NEWSON: REAL COLLINEATIONS. 43 



Tics when r — l = ziz(r — rs). The condition r — rs — 1 -= zb r reduces 

 to rs = — 1 and rs:=2r — 1 ; the condition r — l = ±(r — ^rs) reduces 

 to r8i=l and rs=2r — 1. Hence we have only three relations beween r 

 and s for which at least one family of invariant surfaces are quadrics. 



Putting rs = l in (I), (II), (HI), we get, after reduction, 



(I),xy=:Czw; (II), y'-%=Cx'-^z; (III), xy = Czw. 



Thus the first and third families of surfaces reduce to the same 

 family of quadrics. 



Putting rs=—l in (I), (II), (III), we have 



(I),xz = Cyz; (II), xz = Cyw ; {III), x^-y+'^Cz'-'w'+K 



Here we see that (I) and (II) give the same system of quadrics. 

 Putting rs = 2r— 1 in (I), (II), (III), we get 



(I),x--iy = Czw-^'-i; (II), xw=Cyz; (III), xw=-Cyz. 



From these three cases we see that, if one of our families of in- 

 variant surfaces are quadrics, another is also, and these two families of 

 quadrics coincide. 



The invariant tetrahedron (ABCD) is the common self-polar tetra- 

 hedron of these families of quadrics. All quadrics of the system 

 xy = Czw pass through the edges AC, CD, AB, and BD; the edges 

 AD and BC are reciprocal polars of all quadrics of the family. Simi- 

 lar properties hold for the other two families xz=Cyw and xw = Cyz. 



Each of these systems of quadrics remains invariant under gc- 

 transformations which form a two-parameter subgroup of liGs (ABCD). 



Theorem 11. There are three two-parameter subgroups of liGs 

 (ABCD) each of which leaves invariant a family of quadric surfaces ; 

 these are given by rs = l, rs= — 1, rs=2r — 1. 



Subgroups lohose path curves are all conies. — We now proceed 

 to investigate the one-parameter subgroups of liGs (ABCD) whose 

 path curves are all conies. Since conies are plane curves, such a 

 one-i^arameter group must be of type VIII. We find in §4 that 

 there are six two-parameter subgroups of type VIII in liGs (ABCD) ; 

 hence, a one-parameter subgroup of hGs (ABCD) w^hose path curves 

 are all conies must be a one-parameter subgroup of one of these 

 groups of type VIII. 



To obtain one of these two-parameter subgroups of type VIII let 

 s^=l. The line DB is a line of invariant points, and all planes 

 through the opposite edge AC are invariant planes. The j)alh curves 

 are all alike in these invariant planes, and hence it is sufficient to ex- 

 amine them in one of these planes, as ABC. The path curves in the 

 plane ABC are conies for three different values of r, viz., r= — 1, 2, \. 



