216 THEORY OF COLLINEATIONS. 
THEOREM 39. The fundamental group of type IL consists of all 
collineations of type Il which leave invariant the same figure A A/7; 
it is a two-parameter group, the parameters being / and ¢. 
258. One-parameter Subgroups of G,'(AA‘l). We pro- 
ceed to show that the group G,’(AA'l) breaks up into o/ 
one-parameter subgroups. The two independent parameters 
of G,’!(AA‘l) are k and ¢ and the laws of combinations of 
these parameters are expressed by k,=kk, and t,=t+t,. 
Let us set k = a‘, where a is a constant, and let t be the in- 
dependent variable. In this way we select from the group 
G,’/(AA’l) a set of ~‘collineations. Let T’ and T,’ be two 
collineations of this set, characterized respectively by the 
parameters (a',t) and (a",t,). Their resultant T,’ has the 
parameters (a”,t,), where t,=t-+t,. The inverse of ¢ has 
the parameters (a~‘,—t). Both group properties are satisfied 
by the collineations of the set satisfying the relations k =a‘, 
and the set isa one-parameter group designated by G,'(AA'l),. 
Within the group G,’(AA’l) there are ~‘’ such subgroups, 
one for each value of a (except a = 0). 
THEOREM 40. The fundamental group G2/(44/‘/) breaks up 
into o/ one-parameter subgroups G,/(AA//)q. 
259. Path-curves of G,’(AA‘l),. The effect upon a point 
P of the plane of all the collineations of the group G,/(AA’l), 
is to move it along its path-curve. The equation of the 
family of path-curves of the group G,/(AA’l), is found by 
eliminating ¢ from the pair of equations 
2 x il 1 at—1 & 
0 AN — Tee ee 
y yi pa A a 
Yi 
Eliminating we get 
Xv) i x) x 1 « 
log. nm ie AN = log. a fa _ Ay ells 
vain (56) 
whence « = Cyu 
The curves of this family are transcendental curves; they 
