284 THEORY OF COLLINEATIONS. 
the group G,'/(A) ist, and that the law of combination of 
parameters in this group is expressed by the equation 
t,=t+t,. The transformations in this group are commuta- 
tive, Art. 11, and ¢ assumes in turn all complex values. 
The group contains the identical transformation correspond- 
ing to the valuet=0. By definition, Art. 26, an infinitesi- 
mal transformation is one that differs by an infinitesimal 
value of the parameter from the identical transformation. 
Let us write ¢t in the form re*’ , where r is real and positive. 
The identical transformation of the group is given by r= 0. 
Infinitesimal transformations of the group are given by r=4, 
where 4 is an infinitesimal. Since 4 varies continuously from 
0 to 2x, the relation t= de’ gives us an infinite number of 
infinitesimal transformations in the group G,/(A). If t be 
represented geometrically by the Argand diagram, the values 
of t corresponding to these ~‘ infinitesimal transformations 
lie on a circle about the origin of radius r = 4. 
Let us choose one of these infinitesimal transformations 
corresponding to a fixed value of @, say #,, and designate it 
by J,. If I, be repeated we find the resultant of J, and J, as 
follows: t, =t, +t, = de + de =2se%, In like manner if 
I, be repeated n times, we have t, = nde. The position of 
the point t, on the Argand diagram is at a distance nd from 
the origin and on a line making the angle #, with the axis of 
reals. By a proper choice of » we can make the point t, move 
from the origin along the half-ray /», to any desired position 
on this ray. Consequently every transformation in G,/(A) 
corresponding to a value of ¢ situated on the half-ray /4, can 
be generated by the repetition of the infinitesimal transforma- 
tion I, given by t, =de*. In like manner we see that each 
infinitesimal transformation in G,/(A) can generate those 
finite transformations of the group whose corresponding val- 
ues of ¢ lie in the Argand diagram on a half-ray through the 
origin. It is evident that any given finite transformation in 
G,'(A) can be generated by the repetition of one, and only 
one, infinitesimal transformation. 
