294 THEORY OF COLLINEATIONS. 
hG.(AA’A"’) for which both & and k’ are positive can be gen- 
erated from one and only one infinitesimal collineation of the 
group, while no collineation for which either k or k’ is nega- 
tive can be so generated. 
THEOREM 18. Every real hyperbolic collineation of type I, for 
which # and #’ are both positive, can be generated from one and 
only one real infinitesimal collineation: no such collineation, for 
which either % or & is negative, can be generated froma real infin- 
itesimal collineation. 
348. The Elliptic Case. We turn now to the consideration 
of the real elliptic group eG,(A A’A”’) in which the invariant 
triangle has one real vertex, A, and two conjugate imaginary 
vertices, A’ and A”. It was shown in Art. 312, Chap. III, 
that k and k’ are not independent parameters, but that they 
are conjugate imaginary numbers. Thus the real elliptic 
group eG,(AA’A’’), instead of having two independent para- 
meters k and k’, has only one: viz., k; but this is a com- 
plex number and may assume in turn all possible complex 
values. Consequently the group eG,(AA’A’’) contains a col- 
lineation corresponding to each point on the Argand diagram. 
Therefore the group eG,(AA’A’’) has exactly the same 
structure as the one-dimensional loxodromic group G,(AA’) 
discussed in Art. 27, and we may apply the results of that 
discussion directly to the present case. 
The collineations forming the one-parameter subgroup 
eG,(AA’A”’) correspond on the Argand diagram to the points 
on the logarithmic spirals k = e(¢+*) around the origin. <A 
collineation 7’, corresponding toa point P not on the unit 
circle, belongs to an unlimited number of distinct subgroups 
and may be generated from an unlimited number of distinct 
infinitesimal collineations. If the point P, corresponding to 
T,, lies on the unit circle, T belongs to only one subgroup and 
can be generated from either of two infinitesimal collineations: 
AVA Ge 
