FUNDAMENTAL GROUPS. 215 
merator and odd denominator and odd numerator and even 
denominator respectively. None of these three involutoric 
collineations can belong to a group G,(AA’A”), for which r 
is irrational. 
THEOREM 38. Every one-parameter group @,( 4 BC), for whieh 
ris rational contains an inyolutoric perspective collineation; no 
such group for which 7 is irrational contains an involutorie perspec- 
tive collineation. 
C. FUNDAMENTAL GROUP OF TYPE II AND ITS SUBGROUP. 
257. Fundamental Group of Type II. The canonical form 
of a collineation 7’. of type II has to be, Art. 149: 
————— y, = —— 
= 
x+t il x+t 
o+ty ie ar ew 
(54) 
A 
when the invariant lines of the figure are the lines « = 0 and 
y = 0, and the invariant points are the origin and the point 
(A,0). Equations (54) may also be written: 
x 1 1 k-1«& 
ee GINO) a Lae ae ee eee 
A second collineation T,’ with the same invariant figure may 
be written: 
FS | ees Pg ec BL) SU ae / 
ibe of =p -: and a Te rele (54’) 
Eliminating «x, and y, from the equations of 7’ and T,’ we 
get the resultant 7, in the form 
js Se) RST Agen ep ee "1 
f De re Upia ne are ae yt tes (54’’) 
where k, = kk, and t,=t-+t,. The inverse of T’ is found by 
solving equations (14) for x and y; 
/-1 - ny ps eae 1 =: i IIR = wy 
LOSS ie ctr 2S a Tea ae ot”) 
Hence both group properties are established for the set of 
collineations of type II leaving the figure (AA’/) invariant. 
The set is therefore a two-parameter group designated by 
G,'(AA’l), the parameters being / and t. 
