212 THEORY OF COLLINEATIONS. 
THEOREM 35. The constant 7 in the one-parameter group 
G,( AA’A’’), is the cross-ratio of the pencil of four lines formed by 
a tangent to one of the path-curves and the three lines drawn from 
its point of contact to the vertices of the invariant triangle. 
258. Path-curves are Straight Lines when r=1,0, 0. 
The path-curves of a one-parameter subgroup of G,(AA’A”’) 
reduce to straight lines for three values of 7, viz.: r=(1,0, ~). 
Let r= 1 in x"z*7 = Cy, and we get a pencil of lines « = Cy 
through A; let » =0 and we get z=Cy, a pencil of lines 
through A’; let 7 = © and we get « = Cz, a pencil of lines 
through A”. 
When 7 = 17 the cross-ratio of the one-dimensional trans- 
formations along the line A A’ and A A” are equal and hence 
that along the line A’ A” is unity, thus this last transforma- 
tion is identical; hence every point on A’ A” is an invariant 
point. Consequently every line through A is an invariant 
line; hence the subgroup of G,(A A’ A”) for r = 1 is a group 
of perspective collineations, having A for the vertex and 
A’ A” for the axis. In like manner we see that when r = 0 
and ©, we have subgroups of perspective collineations and 
vertices A’ and A”, and axes A A” and A A’ respectively. 
THEOREM 36. The group G:(4 A’ A”) contains three subgroups 
of perspective collineations, viz., the subgroups for which 7 = 7,0, «. 
For these three subgroups the path-curves are straight lines. 
254. Path-curves are Conics when r = — 1, 2, 1 [2- There 
are three other specially important subgroups of G,(AA’A”); 
these correspond to the values r=—1,2,1/2. Putting 
r=2 in a'z**=Cy, weget yz =Cz’*; the path-curves re- 
duce in this case to a system of conics having double contact 
at Aand A”. A’A and A’A are common tangents to the 
pencil of conics and A A’ A” is the chord of contact. In like 
manner, when 7 = — /, the path-curves are «xy = Cz*; when 
r = 1/2, the path-curves are «z= Cy’. These are also pen- 
cils of conics having double contact; in each case two sides 
