SINGULAR TRANSFORMATIONS. 275 
THEOREM 3. The groups G@s( A’), and Gy/(Al’)a of type IT, 
second class, each contains discrete systems of singular transforma- 
tious of types IIL and V; the groups @,/( Al’), and G,'( AA’), each 
contains discrete systems of singular transformations of type V. 
325. Singular Transformations in G,(AA’l’),. We shall 
next examine the group G,(AA’l’), for singular transforma- 
tions. Its structure was given in Art. 320, thus: G,(A A'l’),= 
o01G,(AA’'A”), + H,'(Al)+S.T. There are four cases to be 
considered, viz.: When * is rational with even numerator and 
odd denominator, 7 rational with odd numerator and odd de- 
nominator, 7 rational with odd numerator and even denomi- 
nator, 7 irrational. We must examine each case separately. 
Let r be rational with even numerator and odd denomina- 
tor. Then according to article 256 each subgroup G,(AA’A”’), 
in G,(AA’l’), contains the same involutorie perspective col- 
lineation S, having its vertex at A’ and its axis along /’; and 
the group G,(AA’l’), contains only this one involutoric per- 
spective collineation S. 
Let S be combined with T any collineation of type I in 
G.(AA'l’),; the resultant one-dimensional transformations 
along both / and /’ are both loxodromie, and hence the result- 
ant of S and T is of type I and is one of the collineations of 
the group G,(AA/l’),. On the other hand, let S be com- 
bined with S’, a collineation of type V in H,/(A1) (which is a 
subgroup of G,(AA’l’),). Along / we have an involutorie 
combined with an identical transformation, and the resultant 
in this direction is involutoric with invariant points at A and 
A’. Along l’ we have an identical combined with a parabolic 
transformation, and the resultant is parabolic. Through A’ 
we have an identical combined with a parabolic transforma- 
tion and the resultant is also parabolic. Hence, the resultant 
of Sand S’is T’, a collineation of type II, whose invariant 
figure is (A A’l’) and whose constants k and t are k = —1 and 
t equal to the t of S. If S be combined in turn with each ela- 
tion of H,'/(Al), we have an infinite system of collineations 
of type II, for each of which k = — 7 while t has all complex 
