42 KANSAS UNIVERSITY QUARTERLY, 



The path curves are evidently straight lines and constitute the con- 

 gruence of lines joining every point on AC to every point on BD. 



Again, let r=ao and 8=0 with the condition that rs = 0; then 

 the transformations along AB and CD are both identical, while those 

 along AC, AD, BC and BD are all equal. The invariant surfaces 

 and path curves are analogous to those above. In like manner, if we 

 make r^O and s=oo, and r8 = — 1, we get identical transformations 

 along BC and AD ; the cross-ratios along AB, AC, DB and DC are 

 all equal, the invariant surfaces are pencils of quadrics and pencils of 

 planes, the path curves are all straight lines. 



Theorem 9. The group hGs ( ABCD) contains three one-parameter 

 groups of type X ; these are given by the following sets of values of 

 r and s : (1, 1), ( go, 0), (0, oo, rs = — 1.) 



§5. Some Special Subgroups of hG3(ABCD). 



Subgroups tvith invariant quadric cones. — For certain values of r 

 and s the invariant cones whose equations are (IV) -(VII) are cones 

 of the second order. One of these families of cones will be of the 

 second order when the plane path curves in one the faces of the tet- 

 rahedron are conies. 



Equation ( IV) represents a family of quadric cones for three values 

 of s, viz., s== — 1, 2, \\ equation (V) represents quadric cones when 

 ^ = — 1, 2, \\ equation (VI) represents quadric cones when r — rs=^ 

 — 1, 2, \ ; equation (VII ) represents quadric cones when r^ — 1, 2, h. 

 Consider the case when r^ — 1 ; s may assume oc^ different values, 

 and hence there are cr? one-parameter groujDS which leave invariant 

 the same family of quadric cones. These form a two-parameter group. 

 The vertex of the cones of the family are at D, and the lines DA and 

 DB are elements common to all the cones of the family. Thus we see 

 that there are twelve two-parameter subgroups of hGa (ABCD) which 

 leave invariant a family of quadric cones. 



Theorem 10. The group liGg (ABCD) contains twelve two-para- 

 meter subgroups each of which leaves invariant a family of quadric 

 cones ; these are given by the values of r and s as follows : r ^ — 1, 2, | ; 

 s = -l, 2,\- r-rs = -l, 2, | ; ^=-1' 2, i 



Subgroups with invariant quadric surfaces. — We now seek the 

 most general conditions under which the invariant surfaces of a one- 

 parameter subgroup of liGs (ABCD) shall be a family of quadric sur- 

 faces. One or more of the three families of surfaces whose equations 

 are (I), (II), (III) will reduce to quadrics for certain values of r and 

 s. The surfaces given by (I) are quadrics when rs=±:l; equation 

 (II) gives quadrics when r — rs — l = ±;r; equation (III) yields quad- 



