THE GENERAL LINEAR GROUP. 307 
Fic. 40. 
Since there are ~¢ triangles in the plane having the line at 
infinity for one side, it follows that the group G,(/.)a con- 
tains / one-parameter subgroups of type I, all of whose 
path-curves are conics having double contact on the line at 
infinity. 
366. Collineations of Type III in G,(l.)a. It was proved 
in Art. 359 that a collineation of type III in G,(/.) has the 
constant ratio of areas R=1. Therefore all such collinea- 
tions in G,(l.) transform areas into equal areas and conse- 
quently belong to the subgroup G,(/.)a. 
There are ~‘ collineations of type HI in G,(/.)a, for there 
are «/ linear elements on./ and each lineal element is the in- 
variant figure of ~* collineations T”. These ~‘ collineations 
T” fall into ©* one-parameter subgroups G,’(Al..), ©* for 
each point on/.... The ~* parabolas of the plane can be ar- 
ranged in ~* pencils of similar and coaxial parabolas; each 
of the pencils constitutes the path-curves of one of the ~*one- 
parameter subgroups G,’(Al..). 
367. Collineations of Type Vin G,(l.)a. All collinea- 
tions of type V in G,(/.) transform areas into equal areas, 
