88 KANSAS UNIVERSITY QUARTERLY. 



put k'=k^"'', and k = a*; i. e., k = a* and k'^a'^""")*, where a and r are 

 two real constants. We consider now the system of collineations in 

 hG3(ABCl) whose parameters satisfy these relations. 



Let T and Ti be two collineations whose parameters are a*, a*^"'")^ t, 

 and a^, a(^"'')^, ti, respectively. Their resultant, To, has the parameters 

 a.h, a'^''")*2, U where t2 = t-|- ti. Since there is only one variable para- 

 meter, t, this system contains cc^ collineations : these form a one- 

 parameter group, since the resultant of any two collineations of the 

 system is again a collineation of the system. Such a one-parameter 

 group is designated by hGi(ABCl)ai-. 



There is a one-parameter group within hG3(ABCr) for each real 

 value of r and each positive value of a; hence hG3(ABCr) contains 

 Qc- one-parameter subgroups. The properties of one of these sub- 

 groups are the same as the jjroperties of a one-dimensional parabolic 

 group. 



Theorem 2. The group hG3(ABCl) contains cc- one-parameter 

 subgroups ; for each of these subgroups a and r have fixed values, and 

 t is the variable parameter. 



Invariant curves and surfaces of ]iGi{ABCl)iiY. — The one-para- 

 meter group hGi(ABCl) leaves invariant, besides the fundamental 

 figure h(ABCl) a system of cc- path curves and certain systems of 

 surfaces passing through these path curves. We find the equations 

 of these invariant surfaces as follows : 



Let (ABCD), where D is some point on 1, be the tetrahedron of 

 reference, and let T be a collineation of the group hGi(ABCl) 

 which transforms a point P whose coordinates are ( x, y, z, w ) to Pi 

 whose coordinates are (xi, yi, zi, wi). Pass planes through PAC and 

 PiAC Writing out the cross-ratio of the four planes through AC we 

 have 



-:- = a', (1) 



since this cross-ratio is the same as that along the line AB. In like 

 manner we derive the equations 



|=a(i-)', (2) 



t = x=a-", (3) 



?f-7 = t. (4) 



Suppose that P is a movable point and Pi fixed, so that any func- 

 tion of the coordinates of Pi only is a constant. Eliminating t from 

 (1) and (4) we get 



za"^=Cx, I 



which is the. equation of a family of invariant cones with vertices at 



