QUADRATIC VECTORS. 383 



(123)2 (456)^:^7 = kMPtPePi) + h^^iP^PiPi) + kMPiPtPi). (29) 



Comparing corresponding components of (jSy) in (28) and (29) we 

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



, t(567) (123)' , Hmn (123)' ^(457) (123)' 



"" " (P.P,P.) ■ ''' = {P>P,P,) ■ ''' = (P,PJ'.) • ^'"^ 



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

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



^( 567) {P,P,Pp) h{^l) {P,P,Pp) 



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



y^(457) (P4P5PP) .... 



+ ^' (456) (P^PsPO ' ^^^^ 



If jSt be written for p, the right side reduces to h^y by the identical 

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

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

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

 j8i, ^2, . • 187 can differ in the constants h, Ci, c-i, and Cz, that is, in regard 

 to h and the form t, but not otherwise. 



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

 if the choice of ^^ 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 (P4P5PP). Expanding by the elements of 

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

 have 



(P4P5PP) = {(31p) (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 



