THE AXES OF A QUADRATIC VECTOR. 337 



the quadratic vector can then be reduced to a binomial in the vectors 

 132, iSs, the four diplanar axes becoming zeros, the value of 8 being 

 unique. 



If it be required to have two sets of three coplanar axes one way is to 

 let An and A22 both vanish. But we have another equally easy way; 

 if we let .421 and ^^32 both vanish, we shall have two sets of coplanar 

 axes, /3i being common to both sets. To prove this, note that, by 

 (25) and (26), if Aoi = Azi = 0, we have two linear equations in the 

 quantities 



(561) (236) (167) and (451) (234) (417) (27) 



the determinant of the coefficients being (314) (126) - (316) (124), 

 which is the same as (123) (146). Now (123) is different from zero 

 by hypothesis, for in assuming the form (2) we assume ^i, ^2, and ^3 

 to be diplanar;^ on the other hand (146) may well be zero; if so we 

 may write /3i = vi^i + Jj/Se and either of the two equations reduces 

 at once to 



m (236) Ce - w (234) (74 = 



by cancelling factors which cannot vanish if the quadratic vector is 

 irreducible. This is the same as 



(176) (236)^6 + (174) (234)C4 = 



which by an identity of the form (23) gives 



(175) (235) = 0. 



Now (235) cannot vanish along with (146) hence (175) = 0, that is, 

 we have two sets of coplanar axes having (8i, in common. 



If the determinant in question does not vanish, the quantities 

 (27) must both vanish. Remembering that we cannot have two^ 

 distinct sets of coplanar axes nor four coplanar axes, if the quadratic 

 vector is irreducible, we see by inspection that neither (234) nor (236) 

 can vanish. Hence we must have the same case as above, viz. two 

 sets including /3i. 



If it be required to have three sets of coplanar axes it is now easy to 

 pick out two quite different cases: 



1°. We may let An, A22, and ^33 all vanish. Three of the four 

 axes ^4, /35, (Se, ^i, lie, respectively, in the faces of the triedron whose 

 edges are 61, ^2, /Sa- 



5 It was shown in C. Q. V. that a quadratic vector possesses two distinct 

 ^ets of three diplanar axes except in certain special cases. 



