52 GEOMETRIC THEORY. 
effect of 7, and T,,. This may be expressed symbolically by 
the equation T;,T;,, = T;,,,.._ In the same way it may be shown 
that the combined effect of any number of transformations of 
the group is equivalent to some single one of the same group. 
hus ee eee sel WNereiS: — COG. ane. “bnerchar- 
acteristic cross-ratio of the resultant transformation is equal 
to the continued product of the characteristic cross-ratios of 
the component transformations T,T,T, ... T,. 
65. Resultant of Two Parabolic Transformations. Let T 
denote a transformation of the group pG,(A) which trans- 
: ea age Sa 
forms P to P,; then iste era Also let T, be another 
transformation of the same group which transforms P, to P,; 
if 
WN Fe = =t,. Eliminating the fraction ae from 
these two equations we have =5 - = = t, where t,=t-+t,. 
In the same way it may be shown that the resultant of any 
number of transformations of the group pG,(A) is another 
transformation of the same group, and that the characteristic 
constant of the resultant is equal to the sum of the constants 
of the components, thus ,=t-+-t,t,+ ... &-,. 
66. The Two-Parameter Group, G.(A). There are o°’ pro- 
jective transformations of the points on a line, and only ~? 
points on the line. Hence any point can be transformed into 
any other point on the line or into itself in ©’ different ways. 
In other words there is a system of * transformations which 
leaves any point A on the line invariant. We shall proceed 
to show geometrically that this system of transformation has 
the fundamental properties of a group, 7. e., that the com- 
bined effect of any two or more of the transformations of the 
system is equivalent to some single transformation of the 
system, and that the inverse of every transformation in the 
system is also in the system. To show this we take a range 
of points in the line / (Fig. 5), and project it by means of a 
conic K into a second range on the line l’. Revolve l’ about 
