288 THEORY OF COLLINEATIONS. 
337. The Groups H,(A,l) and G,(AA'A”),. It was 
shown in Arts. 245 and 250 that the two groups H,(A,/) and 
G,(A A’A”), have the same structure as the one-dimensional 
group G,(AA’). Ineach group the parameter is k, which 
assumes in turn all complex values, and the law of combina- 
tion of parameters is expressed by the equation k,=kk,. 
Hence all the results obtained above for the group G,(AA’) 
apply immediately to each of the groups H,(A,/) and 
G,(AA’‘A”),. Each of these groups contains the identical 
collineation and an infinite number of infinitesimal colline- 
ations. Each of these infinitesimal collineations is the gen- 
erator of those finite collineations of the group whose 
corresponding points on the Argand diagram lie on its half- 
spiral through the unit point. Every finite collineation in 
one of these groups can be generated from an infinite number 
of infinitesimal collineations (except those whose correspond- 
ing points on the Argand diagram lie on the unit circle) ; 
each of these exceptional collineations can be generated from 
two and only two infinitesimal collineations of the group. 
338. r-Parameter Groups of Plane Collineations. The 
structure of all collineation groups of the plane was discussed 
in $$ 1 and 2 of the present chapter. In regard to structure 
the groups of plane collineations may be divided into two 
classes, viz.: those which do, and those which do not, contain 
singular transformations. 
Groups containing no singular transformations are made 
up of one-parameter subgroups, such that every collineation 
in such a group belongs to at least one one-parameter sub- 
group. Consequently every finite collineation in an 7-para- 
meter group, G,, which contains no singular collineations, can 
be generated from one or more infinitesimal collineations of 
the group G,. 
In groups which contain singular collineations it is evident 
that all non-singular collineations of the group can be gene- 
rated from one or more infinitesimal collineations of the 
group; but no singular collineations in such a group can be 
