THE AXES OF A QUADRATIC VECTOR. 335 



(231) (237)C3+ (241) (247)6*4+ (257) (251)C5+ (2G1) (267)6*6 = (18) 



That is, the expression in brackets in the second hne of (17) is equal 

 to —(231) (237)63. In a similar manner, the expression in brackets 

 in the third Hne of (17) is equal to -(231) (237)C2. Therefore (17) 

 becomes 



(561) (567)6*566 + (647) (641)6*66*4 + (451) (457)6*405 



= (231) (237)026*3 (19) 



4. Simplification of the A's. 



Return now to our expression (9) for An, transform the denominators 

 as in (12), and simplify, 



An = 



(234) (235) (236) f-(561) (567)6*5(76- (641) (647) 06(74 -(451) (457)6*405] 



(123)^640506 



= - (234) (235) (236) (237) 77-^^^, by (19). (20) 



Since it is evident that all the A's will have a common denominator 

 this may be rejected, and we may write as the value of An 



An = (234) (235) (236) (237)0203 (21) 



It thus appears that An vanishes when either of the four axes ^i, /Ss, 

 j86, or /St lies in the plane of /So, /S3. The quadratic vector is assumed 

 irreducible, hence O2 and 63 do not vanish, as was proved in C. Q. V. 

 Considering next the simplification of A21, we may first add the 

 three terms, and reject the same factor as above. This gives 



A21 = - (123)-i S(314) (567) (561) (235) (236)0506 (22) 



456 



From the fundamental identity (13), letting FX = (X67) (23X), 



(567) (235)05 + (467) (234)04 + (167) (231)Oi = (23) 



and by letting F\ = (4X7) (23X), 



(457) (235)05 + (467) (236)06 + (417) (231)6*i = (24) 



It is now easy to eliminate 6*5 from the last three identities. The 

 terms in 6*66*4 cancel at the same time, giving 



