Two- AND THREE-PARAMETER GROUPS. 25 
tions can be arranged in inverse pairs; it contains the identical 
transformation, one pseudo, but no involutoric transformation ; it 
contains an infinite number of infinitesimal transformations, and 
every transformation of the group can be generated from an infini- 
tesimal transformation of the group. 
30. Number of One-Parameter Groups. We have thus 
found two types of one-parameter groups of transformations 
of the points ona line, viz., G,(AA’) and G,/(A). Evidently 
there are as many groups of the first type as there are pairs 
of points on a line, viz., ©”. Also, there is a group of type 
II for every point on a line; therefore, ~' in number. It is 
also evident that every transformation of the points on the 
line belongs to one and only one of these one-parameter 
groups (except the identical transformation which is common 
to all). 
$4. Two- and Three-Parameter Groups of 
Projective Transformations. 
We shall now investigate the question of the existence of 
two-parameter groups of projective transformations of points 
onaline. We shall make use of a method which is of great 
importance and will be often used in the following chapters 
to prove the existence of groups of transformations. 
31. The Group G,(A’). We wish to examine the aggre- 
gate of transformations which leave a single point invariant. 
Let us take two transformations, 7 and 7, having one, but 
only one, invariant point A’in common. The point A’ may 
be taken for the origin without loss of generality. Let T 
and T, be taken in the normal form, 
in it 0) v1 1 0 
A1laA Ask dike ob 
A’ 1 kA’ Ay 1 kAy 
: 2,= — - Ti: 2% = ———.. DD 
r t ee On ic Z 2 ian Ree (22) 
Alias PAT ener 
ALT We At f@ ki 
