of the Concomitants of Two Quadrics 203 



should be the polar of x in f Similarly for Uahahx- Again, 

 {AB){Ahu) h^ vanishes if the polar of x in/' is conjugate to u in ;)^, 

 i.e. in {Ahiif = 0. 



The sixteen forms (1, 1, 1) : 



two like ax {Ban) (Bp), two like Ua {Bax) (Bp), 



„ a3^a^{a^p)ua, „ „ ax(abp)baUa, 



„ „ ax{Bau)(AB)(Ap), „ „ u^(Bax){AB)(Ap), 

 „ (abp){Abu)(A/3x)ap, „ „ (oL^p)(Abu)(A/3x)ba. 



§ 16. The polar plane of a point (x), with regard to /, meets 

 a plane (u) in a straight line whose coordinates are (au) a^. If 

 ax(aBu){Bp) = 0, this line cuts the conjugate of p in /'. Let us 

 denote this relation by (f^, 11'). The significance of the reciprocal 

 of this, viz. (Stt, n'), is obvious. This accounts for four forms since 

 either /or/' can be employed. 



Suppose we word this relation differently and say that the 

 plane {u) cuts the polar of (x) in/ in a line which lies in the linear 

 complex polar of (p) in II' : then a like meaning attached to 

 (fx, His) interprets tta;% (^^p) Ua.. So also 



i-u, 'ir^^ = ax{abp)ba.Ua, 

 (/,, {AB) {Ap) (Bp)) = a, (aBu) (AB) (Ap), 



with reducible terms, and 



(tu, (AB) (Ap) (Bp)) = lu (Bav) (AB) (Ap), 



while (kx, TTia), (Xu> II 12) denote the remaining two forms of the 

 above list. To complete the set of sixteen forms we merely write 

 %' for S, k' for k, and so on. 



The polar quadrics (0, 1, 2) and (2, 1, 0). 



I 17. There are nine forms of order (2, 1, 0), any one of which 

 represents a quadric associated with a given line (p) ; or, from 

 another point of view, represents a linear complex associated with 

 a given point (x). The simplest of these is (abp) ajj^. Let this 

 denote the polar quadric of the line (p) with regard to the system 

 /*+ X/'. It is convenient to use the symbol p (ff) for this relation. 



The equation (abp) aj)x = is the analytical condition required 

 when the polar planes of a point (x) with regard to / and /' meet 

 in a line which intersects (p). For the coordinates of these polar 

 planes of x are denoted by a-^a^, bib^ (* = 1, 2, 3, 4). Hence the 

 coordinates of their line of intersection are aj)x(ah)ij; and this 

 line cuts (p) if (abp) axbx = 0. 



