of the Concomitants of Two Quadrics 205 



Correlatively there are nine forms (0, 1, 2), quadratic in u, 

 exactly parallel with the above, of which (a^p) UaU^ is the simplest. 



The four forms (3, 0, 1) and then' correlatives: 



(Abu) (AjSa;) a^axb^, {A^x) (Abu) ba.Ua.u^, 



(AB) (A/3x) (BoLx) a^a.^Ua, (AB) (Abu) (Bau) baiua^, 

 and four similar forms interchanging f and f, 



\ 21. If a^', bx", Cx" signify any three quadrics, then (abcu) a^b^Cx 

 vanishes when the common point of the polars of (x) in the three 

 quadrics lies on the plane (u). Applied to the quadrics ff, k, k' 

 taken three at a time, this condition involves the four forms (3, 0, 1) 

 indicated above. The correlative condition, applied to each set of 

 three from among S, S', %, %', gives rise to the four forms (1, 0, 3). 

 For example, if we select /,/', k as the three quadrics, then the 

 condition is (Abu) (A^x) a^axbx = 0. 



The polars of (x) in all four quadrics/,/', A:, k' meet in a point 

 if (x) lies on any face of the self-conjugate tetrahedron 



(A^x) (BoLx) a^baaxbx = 0. 



The remaining forms of the system. 



I 22. None of the remaining forms appear to have any special 

 geometrical importance : but we give a few examples. First, as to 

 the forms of order (2, 0, 2), we may exhibit them as follows: 



ax (aBu) (Bax) lu and a similar form, 



(Abu)(A^x)baa^iL„,ax „ „ 



(AB) (Abu) (Bau) axbx and a correlative form, 

 and [(AB)J. 



Suppose (q) to denote the common line of the plane (u) and 

 the polar of (x) in / and (q) to denote the line joining (x) to the 

 pole of (u) in /. Then the condition that q, q should satisfy 

 the harmonic relation (Bq) (Bq') = becomes on substitution 



Ux (aBu) (Bax) Ua = 0. 



Thus the first in the above group of forms is interpreted. The 

 second form vanishes if two lines (q), (q) satisfy the harmonic 

 relation (abq) (abq') = 0, where (q) now denotes the intersection of 

 the plane (u) with the polar of (x) in k, while (q^) is the same as 

 before. 



Again, the third form of the set vanishes if the lines in which 



