232 THEORY OF COLLINEATIONS. 
Eliminating «, and y, from these sets of equations, we get the 
resultant of S and S, in the same form as S, viz.: 
y th x ds A (=) A = i « Als = 
yo khy ' B \kk Bee Kaepnty BN 
1 1 (ieee Lin) 1 Pee 
Bykk, By kk, hoy Balkan o 
(67) 
ye ky 
Comparing coefficients of like terms in the two forms of (32’’) 
we have three equations, as follows: 
bs oe DH | EY z= A, /ki=1 
k.=kk,, B» ( ke es, irae aise ( ky Ik 
ky fl k 1 ky 1 
Bila © Bik," Bek” 
The fact that (67) is of the same form as (67), shows that 
the resultant of Sand S, is a perspective collineation S, 
having the axis coinciding with the axis of S and S,. Equa- 
tions (68) give us the values k,, A, and B,, in terms of k, k,, 
A, B, A, B,. Wecan now state the result: 
The resultant of two perspective collineations having the 
same axis, but different vertices, is a perspective collineation 
with the same axis; the cross-ratio of the resultant is the 
product of the cross-ratios of the components. 
278. The Group H,(l). Equations (67) contain three 
parameters k, A, B; hence there are ~* perspective collinea- 
tions having a common axis. It has just been shown that 
this set of perspective collineations has the first group prop- 
erty, viz.: the resultant of any two of the set is one of the 
same set. We can also show that the set has the second 
group property, viz.: the inverse of one of the set is also in 
the set. If (67) be solved for a and = we get the inverse: 
x xy A 1 k k=1 Wh 
Fe ag Aes Pie ear (6) 
This is of the same form as (67) except that & is changed 
into 1/k; hence the inverse of every collineation in the set is 
also in the set and the set is a group. The symbol for the 
group is H,(/). 
(68) 
