NEWSON: REAL COLLINEATIONS. . 41 



la the case that a transformation of type I degenerates to type 

 VIII, one of the families of invariant surfaces degenerates into a 

 family of planes intersecting in one edge of the tetrahedron ; this edge 

 is opposite the edge which is the line of invariant points. All trans- 

 formations of type VIII in hGg (ABCD) leaving the same edge in- 

 variant form a two-parameter group. The path curves of the sub- 

 groups of these two-parameter groups are always plane curves. 



Theorem 7. The group liGs (ABCD) contains six two-parameter 

 subgroups of type VIII; these are given when r=^l, 0, cc; and 



S = l, 0, !^. 



Suhgroups of type VI in hO-i {ABCD). — For certain values of r 

 and s a transformation of type I reduces to type VI. In such a case 

 all points in one of the invariant planes of type I are invariant points, 

 and all lines through the opposite vertex of (ABCD) are invariant 

 lines. 



Let r = and let s have any finite value; then the transformations 

 along BC, CD and DB are all identical ; hence the two-dimensional 

 transformation in the face BCD is identical. At the same time the 

 cross-ratios along AB, AC and AD are all equal to k. Such a trans- 

 formation is evidently of type VI. Let r = l and s = 0; then the two- 

 dimensional transformation in the face ACD is identical, and the 

 one-dimensional transformations along BA, BC and BD are all equal. 

 Again, let r = go and s = 1 ; then the two-dimensional transformation in 

 the face ABD is identical, and the one-dimensional transformations 

 along CA, CB and CD are all equal. Finally, let r=Go and s==go; 

 then every point in the face ABC is an invariant point and the one- 

 dimensional transformations along DA, DB and DC are all equal. 



It is evident in each of the above cases that the invariant surfaces 

 are all planes or cones and the path curves are all straight lines. 



Theorem 8. The group hGs (ABCD) contains four one-parameter 

 subgroups of type VI ; these are given by the following sets of values 

 of rands: (0, s), (1,0), ( cc , 1), ( qd, oo). 



Suhgroups of type X in hGi{ABCD). — For certain values of r 

 and s a transformation of type I reduces to one of type X. From the 

 nature of type X it is evident that it must occur as a special case of 

 type VIII, when the two one-dimensional transformations along op- 

 posite edges of (ABCD) are identical transformations. 



Let r = l and 8 = 1; then the one-dimensional transformations 

 along AC and BD are both identical transformations, and thus every 

 point on each of these edges is an invariant point. The cross-ratios 

 along AB, AD, CB and CD are all equal. The invariant families of 

 surfaces I-VII reduce for r=l and s = l to the following: 

 xy = Czw, xw = Cyz, x = Cz, and y = Cw. 



