274 THEORY OF COLLINEATIONS. 
that t+t,=0 and a‘** = 1, their resultant is parabolic along 
/ and identical along /’.. The resultant is therefore an elation 
belonging to the group H,/( Al’). Every collineation in the 
group H,'( Al’) is contained in G,’(ll’),. But if T’ and T,’ 
are so chosen that t+t,~ 0 and a‘** = 1, then their resultant 
is parabolic along both /’ and /. The resultant is an elation 
but it does not belong to H,’(Al’). The vertex of its inva- 
riant figure is at A and the axis of invariant points is a line 
through A not! orl’. Since 7 has only integral values, this 
system of elations is not continuous and does not form a con- 
tinuous group. The inverse of every elation in the system is 
also in the system. 
Since the group G,’ (/l’), is contained in the groups G,’ (Al’),, 
its system of singular transformations of type V is also con- 
tained in G,’( Al’), ; hence G,’( Al’), contains singular trans- 
formations of two kinds, viz.: types III and V. 
Since the group G,'( Al’), contains 1 subgroups G,’(Il’),, 
one for each line / through A, it must therefore contain a dis- 
crete system of ~’ elations S’(A), selected from the group 
H,'(A), which system has both group properties and thus 
forms a discontinuous subgroup of G,’(Al’),. But G,/( Al’), 
also has H,’/(/’) as a subgroup. If we combine an elation 
from S’(A) with one from H,’(l’), the resultant is a collinea- 
tion T’’(Al’) of type II]. Hence the singular transformations 
of type III in G,’( Al’), appear as the resultants of the ~* 
elations in S’(A) with those of the group H,’(l’). We also 
see that the discontinuous group S’(A) is a subgroup of the 
discontinuous group d G’’( Al’). 
THEOREM 2. The system of singular transformations of type V 
in G,(/l’)q form a discontinuous group dS/( 4). 
In the same manner it may be shown that the groups 
G,/(A’l), and G,'(AA’), dualistic to G,’( Al’), and G,’(Il’), 
contain singular transformations as follows: G,'( Al’), con- 
tains singular transformations of types III and V; G,’(AA’), 
contains a system of singular transformations of type V. 
