QUADRATIC VECTORS. 383 



(123)2 (456)^^i87 = lcMP^P6P^) + kMP^PiP7) + kMP^P^Pr), (29) 



Comparing corresponding components of (jS?) in (28) and (29) we 

 have, as necessary and sufficient conditions that 187 shall be an axis, 



, h{5Q7) {my , _ A(647) (123)'' _ h{457) (123)' 



^' = "WWT' ' " (PeP4P.) ' ^' ~ {P.P.P.) ' ^^'^ 



Allowing for the moment that none of the denominators vanish, we 

 may introduce these results in (26) and have, finally, 



K 567) {P,P,Pp) A(647) (P,P,Pp) 



'^^^ ^'' (456) (P5P6P7) (456) (P6P4P7) 



/^(457) (P4P5PP) ,„.. 



+ ^' \m{p.p.py ^ ^ 



If 187 be written for p, the right side reduces to h^i by the identical 

 relation (28) . h cannot be zero for Po(p) would vanish and F(p) would 

 reduce to the term tp, contrary to the hypothesis that F{p) is of type I. 

 h is otherwise arbitrary and two vectors alike in possessing the axes 

 j8i, 182, •187 can differ in the constants h, Ci, C2, and C3, that is, in regard 

 to h and the form t, but not otherw'ise. 



I shall now show that none of the denominators in (30) can vanish 

 if the choice of /S? is consistent with the hypothesis that no six axes 

 lie on a quadric cone; whence the seven axes of (31) are assignable in 

 any manner consistent with that hypothesis. Consider the determi- 

 nant on the right of (22), or (PiP^Pp). Expanding by the elements of 

 the first row, and developing the minors as in (18), (19), and (20), we 

 have 



(P4PbPp) = K31p) (12p) (234) (235) (415) 

 + (12p) (23p) (314) (315) (425) 

 + (23p) (31p) (124) (125) (435) } (123) 



In the first term on the right, in place of the product of the two factors 

 (31p) (234), write, identically, (314) (23p) + (123) (34p). Then em- 

 ploy successively the two identities 



(235) (415) + (315) (425) = (345) (125) and (31p) (124) - (12p) (314) 



= (123) (41p) 

 and we have 



