ANALYTIC CONDITIONS. 147 
This is called the general collineation group or general pro- 
jective group of the plane. 
THEOREM 6. The system of o* collineations of the pli ne form 
a group Gs. : 
$ Analytic Conditions for a Subgroup of G,. 
181. We shall develop in this section some fundamental 
theorems in the theory of groups of collineations and shall es- 
tablish the necessary and sufficient conditions for the exist- 
ence of a subgroup of G,. 
We shall use throughout this and the next two sections the 
homogeneous form of a collineation T and shall take the pro- 
portionality factor p equal to unity. Thus 
m=acrt+tbyt+ez, 
Ti W=aQr+by+toz, Giles) 
4 =a;u+b3y+c32. 
This form is just as general as that used in Art. 175, and for 
our purpose far more convenient. In this form three definite 
numbers «, y, z, are transformed by T into three other definite 
numbers «,, y,,2,. But if the three equations (11’) be each 
multiplied by p40, we see that T transforms px, py, pz into 
0X1, 0Y:, 92,3 1. €, the ratios «:y:z are transformed into 
(DASA Paeras 
There are two homogeneous forms of linear transforma- 
tion each of which is equivalent to the same Cartesian form, 
viz.: T as above and 7; thus 
nial —%, =ae+by+e2, 
ee = Y, = dou + boy + oz, (7) 
— 2, =a; + bay +¢32. 
But the * transformations 7’ do not form a group. The re- 
sultant of two of them is a transformation of the kind T. 
We shall use only the first kind to represent a collineation and 
care must be taken in any equations that involve the squaring 
of the equation T; for such an operation merely introduces 
