SINGULAR TRANSFORMATIONS. 277 
collineation S. This depends on the manner in which the in- 
variant triangles of the subgroups G,(AA’A”), are put 
together. Thus, when the line /’ passes through A, the 
group G,(AA/l’), contains singular transformations when 7 
has even numerator and odd denominator. On the other 
hand, if /’ passes through A’, in which case it is designated 
by 1’, the group G,(AA‘l”’), has singular transformations 
when r has odd numerator and even denominator. 
THEOREM +. The group @,(4 4//’), where 7 isa rational num 
ber such that the subgroups of G, (4 4//’), all contain the same 
involutorice perspective collineation S, contains singular transforma- 
tions; these are of type II, and all have the same value of /, viz.: 
k=—1., 
326. Other Groups Containing Singular Transformations 
of Type II. Any collineation group of the plane which con- 
tains no subgroup of type II and which contains subgroups of 
the variety G.(AA’l’), such that each contains but a single 
perspective involutoric collineation S, will evidently con- 
tain singular transformations of type Il. In addition to 
G,(AA'l’),, the following groups of type I, second class, 
also contain singular transformations of type II: G,(AA’),, 
G,(Ul’),, G,(Al),. 
In type I, third class, the group G,(A/S) contains ©? sub- 
groups G,(AA’l’),_,. Each of these subgroups contains 
o/ singular transformations, and hence G,(A/S) contains 
o* singular collineations of type I. 
The group G,(A, /’’),__, contains ~’ subgroups G,(A Al’) 
,--1- There is evidently but one involutoric perspective 
collineation S in the group G,(A, /”),-_,, and this has its 
vertex at A and its axes coinciding with /’’; S belongs to 
every subgroup G,(AA/’l”),__, nG,(Al”),__,. S, com- 
bined in turn with each of the ~* elations in G,(A,/’’), gives 
o* singular transformations of type II. 
The group G,(l’’),-_, contains ~* subgroups G,(A,/’’) 
---1, one for each position of A in the plane; hence, it 
contains «* subgroups G,(AA’l”’),__,. Since each of 
