24 ONE-PARAMETER GROUPS. 
1 1 
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Hence the parameters of a pair of inverse transformations are 
numerically equal but of opposite signs. Since the negative 
of every complex number is also a complex number, it follows 
that the inverse of every transformation in the system is also 
in the system. Therefore the system of transformations of 
type II having the same invariant point possesses the second 
group property. This system has both of the defining group 
properties and is therefore a group. This group is continu- 
ous; it contains a transformation for every value ¢ of the 
complex number system. It is designated by G,'(A). 
THEOREM 14. The totality of transformations of type IT which 
leave the same point invariant forms a continuous group; the con- 
stant, t, of the resultant of any two transformations of the group is 
equal to the sum of the constants of the components. 
29. Properties of the Group G,/(A). The resultant of a 
pair of inverse transformations is the identical transformation 
whose constant is t, =t—t=0. The group G,/(A) therefore 
contains the identical transformation. 
The only transformation in the group which is its own in- 
verse is the identical transformation, 7. e., the group contains 
no involutoric transformation. It contains one pseudo-trans- 
formation for which t= ©. This transforms every point on 
the line to the invariant point. 
A transformation of the group whose constant t is infinitesi- 
mally near to zero, 7. e., t=|o\e*", where p is an infinitesimal 
and #@ varies from 0 to 2x, is an infinitesimal transformation. 
If an infinitesimal transformation is repeated n times, the re- 
sultant has the constant nt. By a proper choice of 7 and 6 
this may be made any number we please; hence every trans- 
formation in the group G,/(A) can be generated from an 
infinitesimal transformation of the group. 
THEOREM 15. The resultant of any two transformations of the 
group G,/( A ) is also a transformation of the group ; its transforma- 
