SINGULAR TRANSFORMATIONS. 271 
The group G,( A/S) contains ©” subgroups G,(AA’A’’), 
where A’ is in turn every point on A/ and A” in turn every 
point in the plane. One conic of the pencil S passes through 
A”, and A’A” is tangent to this conic at A’. Take a collinea- 
tion T from the group G,(AA’A”’) and another T, from the 
group G,(AA,'A,’) having cross-ratio constants k and 1/k 
respectively. Their resultant is parabolic along Al and 
through A, and hence is a collineation 7” of type III belong- 
ing to the group G,’’( Al). It can be shown that all the col- 
lineations of the group G,’’( Al) are to be found in G,(S). 
Again, take two collineations T and T,, whose cross-ratio 
constants are k and = from the groups G,(AA’A’’) and 
G,(AA’A,'’), where A and A, are collinear with A. Their 
resultant is identical along A and through A, but is parabolic 
along A, and through A’; hence, it is an elation S’” and be- 
longs to the group H,/(Al). Evidently, all elations in 
H,( Al) are contained in G,(S). 
The remaining formule are easily verified. 
$2. Singular Transformations. 
We come to the consideration of the so-called singular 
transformations, Art. 318, in the collineation groups of the 
plane. These were defined as systems of collineations of one 
type not forming a continuous group, yet occurring in an 
otherwise continuous group of another type. We shall find 
two distinct kinds of singular transformations, viz.: discrete 
systems of collineations of type III or V occurring in groups 
of type II, second class; and discrete systems of collineations 
of type II occurring in groups of type I, second and third 
classes. 
We shall examine our systems of singular transformations 
to see if they have both group properties; we shall find that 
the systems of singular transformations of types III and V 
