36 REAL TRANSFORMATIONS. 
infinitesimal transformation belonging to the other subdi- 
vision. The group pG,(A) contains no involutoric trans- 
formations. 
THEOREM 22. The one-parameter parabolic group of real trans- 
formations on a line contains one identical, one pseudo, and two in- 
finitesimal transformations, but no involutoric transformation ; it 
consists of two subdivisions each of which is generated by its own 
infinitesimal transformation. 
42. The Group G,(A). The theory developed in $4 for 
complex constants and variables holds also for real transfor- 
mations. Equations 31-34 inclusive may be interpreted in real 
transformations as follows. The resultant of two hyperbolic 
transformations with one invariant point in common is gen- 
erally a hyperbolic transformation having one of its invariant 
points at the common invariant point of its components ; and 
the cross-ratio of this resultant equals the product of the cross- 
ratios of the components. Thus the resultant of hT(AA’) 
andhT,(AA”) ishT,(AA’’) and k,=kk,. The resultant will 
be parabolic in case k and k, have reciprocal values and TJ and 
T, are from different one-parameter groups. (See equation 
33.) 
The * real transformations leaving A invariant is made up 
of ~’ one-parameter hyperbolic subgroups, hG,( AA’), where 
A’ is in turn every point on the line except A, and one para- 
bolic subgroup, pG,(A), the limiting case of hG,( AA’) when 
A’ coincides with A. There are no elliptic transformations 
leaving A invariant. These * transformations leaving A 
invariant form a two-parameter group G,(A). 
THEOREM 23. The group G.(A) contains oo? hyperbolic sub- 
groups hG,( AA’), one parabolic subgroup pG;(A ), but no elliptic 
transformations. 
43. The Group G,. The aggregate of all real transforma- 
tions of the points on a line forms a three-parameter group, 
designated by G,. It contains * one-parameter hyperbolic 
subgroups, one for each pair of real points on the line; it con- 
