102 KANSAS UNIVERSITY QUARTERLY. 



That T is correctly expressed by these equations is shown as fol- 

 lows : (1) shows that the one-dimensional transformation in the pencil 

 of planes through 1 is parabolic; (1) and (2) show that the two- 

 dimensional transformation in the bundle of rays through A is of type 

 III ; (4), (5) and (6) show that A^ (0, 0, 0, w) is an invariant point ; 

 (G) shows that z = is an invariant plane; making z^=0 in (4), (5), 

 and (6), the modified form of (4) and (5) shows that, in the plane p, 

 1 is an invariant line and that the collineation in p is of type III. 



Six-parameter group Gii{Apl). — The numbers m, n, h, k, g, t may 

 vary independently, and hence there are oo*^ collineations of type V, each 

 leaving (Apl) invariant. Let Ti be a second collineation of the same 

 system, whose constants are mi, m, hi, ki, gi, ti, and which transforms 

 Pi to Po. Eliminating xi, yi, Zi, wi from T and Ti, we get To, whose 

 equations are of the same form as those of T and whose constants are 

 m2; ns, h2, k2, g2, to. We find the following values of to, etc.: 



+ 



These six equations show that all the parameters are essential, and 

 that we have a six-parameter group; thus verifying theorem 1. 



§2. OXE-PARAMETER SUBGROUPS OF G6(Apl). 



The one-parameter group G\{Apl). — If we keep m, n, h, k and g 

 fixed and let t alone vary, we select thus from G6(Apl) cc^ collinea- 

 tions which form a one-parameter subgroup of G6(Apl). This follows 

 from the fact that under these conditions there are no longer six inde- 

 pendent equations (7)-(12), but only one, viz., (7). The parameter 

 of the group is t, and the equation t-p ti = t2 tells us that its properties 

 are those of a one-dimensional parabolic group. Evidently there are 

 ay" such subgroups of G6(Apl), one for each value of m, n, h, k, g. 

 Such a group is designated by Gi(Apl). This is the only variety of 

 one-parameter subgroups contained in G6( Apl) ; for if any other para- 

 meter besides t be made to vary alone and the other five be kept 

 fixed, the resulting oc^ collineations do not form a group. Equations 

 (7)-(12) confirm this statement. 



Invariant curves and surfaces of Gi(ApI). — The families of sur- 



