202 Mr T'urnbull, Some Geometrical Interpretations 



The eight cubic complexes : 



F, (ahp) K (Bp), F, (oL^p) ap (Bp), 



F, (ahp) K (AB) (Ap), F, {oc^p) a^ (AB) (Ap) ; 

 and four involving F^. 



§ 13. If aj^, hx, Cx are three quadrics, the lines p cutting them 

 in involution are given by the cubic complex 



{hep) (cap) (ahp) = 0. 



Let us denote this complex by the symbol {a^, h^', c^)- Then 

 {f,f', k') may be formulated, and we shall have 



(ax% hx^, k') = {{ahp)axb^, {Baxff 



= ((abp).axbx, - 2b:bx"ba"bx + "^^'b^'J 



= - 2 {abp) {ab"p) (bb'p) bjb^' + 2 {abp) (ab'p) (bb'p) b^"\ 



The second term is zero, since b, b' are interchangeable. The first 

 term is F^ (abp) ba. (Bp) to a constant coefficient. 



Reciprocally (S, ^',x) represents F.2 (a^p) a^ (Bp) ; and there 

 are two like forms involving Fi. 



§ 14. This leaves four complexes such as F2 (abp) ba (AB) (Ap) 

 to be interpreted, but the geometrical significance is not at all 

 immediate. If however we write (/, /', k') as (Bpf, then the line 

 (p) has a polar linear complex 



(npf(Dq) = 0. 



And if q = (Ap)(A), i.e. if q is the conjugate line of p in the 

 quadricy, then 



(Dpy(DA)(Ap) = 0. 



This latter form is equivalent to F2 (abp) ba. (AB) (Ap) : and similar 

 results follow for the other three forms, as in § 13. 



The mixed concomitants. 



§ 1.5. To denote the order of a form, let (i,j, k) mean that the 

 order is i in x, j in p, and k in u. Then there are three linear 

 forms (1, 0, 1) and sixteen linear forms (1, 1, 1). 



The three linear forms (1, 0, 1) : 



Up % ax, Uahabx, (AB)(Ab u ) bx'. 



If (v) is the polar plane of a point (x) in /, then (v) = ax(a). 

 Hence Usa&av. = is the condition that a conjugate plane of u in/' 



