Two- AND THREE-PARAMETER GROUPS. Peel 
in S,. The set of transformations S, (A) has therefore both 
group properties; (1) the resultant of two transformations of 
the set is in the set; (2) the inverse of every transformation 
in the set is also in the set. Hence the set S,(A) is a group 
G, (A). 
THEOREM 16. All transformations which have a common inya- 
riant point forma two-parameter group; the cross-ratio of the result- 
ant of any two transformations of the group is equal to the product 
of the cross-ratios of the components. 
32. Properties of G.(A). From the continuity of the point 
system on a line and from the known continuity of each sub- 
group, we infer the continuity of the group G,(A). The 
transformations of the group G,(A) are not commutative. 
Since k, = kk,, it is evident that the cross-ratio of the result- 
ant is independent of the order of the components; but the 
position of the second invariant point of 7, is not independent 
of the order of T and T,. Forif A and A, are interchanged 
in (82), the value of A, is changed, thus showing that T and 
T, are not commutative in G,(A ). 
When T and T, have both invariant points in common and 
em : , their resultant is the identical transformation (art. 27); 
but when 7 and T, have only one invariant point in common 
and k, = 2 the resultant is of type II. For putting k, =1/k 
in (32) we get 
L— 7 k-1 1 1 6 
a reas Ea (33) 
whence A, must equal zero, since neither factor on the right 
can be zero. Thus the two invariant points of 7, coincide and 
it is of type II. The value of the constant t of T, is found 
as follows: 
lim 1-— ke k—1 /1 1 
= ————— 4 
t kis A2 k G A ) (34 ) 
THEOREM 17. The group G.(A) contains o/subgroups G,(AA’) 
and one subgroup G,;/(A). The transformations in G.(A) are not 
commutative. The resultant of two transformations of type I in 
1 
G.(A), for which k, = = and A,’ not equal to A’, is of type I. 
