100 KANSAS UNIVERSITY QUARTERLY. 



Subgrovps of G6{Apl) —T\\exQ are two varieties of five-parameter 

 subgroups of G6(Ai3r). One of these subgroujjs consists of all col- 

 lineations in G6(Apl) which leave invariai.t a net of oc^ conies in p 

 and the system of crJ' cones through A. A sul)group of the other va- 

 riety consists of all coUineations in GelApl) which leave invariant the 

 system of cc^ conies in p and a net of cc- cones through A. These 

 two varieties of five-parameter groups form a dualistic pair. There 

 are oc^ groups of each variety. 



There are three varieties of four-parameter subgroups in GelApl), 

 viz.: (1) All those coUineations in Gu(Apl), which leave invariant anet 

 of Go^ conies in p and a net of cc'^ cones through A, form a subgroup; 

 (2) all those coUineations, w^hich leave invariant a pencil of cc^ conies 

 in 13 and all cones through A, form a subgroup; (3) all coUineations 

 in G6(Apl), which leave invariant the cc' conies in p and a pencil of 

 cc^ cones through A, form a subgroup. The second and third varieties 

 form a dualistic pair and the first is self-dualistic. 



There are two varieties of three-parameter subgroups in Gg(ApI), 

 viz.: (1) All coUineations in Gg(ApI), which leave invariant a net of 

 00^ conies in p ard a pencil of oc^ cones through A, form a subgroup ; 

 (2) all coUineations in G6(Apl), which leave invariant a pencil of cr} 

 conies in p and a pencil of cc- cones through A, form a subgroup. 

 These two group varieties form a dualistic pair. 



There are cc* two-parameter subgroups of G6(Apl). One of these 

 subgroups consists of all coUineations which leave invariant a pencil 

 of conies in p and a pencil of cones through A. There is only one va- 

 riety of such two-parameter groups. 



In the above two-parameter group the joarameters of the two-di- 

 mensional transformations in p and through A are t and t', respectively. 

 If we set t'=gt and keep g constant, we obtain a one-parameter sub- 

 group of GelApl). 



Theorem 2. There are nine varieties of subgroups of type V in the 

 group G6(Apl), viz.: Two varieties of five-parameter subgroups, three 

 of four-parameter subgrou^DS, two of three-parameter subgrouj)S, one 

 of two-parameter subgroups, and one of one-parameter subgroups. 



Special suhgroups of G<i{Apl). — The group Gg(ApI) contains a 

 number of subgroups composed of coUineations of lower types than 

 type V, viz., groups of type XIII, XII, and VII. The three-param- 

 eter group of type III in the plane p contains two two-parameter 

 subgroups of elations, viz., H'2(A) and H'i(l). The coUineations in 

 G6(Apl), whose plane coUineations in p are of type V, are generally of 

 type XIII. Hence, corresponding to the two groups H'2(A) and 

 H'2(l) in p, the group GeCApl) contains two five-parameter subgroups 

 of type XIII. These five-parameter groups contains four- and three- 



