98 KANSAS UNIVERSITY QUARTERLY. 



stant and n, h, t variable, we have a three-parameter subgroup of 

 G4(ABlp). All collineations of this group GsfABlp) leave invariant 

 the family of surfaces given by 1. If n is constant while a, h and t vary, 

 we have another three-parameter group G3( ABlp)n. This group leaves 

 invariant a system of oc- quadric cones. There are thus two systems 

 of three-parameter subgroups of G4( ABlp), one for each positive value 

 of a and one for each real value of n. 



If a and n are both constant while h and t vary, we have a two- 

 parameter subgroup of G4(ABlp). If h and n are both constant and 

 k and t vary, we have another two-parameter group. The invariant 

 surfaces are easily determined. Thus there are two systems of two- 

 parameter subgroups of G4(ABlp). 



Theorem 19. The group G4(ABlp) contains two singly infinite 

 systems of three-parameter subgroups and two doubly infinite systems 

 of two-parameter subgroups. 



Since k is both positive and negative, and since only these collinea- 

 tions with positive k can be generated from infinitesimal transforma- 

 tions of the group, it follows that only one-half of the collineations in 

 the group G4(ABlp) belong to its one-parameter subgroups. 



§2. Some Special Subgroups of G4(ABlp). 



Three-parameter subgroups of type XIII. — When k = l, or, what 

 amounts to the same thing, when a;=l, the one-dimensional trans- 

 formation along the line AB is identical, and hence every point on the 

 line is an invariant point ; dualistically every plane through the line 

 1 is an invariant plane. Therefore, the resulting transformations are 

 of type XIII. Since there are three remaining parameters, n, h, and 

 t, we have a three-parameter subgroup of type XIII. 



Three-parameter suhgrovp of type XI. — If n^O, the three-para- 

 meter group of type III in the plane p reduces to a two-parameter 

 group of type V. The resulting collineations in space are of type XI, 

 since every point on 1 is invariant, and form a three-parameter group 

 whose parameters are k, h, and t. Also, if we put t=0, nt±0, and 

 htdzO, the Of? collineations in the plane p are again of type V and 

 form a two-parameter group. Thus we have another three-jDarameter 

 group of type XI. 



Two-parameter suhgroup of type VII. — If a = l and nr=rO, all 

 points in the plane ABl are invariant and the collineations are of 

 type VII. They form a two-parameter group of type VII. Also, if 

 a = l and t = 0, but nt±0 and ht±0, we have left a two-parameter 

 group of type VII, dualistic to the last. 



Theorem 20. The group G4(ABlp) contains one three-parameter 

 subgroup of type XIII, two three-parameter subgroups of type XI, 

 and two two-parameter subgroups of type VII. 



