290 THEORY OF COLLINEATIONS. 
in the group may be generated from either of the infinitesimal 
transformations of the group. Indeed, since e*” isa periodic 
function of period 2x, T may be generated an infinite number 
of times from each infinitesimal transformation of the group. 
THEOREM 13. Every real elliptic projective transformation in 
one dimension can be generated from two distinct infinitesimal 
transformations. 
341. The Group hG,(AA’). The parameter k of the 
hyperbolic group hG,( AA’) is real and varies from — ~ to 
+o, This group contains the identical transformation cor- 
responding to k=7 and two infinitesimal transformations 
corresponding to the values k= 14 where 6 is an infinitesi- 
mal. Since k,=kk,, it follows that every finite transformation 
of the group, for which k is positive and greater than unity, 
can be generated from the infinitesimal transformation 
k=1+56; every transformation, for which k is positive and 
less than one, can be generated from the other infinitesimal 
transformation k= 1—3; the transformation of the group for 
which k is negative can not be generated from either infini- 
tesimal transformation of the group. 
We thus see that the group hG,( AA’) is composed of three 
subdivisions as follows: All transformations for which 
1<k<o form subdivision I, and are generated from 
k=1+6; all for which 0<‘’<1 form subdivision II, and are 
generated from k=1—5; all for which —o <k<0O form 
subdivision III and can not be generated from any real in- 
finitesimal transformation. 
THEOREM 14. Every real hyperbolic projective transformation 
in one dimension, for which / is positive, can be generated from one 
and only one real infinitesimal transformation; no real hyperbolic 
transformation with negative 4 can be generated from areal infini- 
tesimal transformation. 
342. Real Collineations in the Plane. We now take up 
the question of the generation of real finite plane collinea- 
tions from real infinitesimal collineations. We must exam- 
