40 KANSAS UNIVERSITY QUARTERLY. 



We observe that the points (0, 0, 0), ( cc, oc, oo ), and (1, 1, 1) be- 

 long to every curve of the family ; hence every one-parameter sub- 

 group of liGa (ABCD) contains the identical transformation (1, 1, 1 ) 

 and the two pseudo-transformations (0, 0, 0) and (go, go, go). 



From the above equations we have x'^'"''=l, and x'''''+'''^"'''*'=l or 

 ^rs-rv__]^_ Since X is real, it can have only the values ±1 ; substituting 

 these values of x in y=x^"'' and z=x^"'+'"-, we see that the real values of 

 y and z are limited to the numbers ±1. Hence, the only points com- 

 mon to two curves of the family in addition to those mentioned above 

 are (1,1,-1), (1,-1,1), (-1,1,1), (1,-1,-1), (-1,1,-1), (-1,-1,1), 

 (-1,-1,-1). The point (-1,1,1) is common to every curve of the 

 family which lies partly in the second octant. The corresponding 

 transformation is an involutoric perspective transformation of type 

 VI, having its vertex at B and the plane ADC for its axial plane. 

 The transformations corresponding to the points (1, 1, -1), (1, -1, 1), 

 (-1,-1,-1) are also involutoric perspective transformations of type 

 VI, with vertices at D, C, and A, respectively, and whose axial planes 

 are the opposite faces of the tetrahedron (ABCD). 



The transformations corresponding to the points (1, -1, -1), (-1, 1, 

 -1), and (-1,-1,1) are involutoric skew perspective transformations 

 of type X, whose skew axes are respectively the edges of AD and 

 CD, AC and BD, AD and BC of the invariant tetrahedron (ABCD). 

 Each of these transformations belongs to every one-parameter sub- 

 group of liGs (ABCD) whose representative curve lies partly in the 

 eighth, sixth and third octants, respectively. 



Theorem 6. The go'--' one-parameter subgroups of hGa (ABCD) do 

 not include all the transformations in liGs (ABCD) ; a transformation 

 not belonging to a one-parameter subgroup has one or more of its 

 cross-ratio parameters negative. Every subgroup hGi (ABCD)i-3 for 

 which r and s are rational contains one involutoric j^erspective trans- 

 formation either of type VI or X. 



§4. Subgroups of Types VI, VIII and X in hGg (ABCD). 



Suhgrovps of type VIII in hGi {ABCD). — The constants r and s 

 may have such values that all the one-dimensional transformations 

 along the same edge are identical transformations. This may occur 

 in six different ways, since there are six edges of the tetrahedron. 



If r=-l while s remains finite, we have an identical transformation 

 along AC. If r=0 while rs remains finite, we have an identical 

 transformation along BC. If r = go, we see that the transformation 

 along AB is identical. If s=l and r is finite, we have an identical 

 transformation along BD. If s=0 while r remains finite, the trans- 

 formation along CD is identical. If rs=r — 1, the transformation 

 along AD is identical. 



