NEWSON : REAL COLLINEATIONS. 95 



group, leaving invariant the family of quadric surfaces given by 

 equation III. There is one such subgroup for each value of n. 



The lines 1 and 1' are two generators of the same system and AB a 

 generator of the other system on every invariant quadric. These in- 

 variant quadrics always have both systems of generators real. 



Theorem 12. The three-parameter group G3(ABir) contains 

 three singly infinite systems of two-parameter subgroups. These are 

 given by a = const. ; a" = const.; and n = const. Every collineation of 

 type III leaves invariant a family of quadric surfaces. 



§3. Some Properties of the Subgroup of G3(AB]r). 



Graphic representation of suh groups of Gz{ABU'). — The three 

 parameters t, t', k of G3(ABir) are all real and may be taken as the 

 coordinates x, y, z, respectively, of a point in a space S. There is a 

 collineation of the group for each real point in S. The one- and two- 

 parameter subgroups of G3(ABir) are represented by curves and 

 surfaces respectively in S. 



The surfaces given by the equations 



z = a'' and y = nx 



represent the two-parameter subgroups, and their curves of intersection 

 represent graphically the one-parameter subgroups of G3(ABir). 



These curves representing the one-parameter subgroups lie entirely 

 in the space above the plane z=0. No collineation in G3(ABir) 

 with negative value of k belongs to one of its one-parameter sub- 

 groups. Consequently one-half of the collineations of the group can- 

 not be generated from infinitesimal collineations of the group. 



The point (0, 0, 1) is on every curve of the system representing 

 the subgroups ; hence the identical collineation is common to every 

 one-parameter subgroup of G3(ABir). Other properties of these 

 groups are easily deduced from the properties of this family of curves. 



Theorem 13. Only one-half of the collineations in G3(ABir) be- 

 long to its one-parameter subgroups and can be generated from the 

 infinitesimal collineations in Gs(ABir). 



§4. Some Special Subgroups of GslABU'). 



Two-parameter subgroups of types IX and XI I. — ^The invariant 

 figure of a collineation of type III has three invariant lines, Al, Bl', 

 and AB. If t or t' is zero, the one-dimensional transformation along 

 Al or Bl', respectively, is identical, and the resulting collineation is 

 of type IX. If k and one of the t's vary while the other t is zero, we 

 have a two-parameter subgroup of GsfABU'). Thus G3(ABU') con- 

 tains two two-parameter subgroups of type IX. 



If k = l, the one-dimensional transformation along AB is identical 



