26 - Two- AND THREE-PARAMETER GROUPS. 
Making A’=0 and A,’ = 0 in (22) these simplify to 
x v1 
hi and ¢— 
1—k 1-h 5 31 
(Zjete ( Ar Jatha ( ) 
Eliminating x, from these two equations we get 
ES = —— ‘ 
( a , 2 ce 2) Me (81a) 
But this is the same form as (31), viz. : 
T,: «= ——— 
(eas eP 
Comparing coefficients in (31a) and (31b), we have 
ke => kk 
a Be: ae mh ue» (32) 
These two equations enable us to express k, and A, in terms 
of k, k,, A and A,. 
From these results we see that the resultant of two trans- 
formations of type I, having one invariant point in common, 
has for one of its invariant points the common invariant point 
of the components, in this instance the origin. The first of 
equations (32) shows us that the cross-ratio of the resultant 
is also equal to the product of the cross-ratios of the compo- 
nents, viz.: k, = kk,, just as in the case where the two inva- 
riant points are common to the two transformations. 
Since (31a) is of the same form as (31), we see that the 
first group property is satisfied, 7. e., in the set S, (A) of «0° 
transformations given by (81) the resultant of any two of the 
set is also in the set. The two parameters of the set are the 
cross-ratio k and the abscissa of the other invariant point A. 
The structure of the set is evident; the origin A’ may be 
taken in turn with every other point on the line to form the 
invariant points of a group G,(A’A) and once with itself to 
be the invariant.point of G,’(A’). Hence it contains ~’ one- 
parameter groups of type I and one of type Il. Every trans- 
formation in the set S,(A) belongs to one of these one-para- 
meter groups; its inverse is in the same group and hence also 
