94 KANSAS UNIVERSITY QUARTERLY. 



Invariant curves and surfaces of G\{ABll'). — We shall now de- 

 termine the systems of invariant surfaces of the one-parameter group 

 Gi(ABir), whose intersections are the oc^ path curves of the group. 

 Let (ABCD), where C is some point on 1' and D some point on 1, be 

 the tetrahedron of reference ; and let T be a collineation of the group 

 Gi(ABir) which transforms a point P, whose coordinates are (x, y, 

 z, w) to Pi whose coordinates are ( xi, yi, zi, wi). Pass planes through 

 PAl, PiAL PAl', P:A1', PAB, and PiAB. We obtain at once the 

 following equations : 



^' — -^ = t' = nt, (1) 



^ — ^ = i, (2) 



^:^ = a*. (3) 



z, z ^ ' 



By eliminating t from these equations of transforn^aHon we obtain 

 the following equations of invariant surfaces of Gi(ABir): 



y = Ca^z, I 



t 

 y = Ca""% II 



(x — cy)z = nwy. Ill 



Equations I and II represent families of transcendental ruled 

 surfaces while equation III always represents a family of quadric 

 surfaces. The intersections of these three systems of surfaces are the 

 path curves of the group Gi(ABir). 



Theorem 11. There are three distinct families of ruled surfaces 

 invariant under all the collineations of the group Gi(ABir); two of 

 these families are transcendental surfaces and one is a family of 

 quadrics. These surfaces intersect in the path curves of the group. 



§2. Two-parameter Subgroups of G3(ABir). 



Subgroups imth transcendental invariant surfaces. — Let a, n and 

 t be the three parameters of Ga(ABir). If a remains constant 

 while n and t vary, we have a one-parameter subgroup all of whose 

 transformations leave invariant the family of transcendental surfaces 

 given by equation I. These form a two-parameter group, and there 

 is one such group for each positive value of a. 



If a and n vary in such a manner that a" remains constant and t 

 varies independently, we get a two-i^arameter subgroup of G.;( ABU'), 

 leaving invariant the family of surfaces given by equation II. There 

 is one such subgroup for each value of the constant a". 



Subgroups with invariant quadric surfaces. — If n remains con- 

 stant and k and t vary independently, we have a two-parameter sub- 



