264 THEORY OF COLLINEATIONS. 
$1. Structure of the Collineation Groups of 
the Plane, 
Having found a complete list of the groups of collineations 
in the continuous plane, we must now examine more closely 
into the structure of each group. Many of these groups will 
be found to contain collineations of one type only, while oth- 
ers will be found to contain collineations of two or more types. 
In every case, there are collineations of a characteristic type 
which make up all or the greater part of the group, and 
among these are to be found in many cases a smaller number 
of collineations of one or more lower types. In most ir. 
stances, the collineations of these secondary types form con- 
tinuous subgroups of the given group; but sometimes these 
secondary collineations in a given group do not form a con- 
tinuous subgroup, in which case they are called Singular 
Transformations. 
The complete structure of some of these groups has already 
been given, while in other cases only the characteristic col- 
- lineations of the group have been indicated. The entire list 
of these groups should be examined; usually the structure of 
a group will be given without proof, but in a few typical 
cases where the structure is not at once evident, or where the 
group contains singular transformations, the proofs will be 
indicated. The verification of the structural formulas in the 
remaining groups will be left as exercises for the reader. 
315. Structure of the Perspective Groups. We found in 
Chapter III, Art. 284, eight varieties of perspective groups, 
viz.: three of type V and five of type IV. The group H,'(Al) 
contains only collineations of type V, and H,(A,l’’) only 
those of type IV. The structure of the other six perspective 
groups is here indicated by symbolic equations 
H/(l) = 0'H'(Al), 
H(A), eo ECA) 
(UL) = co" Ee (AR en CAGE 
