132 THEORY OF COLLINEATIONS. 
ete., and the lines t,;U,’; l/’, ete:, into P); Pi; Py’ petc:, and 
l,, l,’, l./’, respectively. The two collineations T and T, act- 
ing in succession are equivalent to a single transformation U, 
which transforms the points P, P’, P’’, directly to P., P.’, P.”, 
etc., and the lines 1, l’, l’’, etc., directly to l,, l,’, 1,/’, ete. 
Since U transforms points into points and lines into lines, it 
is a collineation* which we may designate by T,. We say 
that T, is the resultant of T and T,, and that T and T, are 
components of T,. This relation may be expressed in the 
form of an equation as follows: 
Teale 
where the two components 7 and T, operate in the order 
named. 
If the two component collineations T and T, operate in the 
reverse order, first 7, and then 7, their resultant T,’ is not 
always the same as T,. Thus T,T=T-,' and in general 
TT,#~T,T. The two resultants T, and T,’ are called conju- 
gate collineations. When T, and T.’ are the same, that is 
when 7'T,= T,7, the two collineations T and T, are said to 
be commutative. 
162. Groups of Collineations. A set or system of collinea- 
tions is called a group, as in Chap. I, Art. 26, when it has the 
following properties : 
First group property. The resultant of any two collinea- 
tions of the system, taken in either order, is also in the system. 
Second group property. The inverse of every collineation 
in the system is also in the system. 
Unless a system of collineations possess both group proper- 
ties, it is not entitled to be called a group. A system may 
be so selected that it has the first group property, but not the 
second. As an example of this we may select the system of 
one-dimensional transformations given by the equation 
v,=kx, where k has all complex values consistent with the 
condition |k|= <1. This system has the first group prop- 
* For a general analytic proof see § 2 of the present chapter. 
