INFINITESIMAL COLLINEATIONS. 289 
generated from an infinitesimal collineation belonging to the 
group. 
Every finite collineation in the group G,, the group of all 
plane collineations, belongs to at least one one-parameter sub- 
group and hence can be generated from one or more infini- 
tesimal collineations of the plane. 
THEOREM 11. Every finite collineation in the plane belongs to 
at least one one-parameter subgroup of Gs and can be generated 
from one or more infinitesimal collineations. 
B. GENERATION OF REAL GROUPS. 
We turn now to the question of the generation of real col- 
lineations in one and two dimensions from real infinitesimal 
collineations. We shall first discuss the question in one 
dimension. The real group FRG, contains three types of one- 
parameter subgroups: viz., pG,(A), eG,(AA’) and hG,(AA’), 
which require separate consideration. 
339. The Group pG,(A). The parabolic group pG,(A ) 
contains the identical transformation for which t = 0 and two 
infinitesimal transformations for which t= + 5, where Sisa 
real infinitesimal. Since t,=¢-+t,, we see that every finite 
transformation in pG,(A), for which ¢ is positive, can be 
generated from the positive infinitesimal transformation of 
the group; in like manner every finite transformation in 
pG,(A), for which ¢ is negative, can be generated from the 
negative infinitesimal transformation of the group. This rea- 
soning applies to every real parabolic transformation group of 
one-dimensional projective transformations. 
THEOREM 12. Every real parabolic projective transformation 
in one dimension can be generated from one and only one real in- 
finitesimal transformation. 
340. The Group eG,(AA’). The parameter k of the 
elliptic group eG,(AA’) is of the form k=e*’, where ¢ 
varies from —~ to-+. This group contains the identical 
transformation for which #=0 and two infinitesimal trans- 
formations for which 6= +5. Every finite transformation T 
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