INFINITESIMAL COLLINEATIONS. 283 
$4. Generation of Finite from Infinitesimal 
Collineations. 
332. In the theory of continuous groups as developed by 
Prof. Sophus Lie, the infinitesimal transformation plays the 
most important part. The generation of finite transforma- 
tions and whole groups of transformations by the repetition 
of infinitesimal transformations of the group, is a fundamen- 
tal part of his theory. In the theory of continuous groups of 
collineations in one and two dimensions, developed in the 
preceding pages, the infinitesimal collineation plays no such 
important role. It is, however, of prime importance for us 
to know in what manner, subject to what conditions, and un- 
der what limitations, the finite collineations of a continuous 
group may be generated by the repetition of infinitesimal 
collineations of the group. In this way the points of contact 
of the present theory with Lie’s theory will be most forcibly 
exhibited. The theory of the generation of real continuous 
groups of collineations from real infinitesimal collineations 
differs so markedly from that of the generation of complex 
groups from complex infinitesimal collineations that the two 
cases are best treated separately. 
333. A. Generation of Complex Groups. It has been 
shown in the previous pages that every n-parameter group of 
collineations x <8, is composed of one-parameter subgroups, 
and that every one-parameter group contains at least one 
infinitesimal collineation. Our immediate problem is to dis- 
cuss the generation of one-parameter groups from their infini- 
tesimal collineations. We found in one dimension two types 
of one-parameter groups, viz.: G,(AA’) and G,/(A). In 
two dimensions we found five types of one-parameter groups, 
Wiz) 6G (ALAVA') G, (CAA), G.” (ALS), H(A, 1) and 
H,/(Al). Each of these types must be discussed separately. 
334. The One-dimensional Parabolic Group G,/(A). It 
was shown in Chap. I, Art. 28, that the variable parameter of 
