ONE-PARAMETER GROUPS. 23 
every transformation in G,(AA’) may be generated from an 
infinitesimal transformation of the group. The chief prop- 
erties of the group G,(AA’) may be summed up as follows: 
THEOREM 13. The resultant of any two transformations of the 
group G;(AA’) is a third transformation of the same group; the 
transformations of the group can be arranged in inverse pairs; it 
contains the identical, one involutoric, two pseudo, and an infinite 
number of infinitesimal transformations; every transformation of 
the group can be generated from an infinitesimal transformation of 
the group. 
28. One-Parameter Group G,/(A). Let T and T, be two 
transformations of type I having the same invariant point 
A. They may be written: 
1 1 1 1 
m—A < TA, a t and t2—A 7h m—A +t. (18) 
T transforms « to x,, and 7, transforms x,tow,. Their result- 
ant T, is obtained by eliminating «, from these two equations 
by addition, giving us: 
1 1 
i wee Oe 
Thus, ¢,=7t--¢,.. The resultant, T,, is of type II (thus 
completing theorem 10), has the same invariant point A, and 
its constant, t., is equal to the sum of the constants of T and 
IP. 
The parameter, t, being a complex number, may have any 
one of a doubly infinite number of values ; and hence there are 
a doubly infinite number of transformations of type II having 
the same invariant point: This system of transformations of 
type II having the same invariant point possesses the first 
group property, as has just been shown. That it also pos- 
sesses the second group property we proceed to show. Let T 
be the transformation : 
~ = 1 = — 
a —A = x 
I O 
Sein Os (18) 
its inverse, 7‘, which transforms x, back to 2, is 
