QUADRATIC VECTORS. 395 



(123) [(124) (345) (316) (25G) + (126) (365) (314) (245)] 

 + (314) (316) (235) { (124) (256) + (125) (264) + (126) (245)}. 



The expression in braces vanishes, for it is identically equal to 



(122) (456) 



by a transformation like (64). The coefficient of (123) may be 

 written 



(453) (613) (562) (142) - (452) (612) (563) (143) (72) 



by a mere rearrangement. But this is the same as C4561 (23) by the 

 notation of Art. 6. 



Collecting the coefficients of —Sk^2 we have precisely similar trans- 

 formations to make, except that, in the last factor of every term, (3i 

 is written for ^2, i- e. 1 for 2. Thus the coefficient of (123) in the result 

 is 



(124) (345) (316) (156) + (126) (365) (314) (145), 



while the other terms contain the factor (121) (456) and vanish. 

 By a slight rearrangement, the two above terms may be written 



(453) (613) (561) (142) - (451) (612) (563) (143) (73) 



which may be regarded as derivable from (72), symbolically, by the 

 operation 1 — ; this is the same as saying, geometrically, that a 



quadric cone through the vectors ^i, 185, /Se, /Si, and 0z is denoted by 

 C456i(p, 3) = 0, and that the polar of j8i with respect to C be taken 

 at 02- If the tangent plane to this cone at /3i, obtained by polariza- 

 tion of C, be denoted by r466i(pi, 3) = 0, we shall naturally write (73) 

 as 7'456i(i2, 3). The relation between T and C is most easily expressed 

 by the operator V, thus 



^4561(12, 3) = S^iV' -(74561(2', 3), (74) 



where, as indicated by the accents, V operates on ^2 alone, or, if we 

 prefer, p is written for ^2 before the operation. These results enable 

 us to write, from (70), (cancel (123)), 



^23 (456) = S/c/3i-C456l(2, 3) - Sk^2- 7^456l(l2, 3) (75) 



