136 THEORY OF COLLINEATIONS. 
But T, may also be written in the form 
fox2e = x+ fez, 
ie c p2y2=key, (4”) 
p2%2 =2. 
Comparing coefficients of corresponding terms in the two 
forms of T, we get 
t=t+t, 
k= kk, ©) 
Since T, is of the same form as T and T,, T, belongs to the 
same system as T and T,; hence the given system has the 
first group property. 
The inverse of T is given by the equations 
Plea —t 21, 
TH! y=, (7) 
ei eie 
Hence the inverse of T is a!s9 in the system and the system 
has the second group property. The system is therefore a 
group. It has two parameters, k and ¢, and is thus a two- 
parameter group. Every collineation in the group leaves in- 
variant the figure (A, A’,/); the appropriate symbol of the 
group is therefore G’,(AA’l). 
The two equations (6) express the constants of T, in terms 
of those of T and T,. They are called the equations of con- 
dition or conditional equations of the group. The number of 
conditional equations is always just sufficient to determine the 
constants in 7’. 
169. Subgroups of a Given Group. Let G,, r<9, be an 
r-parameter group of collineations, defined by a set of equa- 
tions involving 7 independent parameters. It frequently hap- 
pens when one or more of these parameters is kept constant 
and the others are made to vary, that the system of collinea- 
tions thus selected from G, has both group properties and is 
therefore a group within a group, or as it is called a subgroup 
of the larger group. Subgroups of a given group may often 
be obtained by setting up a constant relation between two or 
more of the parameters of an r-parameter group and thus 
