22 THEORY OF COLLINEATIONS. 
occurring in groups of type II, second class, have both group 
properties, and hence are discontinuous subgroups of continu- 
ous groups. We shall also find that the singular transforma- 
tions of type II occurring in groups of type I, second and third 
classes, have the second group property but not the first, 
hence they do not form discontinuous subgroups of continuous 
groups. 
322. Singular Transformations in G’,(Al’),. Take two 
collineations of type II, T’(AA'l’) and T’ (AA,'l'), belong- 
ing to the group G,'(Al’), and let their constants k and t be 
(a‘,t) and (a",t,). For every value of a#0 or o, tand t, 
may be so chosen that t+t,40, whilea't*=1. To prove 
this, put a= re?’ and t+t,=p-+1q; we then have 
(neu) ex) = 1 : (70) 
Taking logarithms of both sides we get 
(p+iq) (log. r +79) =2na21; 
whence plog.r—qi=0 and qlog.r+p0=2nz2. Solving 
for p and q we find 
2n70 2nn log.r 
—— =, 8 
JD log.2r + 0 2 dl log 2r st “2 ( ) 
n has only integral values and hence, for values of a+ 0 or 
o,andn+0, t and t, can always be chosen so as to satisfy 
the conditions 
tai, Onand. G7 — she (9) 
The group G,’( Al’), has the following structure: G,’( Al’), 
= «0 G (ll), A (UV) +S. 1. Te T"( AA) and (Az) 
be so chosen that t+ t,=0, then a‘ =1; their resultant 
is identical along /’ and parabolic through A; hence, it is 
an elation S’(X/’) where X is some point on/. This elation 
belongs to the group A,/(l’). All elations of the group 
A,'(l’) are present in G,/(Al’),. But if T’ and T;’ be so 
chosen that t+ t,~0 and a‘** = 1, then their resultant is 
parabolic along the invariant line /’ and also parabolic through 
the invariant point A; hence, it is a collineation of type III, 
TUACAC a: 
