20 ONE-PARAMETER GROUPS. 
$3. One-Parameter Groups of Projective 
Transformations. 
26. Resultant of T and T, with Common Invariant Points. 
Let T and T, be two transformations of type I having the 
same invariant points A and A’, and let T transform the point 
2 to.x,, and let T, transform x, to x,. The resultant of T 
and T, also leaves A and A’ invariant and transforms « di- 
rectly to z,. Let T and T, be given in the implicit normal 
forms : 
m1 —A a—A w2— A am—A , 
5 ee ah SONG ery oer (14) 
We eliminate x, from these equations by multiplication, and 
obtain T,: 
we — A a —A 
ey Oo ema 
The cross-ratio of T, is therefore k,=kk,. Since kk,=k,k, 
it follows that T and T, are commutative, thus completing 
theorem 7. 
In the same way it may be shown that the resultant of any 
number of transformations with the same invariant points has 
its cross-ratio equal to the continued product of the cross- 
ratios of the components. 
The cross-ratio k is a complex number and may have a 
doubly infinite number of values; hence there are a doubly in- 
finite number of transformations, leaving two given points A 
and A’ invariant. In fact the system of transformations 
leaving A and A’ invariant contains a transformation corre- 
sponding to each number of the complex number system. 
Certain transformations of this system, corresponding to cer- 
tain special values of k, have received special names. Thus 
the transformations of the system corresponding to k= 1,— 1, 
0, © are called the identical, the involutoric, and the two 
pseudo-transformations, respectively. Any transformation of 
