NEWSON : REAL COLLINEATIONS. 93 



B.^On the Group of Col/ineations Gz(ABIl') of Type III and its Sub- 

 groups. 



§1. The GrRoup G3(ABir) and its One-parameter Subgroup. 



77ie group Gz{ABll'). — A real coUineation in space of type III 

 leaves invariant a figure (ABU') real in all of its parts, consisting of 

 two points A and B and their join ; two lines 1 and 1', the first 

 through A and the second through B ; and hence also the two planes 

 ABl and ABl'. The one-dimensional transformations along 1 and 1' 

 are both parabolic ; that along AB hyperbolic. The plane collinea- 

 tions in the invariant planes ABl and ABl' are both of type II. 



A coUineation T of type III is completely determined by the posi- 

 tion of its invariant figure (ABU') and three parameters k, t, t'; k is 

 the constant cross-ratio along AB, t is the parabolic parameter along 

 Al, and t' that along Bl'. These three parameters are all real and 

 vary independently ; hence there are oc^ coUineations of type III, 

 leaving the fundamental figure (ABU') invariant; these form a 

 three-parameter grouj^ CT3(ABir). 



Theorem 9. The aggregate of all coUineations of type III in space 

 having the same invariant figure (ABl'l) forms a three-parameter 

 group G3(ABir). 



One-parameter sulxjrovps of Gz{ABU'). — It will now be shown 

 that the group G3(ABir) contains cx"^ one-parameter subgroups. 

 Let k = a* and t'=nt, where a and n are constants; a is necessarily 

 positive. By imposing these conditions on the jDarameters k and t', 

 w^e select from G3(ABir) a system of cc^ coUineations. The proper- 

 ties of this system are now to be examined. 



Let T and Ti be two coUineations whose parameters are respectively 

 a*, nt, t, and a'l, nti, ti. Their resultant. To, has the parameters a*2, n2t2, 

 U. For along Al we have t2=t+ tr, "along Bl' we have n2t2=nt + mtr, 

 along AB we have k-2=kki=a^ a*i = a' + *i. Hence the system of co^ 

 coUineations, whose parameters are a*, nt, t, forms a one-parameter 

 continuous group whose parameter is t. This group is designated by 



Gl(ABll')a„. 



There is a one parameter subgroup within G3(ABir) for each 

 value of n and each positive value of a; hence G3(ABU') contains 

 00^ one-parameter subgroups. The properties of one of these sub- 

 groups are the same as the properties of a one-dimensional parabolic 

 group. 



Theorem 10. The group G3(ABir) contains tx- one-parameter 

 subgroups ; for each of these subgroups a and n have fixed values and 

 t is the variable parameter. 



7-K.U.Qr. A-x3 



