46 KANSAS UNIVERSITY QUARTERLY. 



AC are conjvigate imaginary quantities ; also those along DC and DB 

 are conjugate imaginary quantities. The cross-ratio k" along AD is 

 real, and that along BC is of the form exp. ni 6. As in the hyperbolic 

 case, we may put k'^k^"'' and k"=k^"'^+''^; r and s are usually both 

 complex quantities. It can be shown without difficulty that the con- 

 dition that the collineation shall be real requires that r shall be of 



exp. 2iii' 



the form 1 -f exp. 2i c^, and s of the form ' . , , • In the com- 



exp. 2n/'+ 1 



plex plane the locus of r is the unit circle about the unit point, and 

 the locus of s is the line x=l/2. 



The cross-ratios along the six edges of the tetrahedron may be 

 written : 



AB : exp. (tan <^ + i) 6, 



BC : exp. — 2i 6, 



CD : exp. (tan ^^i) ^, 



DB : exp. ( — tan i/'+i) 6, 



AC : exp. (tan <^-f i) ^, 



AD : exp. (tan <^ + tan \j/) 0. 



In these expressions 6 is the variable and assumes in turn all real 

 values from — oc to -f ^- Since k is real, both positive and negative, 

 while the value of exp. (tan ^ +tan i/') 6 is only positive, it follows that 

 half of the transformations in eGs (ABCD)are not contained in its 

 one- and two-parameter subgroups, viz., those for which k is negative. 



The group eGs (ABCD) contains two real subgroups of type VI, 

 the vertices being A and D; it also contains two real subgroups of 

 type VIII, one hyberbolic and the other elliptic, whose axes of invariant 

 points are, resj)ectively, BC and AD ; it also contains one subgroup 

 of type X, which is common to the two subgroups of type VIII. 



The group eGs (ABCD) has only two real two-parameter sub- 

 groups leaving invariant a family of quadric cones. These two fami- 

 lies of cones do not have an element in common ; hence there are no 

 real subgroups leaving invariant a family of cubic path curves. When 

 <P and </' are supplementary, one of the families of invariant surfaces 

 consist of quadrics. These statements may be easily verified by the 

 reader. 



The group eeG?, {ABCD). — Let AD and BC be taken as the two 

 real edges of the invariant tetrahedron (ABCD) in the double elliptic 

 case. Along these real edges the one-dimensional transformations 

 are elliptic, so that the cross-ratios are both of the form exp. ni 6. It 

 follows from these conditions that k must be assumed in the form of 



