96 KANSAS UNIVERSITY QUARTERLY. 



and all points on the line are invariant ; thus a transformation of this 

 kind is of type XII. If t and t' vary while k=l, we have cr? collinea- 

 tion of type XII, all having the same invariant figure and forming a 

 two-parameter subgroup of G3(ABir). 



Theorem 14. The group G3(ABir) contains two two-parameter 

 subgroups of type IX and one of type XII. 



One-par aiyieter subgroups of types VII and X. — If t=0 and k=l, 

 the coUineations in the plane ABl are identical, and the resulting 

 collineations in space are of type VII and form a one-parameter sub- 

 group. In like manner, if t'=0 and k = l, we have another one-para- 

 meter subgroup of type VII, ABl' being the axial plane and B the 

 vertex of its invariant figure. Thus CT3(ABir) contains two sub- 

 groups of type VII. 



If t=^0 and t'=0 while k varies, the resulting oo^ collineations are 

 of type X and form a one-parameter subgroup of this type. Al and 

 Bl' are the two axes of the skew perspective collineation and k is the 

 parameter. 



Theorem 15. The group G3(ABir) contains two one-parameter 

 subgroups of type VII and one of type X. 



0,—On the Group of Collineations Gi(ABIp)of Type 11/ and its 



Subgroups. 



§1. The Group G4(ABlp) and its Subgroups. 



T/ie group Gi{ABlp). — A real collineation in space of type IV 

 leaves invariant a figure (ABlp) real in all of its parts, consisting of 

 two planes p and p' intersecting in a line 1, two points A and B and 

 their join 1' in the plane p, A being on 1. The one-diraensional 

 transformations along 1 and 1' are parabolic and hyperbolic, re- 

 spectively. The two-dimensional transformations in the planes p and 

 p' are of types III and II, respectively. 



A collineation T of type IV is completely determined by its in- 

 variant figure (ABlp) and four parameters k, t, n, h ; k is the constant 

 cross-ratio along AB, and t, n, h are the three parameters of group of 

 plane collineations of type III in the plane p. These four parameters 

 are all real and may vary simultaneously, thus giving go* collineations, 

 all having the same invariant figure. These form a four-parameter 

 group G4(ABlp). 



Theorem 16. The aggregate of all collineations of type IV having 

 the same invariant figure (ABlp) forms a four-parameter group 

 G4(ABlp). 



One-parameter subgroups of Gi{ ABlp). — The group contains 



