and only when, 

 this condition is satisfied. 



We may now suppose ^i, /Ss, and /Se to be three more axes, as in Art. 5. 

 If we assume, as before, that no quadric cone can be passed through 

 the six vectors |Si. . ./Se, and that ^i, /Ss and iSe are diplanar, the investi- 

 gation of Art. 5 is valid through (27). The vanishing of (63) must 

 therefore be equivalent to a relation between the constants of pro- 

 portionality ki, ki, and k^. To obtain this relation we have first to 

 write a2 in terms of ^i, ^5, jSe, by an identity like (28), 



(456)a2 = /34(56a2) + /35(64a2) + ^ei'iBa^) (64) 



For A22 we then have 



(123) (456)^22 = (456) (31a2), by (60), 



= (314) (56a2) + (315) (64a2) + (316) (45a2), by (64), 



= (314)A:4(123) (315) (316) (526) 



+ (315)A:5(123) (316) (314) (624) 



+ (316)^-6(123) (314) (315) (425), (65) 



by using the value of (45a2) from (19), with two similar expressions for 

 (56a2) and (64a2)- The result may be most simply expressed by 

 taking a vector k such that its components along F/Ss/Se etc. are k4, k^, 

 ke, that is 



(456) K = ^4^/35/36 + kiV(3e^i + k.V^i^-, (66) 



We then have, multiplying both sides by ^2 and taking scalars, 



(456)Sk^2 = 5(^4/32^5^6 + kr,^2^e^i + ke^2^4^5) ■ 

 = - [k,{562) + A:5(642) + A:6(452)] 



because the scalar of the product of three vectors is the negative of 

 their determinant. (65) may now be written 



A22 - - (314) (315) (S1Q)Sk^2 (67) 



By similar reasoning 



A33 = - (124) (125) (126)Sk/33 (68) 



To obtain A23, it is more practicable to use, in (20), the determinant 



