276 THEORY OF COLLINEATIONS. 
values. The parameters k and ¢ of this system of collinea- 
tions of type II do not satisfy the relation k =a‘, and hence 
(article 258) do not form a continuous group. These collinea- 
tions of type II in G,(AA’l’), are singular transformations. 
The structure of G,(AA’l’), is thus seen to be 
G,(AA'l’),= 0 G,(AA'A”),+ H, (Al) + @'T’. 
Let us next examine for singular transformations the group 
G,(AA'l’), , where r is rational with odd numerator and even 
denominator. Each subgroup G,(AA’A”), of G,(AA/l’), 
contains one perspective involutoric collineation S; these are 
all different, and form a system (S) whose common axis is l 
and whose vertices are in turn every point onl’. Any one 
of these perspective collineations, combined with a collinea- 
tion of type I in G,(AA’l’),, results in a collineation of type 
Lalso belonging to G,(AA’l’),. The resultant of any collinea- 
tion of the system (S) with any elation of the group H,/(A1) 
is another perspective collineation S, of the system (S). The 
resultant of any two perspective collineations of the system 
(S) is an elation belonging to the group H,'(Al). Thus the 
group G,(AA'l’),, where r is rational with odd numerator 
and even denominator, contains «’ involutoric perspective 
eollineations, but no singular transformations of type II. 
These perspective collineations are not singular transforma- 
tions in the sense of the definition, for each of them belongs 
to a subgroup of G,(AA’l’).. 
In like manner, it may be shown that the group G,(AA’l’),, 
where ¢ is rational with odd numerator and odd denominator, 
contains © involutoric perspective collineations having a 
common vertex at A and axes in turn every line through A’ ; 
but it contains no singular transformations. The group 
G,(AA’l’),, where ¢ is irrational, contains (Art. 256) no in- 
volutorie perspective collineation and hence no singular trans- 
formations. 
From these results, we conclude that the group G,(AA'l’), 
contains singular transformations of type II when, and only 
when, its subgroups have one common involutoric perspective 
