34 REAL TRANSFORMATIONS. 
inverses of those in the other part. Subdivisions I and II 
contain each an infinitesimal transformation. All transforma- 
tions in I may be generated by repetitions of the infinitesimal 
transformation T,-5; all transformations in II can be 
generated by repetitions of the other infinitesimal transforma- 
tion T,4;. The transformations in III, for which k is 
negative, cannot be generated from either infinitesimal 
transformation of the group. 
THEOREM 20. The hyperbolic group hG,(AA’) contains one 
identical, one involutoric, two pseudo, and two infinitesimal trans- 
formations; it consists of three subdivisions: subdivisions I and II 
contain each its generating infinitesimal transformation; the trans- 
formations in subdivision III cannot be generated from either infin- 
itesimal transformation of the group. 
40. The Elliptic Group eG,(AA’). The oné-parameter 
elliptic group, designated by the symbol eG,( AA’), consists 
of all real transformations having the same pair of conju- 
gate imaginary invariant points A and A’. The parameter, 
k = e*", is a complex number and its variation in the complex 
plane is confined to the unit circle about the origin. The 
group contains a transformation corresponding to each point 
on the unit circle. This circle cuts the axis of reals in only 
two points, viz.: when k = 7 and k = — 17; hence the group 
contains only two transformations for which k is real. These 
are respectively the identical and the involutoric transforma- 
tions of the group. Since e*’ cannot assume either value 0 or 
o, it follows that the group contains no pseudo-transforma- 
tions. The group contains two infinitesimal transformations, 
for which k = e* and k = e—**, where 4 is an infinitesimal. 
The elliptic group eG,(AA’) contains two subdivisions; sub- 
division I consists of all transformations in the group for 
which 6 is positive between 0 and 2; subdivision II, of all for 
which @ is negative between 0 and —z. The identical and 
the involutoric transformations of the group form the bounda- 
ries of these subdivisions. The transformations of one sub- 
division are the inverses of those in the other. Each subdi- 
