of the Concomitants of Two Quadrics 201 



pola7' to (p) in (Dpf. If (p) is a member of the complex (Dp)-, 

 the polar is called the tangential linear complex. 



The complex (Dp)Dij is not usually a special linear complex. 

 The preceding case was exceptional. For in that case the quad- 

 ratic complex was (Apf = 0, and all the rays touched the quadric/. 



The comptlexes tt,., IIi.,, C, (ahp){oL^p)a^ba. 

 § 11. The principal quadratic complexes which occur are 

 'jr,, = (abpy, U,,= ial3pY, G^{AB){Ap){Bp). 



The two former are well known, ttjo being the aggregate of lines 

 cutting the quadrics harmonically, and rTi. being the correlative 

 complex. The third, G, is the complex of lines whose conjugates, 

 in /and /' respectively, intersect. For the conjugate of -p in / is 

 {Ap){A) and in /' is {Bp){B). Again, G is satisfied too by the 

 singular lines of the complex ttj.,. For if p is a line of {ahp)- = 0, 

 its tangent linear complex (§ 10) is {ahp) (abq) = , q representing 

 current coordinates : further, jj is a singular line if this tangent 

 linear complex is special, i.e. if 



(abp){aba'b')(a'b'p) = 0, 



which reduces to (AB) (Ap) (Bp) = 0. Correlatively C also contains 

 the singular lines of the complex ITi.,. 



Again, the singular lines of the complex G belong to the com- 

 plex (abp) {a^p) Ufiba. This follows in the same way as in the above 

 case. But a more direct interpretation of this last form arises from 

 the apolar* condition for two linear complexes ; if the polar linear 

 complexes of a line (p) with regard to ttjo and ITis are apolar, then 

 (abp) (a^p) cipba vanishes. 



The complexes F{~, Fi. 



§ 12. Besides the original complexes {Apy and {Bpy, and the 

 four complexes of § 11, there remain two more quadratics, i'V and 

 F^. Just as {abpf is the harmonic complex between / and /', so 

 F^^ is the harmonic complex between /' and k, while F^ is that 

 between / and k'. To prove this we build up a form (/', ky from 

 /' and k, in the same way as (/ f'f, i.e. {ahpY, is built from / and 

 /. Then 



{f',ky^{b,^,{ABxrf 



= (bx', 2ft^"^aa;'-— la^a^'axaxY 

 = 2ap^ (abpf — 2a^a^' {abp) {abp) 

 = [{abp) a/ - {a'bp) a^J = F,^ (| 2). 

 * The linear complexes {Dp) — 0, {Ep) — are apolar if {DE) - . 



