204 Ml' Turnhull, Some Geometrical Interpretations 



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Forming the invariant of the polar quadric, we obtain an ex- 

 pression which reduces directly to {{AB)(A2)){Bp)\^. Hence if j9 

 belongs to the complex G, its polar quadric is a cone. 



§ 18. Again, the tangential equation of the polar quadric 

 (abp) cij)y; = is formed in the same way as Ua~ is formed from a^-. 

 A simple reduction leads to 



{Ap) (Bp) (abp) (aBu) (bAii). 



Likewise the point equation of {ajBp) u^ a^ involves the form 



(Ap) (Bp) (a^p) {AjSx) {Box). 



This interprets the two forms of orders (0, 3, 2) and (2, 3, 0). 



1 19. Again, if we form the polar quadric oi {p) with regard to 

 each pair of quadrics /, /', h, k', we obtain the following results : 



p (/, k) equivalent to (Ap) (A /3a;) a^a^, with a like form for p (f, k'), 



p{f,k') „ F,a^(Bax), „ „ p(f',k), 



p (k, k') „ (A^x) {cc/3p) (Bax) [AB). 



If, further, (q) is the conjugate line of (p) in {Bpy,i.e. in/', 

 then 



q (/, k) is equivalent to a^a^ (A/3x) (AB) (Bp), 

 and q' (/', k') „ b^^b, {Bax) (AB) (Ap). 



All these equivalences are readily verified, but we give a 



special proof for the case of p (k, ¥). In fact, the polar plane of 



dk 

 X m. k = 0, i.e. in (A^xy = 0, has coordinates — , which may be 



OXi 



symbolised as (A^x)(A/3)*. So also the coordinates of the polar 

 of X in k' are denoted by (Bax) (Ba). Hence the line of inter- 

 section of these polars is denoted by (Aj3x)(Bax) [ABa^], which 

 is equal to (A ^x) (Bax) (AB) (a^)* ; and the line cuts p if 



(A^x) (Bax) (AB) (aj3p) = 0. 



§ 20. These eight polar quadrics now enumerated, viz. p (f, f), 

 p (f, k), ..., q (f, k'), must be supplemented with one more form, 

 (a^p) cipbaa^bx, to complete the set of nine forms (2, 1, 0) belonging 

 to the irreducible system of two quadrics / and f. The geometrical 

 significance of this last form is as follows : the line joining the 

 two points, Xi and x^, cuts p ; Xj^ being the pole in / of the plane 

 whose pole in /' is x, and x^ being the pole in /' of the plane whose 

 pole in / is x. 



* {A^) = a a' -a' a, and the combination of (A^) with (Ba), as a transvectant, 

 into [ABa^] is essentially the reduction of Ch. ii, § 15 in the paper of Gordan. 



