REAL TRANSFORMATIONS. 33 
From the fact that there are three varieties of real project- 
ive transformations on a line we may safely infer that there 
are three varieties of one-parameter groups of such transfor- 
mations, viz.: hyperbolic, elliptic, and parabolic groups. 
These three types of groups must be studied separately. 
39. The Hyperbolic Group hG,(AA’). The hyperbolic 
group of one parameter, which is designated by the symbol 
hG,(AA’), consists of all hyperbolic transformations which 
have the same pair of real invariant points, A and A’, but dif- 
ferent real cross-ratios, k. The group hG,(AA’) contains a 
transformation corresponding to each value of k in the real 
number system. Hence the group contains an identical 
transformation for which k = 1, an involutoric transformation 
for which k = — 1, two pseudo-transformations for which 
k=0 and k= o, two infinitesimal transformations for which 
k=1+6 and k=1—6. 
From the law of the combination of the cross-ratios in the 
group, viz.: k,=kk,, we learn that the group hG, (AA’) con- 
tains three distinct subdivisions. Subdivision I consists of all 
transformations for which k is between 0 and 7; subdivision 
II, of all for which k is between 7 and ~; subdivision III, of 
all for which & is negative. The pseudo-transformation, 
k = 0, separates subdivision III from 1; the identical trans- 
formation, k = 1, separates I from II; the other pseudo- 
transformation, k = ~, separates II from III. 
The combination of any two transformations of subdivision 
I gives rise to a transformation belonging to the same 
subdivision; for the product of two positive proper fractions 
is a positive proper fraction. The inverses of all transforma- 
tions in subdivision I are in II. The combination of any two 
transformations in II gives also a transformation in II; but 
the inverses of those in II are in I. The combination of any 
two transformations in III gives one either in I or II. The 
involutoric transformation divides subdivision III into two 
parts; all the transformations in one of these parts are the 
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