NEWSON: REAL COLLINEATIONS. 45 



having double contact along DC and DA. Since every cone of each 

 family has the line AD in common, their curves of intersection are a 

 system of go- twisted cubics passing through A and D. For these 

 values of r and s the invariant systems of surfaces given by (1), (II), 

 (III) become 



(I),xy = Czw; (II), x=^z = Cy%; (III), xy = Czw. 



Thus we see that the cubic path curves also appear as the intersec- 

 tion of a family of quadrics with a family of quartics. 



Again, let us put r^^l 2 and s = 2; equations (IV) and (VII) re- 

 duce respectively to 



yz = Cx- and xw = Cy^. 

 These two families of quadric cones have also the element AD in 

 common. Equations (I), (II), (III) become for these values of r 

 and s 



(I),xy = Czw; (II), x^w^Cy^^z; (III), xy = Czw. 



The path curves are again twisted cubics through A and D and lie 

 on the family of quadrics xy = Czw ; but they now appear as the inter- 

 section of this family of quadrics with another family of quartics. 



In like manner it can be shown that the path curves are twisted 

 cubics for ten other pairs of values of r and s, as follows: r=3, s = 

 1/3; r = 3/2, s=l/3; common chord BC : r=— 2, s = l/2; r=— 1/2, 

 8=2; common chord AC: r = 2, s=3/2; r=— 1, s=3; common 

 chord CD: r=2, s=— 1/2; r=l/2, s=— 2; common chord BD : 

 r=l/3, s= — 1; r=2/3, s=l/2; common chord AB, 



Theorem 13. — There are twelve one-parameter subgroups of hGa 

 (ABCD) for which the path curves are twisted cubics; these are 

 given by the following values of r and s .• (r;= — 158= — 1 ), (r=l/2, 

 s=2),(r=3,s=l/3),(r=3/2,s=l/3),(r=— 2,s=l/2),(r=— ]/2, 

 s=2), (r=2,s=3/2), (r=-l, s=3), (r=2, s= -1/2), (r=l/2, 

 s= -2), (r=l/3, s= -1), (r=2/3, 8=1/2). 



§6. The Groups eGa (ABCD) and eeGs (ABCD) and their 



Subgroups. 



We shall now consider briefly the two cases where the invariant 

 tetrahedron (ABCD) is not real in all of its parts. The first, called the 

 single elliptic case, is w^here the tetrahedron has two real and two 

 conjugate imaginary vertices ; the second, called the double elliptic 

 case, is where the tetrahedron has two pairs of conjugate imaginary 

 vertices. These three-parameter groups are designated bj' eGa 

 (ABCD) and eeGa (ABCD), respectively. 



The group eG-s {ABCD) and its subgroups. — Let B and C be a 

 pair of conjugate imaginary vertices of the tetrahedron (ABCD). 

 The cross-ratios k and k' along the conjugate imaginary lines AB and 



