ONE-PARAMETER GROUPS. 21 
the system for which k = 1-4, where 4 is an infinitesimal, is 
ealled an infinitesimal transformation. 
The transformations of this system have the property that 
the resultant of any two of them is a transformation of the 
same system; and the inverse of every transformation of the 
system is also in the system, as is shown below. Every sys- 
tem of transformations having these two properties is called 
a group of transformations. This group of transformations 
leaving A and A’ invariant is called a continuous group, since 
continuous variation of & gives rise to transformations, all of 
which belong to the group. This group is designated by the 
symbol G,(AA’) and is called a one-parameter group, the 
cross-ratio k being the variable parameter of the group. 
THEOREM 12. The totality of projective transformations which 
leave the same two points of a line invariant forms a continuous 
eroup; the cross-ratio of the resultant of any two transformations 
of this group is equal to the product of the cross-ratios of the com- 
ponents. 
27. Properties of the Group GAA’). The fundamental 
property of the group G,(AA’) is that the, resultant of any 
two transformations of the group is another of the same 
group. This is called the first group property. Other prop- 
erties of the group will now be developed. 
The inverse of 7, any transformation in G,(AA’), is 
also to be found in G,(AA’). To show this let T be the 
transformation which transforms the point x into «#,; then T 
is given by the equation 
Cae ee) 
The inverse of 7 transforms w, back into x; then T~‘ is given 
by 
DOS A Ser A 
ap JNO TT fhe ae ANOS 
Hence the cross-ratio of the inverse of T is given by 1/k; in 
other words the cross-ratios of a pair of inverse transforma- 
tions have reciprocal values. Since k is any number in the 
