THE AXES OF A QUADRATIC VECTOR. 333 



The factor (P4P5O is one of the minors of the determinant (22) p. 381 

 of C. Q. v., which, developed by the method there given, yields 



(P4P5O = (123) (234) (235) (415) (7) 



Expanding the factors (P^Pei) and (P^Pd) in the same way and 

 dropping the common factor (123) we have from (6) 



_ ^ (567) (235) (236) (516) , . (647) (236) ( 234) (614) 



, ^ (457) (234) (235) (415) ,.. 

 + ^^ iP.P.P.) ^'^ 



From (4) we see that An= (/So/Ssai) if we neglect the factor (123), 

 We therefore have from (8) 



• An = (234) (235) (236) j ^'''^ ^'''^ + ^'^^'^ ^'''^ + ^'''^ ^'''^ ] (9) 

 ^ ^ ^ ( (P5P6P7) (P6P4P7) (P4P5P7) ) ^ ^ 



Similarly A2i= (/33|8iai) giving 



. ^ 2 ^314) (567) (235) (236) (516) 



'' 456 (P5P6P7) ^ ^ 



where the terms of the sum are obtained from one another by cyclic 

 advancement of the numbers 4, 5, 6. Also .431 = (/Si/Soai), giving 



J _ ^ (124) (567) (235) (236) (516) ,... 



456 {Pf>P(,P7) 



3. Identities on which Depends the Simplification of the A's. 



The other six ^'s are at once obtained from the first three by cyclic 

 advancement of the numbers 1, 2, 3. Before doing this however, we 

 may simplify the expressions just obtained, by means of identical 

 relations connecting the factors which enter the numerators with the 

 determinants which occur in the denominators. 



The denominator (P4P5P7) is the determinant of the coefficients of 

 the three vectors P4, P5, and P7, and aside from a factor (123)^ is 

 equal, by C. Q. V. page 384, to the expression C1234 (5, 7), which 

 vanishes when the six vectors lie on a quadric cone. For our present 

 purpose we may use a simpler notation and write 



(P4P5P7) = (123)- Ce (12) 



