INFINITESIMAL COLLINEATIONS. 287 
sponding point in the Argand diagram does not lie on the 
unit circle, can be generated from an infinite number of 
distinct infinitesimal transformations of the group. If the 
point P lies on the unit circle about the origin, the correspond- 
ing transformation T may be generated from either of the 
two infinitesimal transformations, k = e+’ ork=e-??, 
The solutions of our two problems are stated in the follow- 
ing theorem: 
THEOREM 10. Every finite transformation of the group @, (4 A’) 
can be generated by the repetition of aninfinitesimal transformation 
of the group; every finite transformation 7 of the group for which 
k=ret#,r 47, can be generated from an infinite number of discrete 
infinitesimal transformations of the group: if r=7, /’can be gener- 
ated from only two infinitesimal transformations of the group. 
336. The Two-dimensional Groups H,'( Al), G,/’ (ALS) and 
G,(AA'l),. Having discussed in detail the generation from 
infinitesimal transformations of finite one-dimensional projec- 
tive transformations, we turn now to apply these results to 
the generation of finite collineations in two dimensions from 
infinitesimal collineations. 
It was shown in articles 247, 269 and 261, that the three 
groups H,/( Al), G,’(AlS) and G,’(AA’l),, all have the same 
structure; 7. e., in each of these groups the parameter is ¢t and 
the law of combination of parameters is t, = ¢-+t,; t assumes 
in turn all complex values. It is evident at once that the re- 
sults obtained for the one-dimensional group G,’ (A) hold also 
for each of the two-dimensional groups H,/( Al), G,/’( AlS) 
and G,/(AA‘l),. Each of these groups contains the identical 
transformation and an infinite number of infinitesimal trans- 
formations. Each of these infinitesimal transformations is 
the generator of those finite transformations of the group 
whose corresponding points on the Argand diagram lie on its 
half-ray through the origin. Every finite transformation in 
one of these groups can be generated from one, and only one, 
infinitesimal transformation of the group. 
