106 KANSAS UNIVERSITY QUARTERLY. 



all of type V; hence the collineations in space are of type XII. There 

 are oc* such collineations in G6(Ai3l), and they form a four-parameter 

 subgroup of type XII. 



Subgroxips of type VII in GeiApl). — When n, m and k are all 

 zero, the transformation in the plane p is identical, and the remaining 

 collineations are of type VII and form a three-parameter group. 

 Dualistically there is a three-j)arameter group of type VII which 

 leaves invariant every ray through A. This results when m=0, 

 n=0, and h=0. The subgroujDS of these two three-parameter groups 

 will not be discussed here. 



Theorem 4. The group Go(Apl) contains two five-parameter sub- 

 groups of type XIII, one four-parameter subgroup of type XII, and 

 two three-parameter subgroups of type VII. 



The theory sketched in this jDaper holds equally well whether the 

 collineations are real or complex. 



Table of groups of type V. — The following is a complete list of the 

 continuous groups of collineations of type V: 



(1) G6(Apl). 



(2) G5(Apl)m. 



(3) G5(Apl)n. 



(4) G4(Apl)mn. 



(5) G4(Apl)nh. 



(6) G4(Apl)mk. 



(7) G3(Apl)mnh. 



(8) G3(Apl)mnk. 



(9) G2(Apl)mnhk. 

 (10) Gi(Apl). 



