INFINITESIMAL COLLINEATIONS. 293 
the plane of our transformation). It is evident, since / and 
k’ are independent parameters, that there is a point in the 
plane corresponding to every collineation of the group 
hG.(AA’A”). Since all collineations, whose parameters k 
and k’ satisfy the relation k’ = k", form a one-parameter sub- 
group of hG,(AA’A’’), we see that the curve y= 2” corre- 
sponds to this subgroup and the individual points of the curve 
correspond to the individual collineations of the group. If 
we give to r all real values, we have a family of curves which 
corresponds to the system of subgroups of hG,(AA‘A’’). 
From the properties of this system of curves we deduce the 
following results: If 7 is an irrational number, the curve 
y=«" contains no real point for which either coordinate is 
negative ; the curve lies entirely in the first quadrant. If r 
isa rational fraction with even numerator and odd denomi- 
nator, y can not be negative, and the curve lies above the axis 
of x in the first and second quadrants. If r is rational with 
odd numerator and even denominator, the curve lies in the 
first and fourth quadrants. If 7 is rational with odd numera- 
tor and odd denominator, the curve lies in the first and third 
quadrants. 
Every curve passes through the point (1,1), which shows 
that the identical transformation belongs to every subgroup. 
The curves of our family contain every point in the first quad- 
rant, but not every point in the second, third and fourth quad- 
rants. Consequently our two-parameter group 1G,(AA’A”’) 
contains collineations which do not belong to any of its sub- 
groups. Such a collineation has one or both of its cross-ratio 
constants negative, and their values are such that they do not 
satisfy an algebraic equation of the form of k’ =k", where 
m and n are integers. 
The variable parameter of every one-parameter group in 
hG,(AA’A”) isk; and every one-parameter group contains 
two real infinitesimal collineations: viz., when k=1+ 6. 
Each infinitesimal collineation generates its corresponding 
portion of the group. Every collineation in the group 
