PERSPECTIVE GROUPS. 227 
S,/on a line g parallel to 1. S’ transforms g to g, and S,’ 
transforms g, to g., both parallel to /. Let the perpendicular 
distances of g, g,, and g, from | be respectively p, p,, and p,. 
By the equations of Art. 145, we have 
ey and = ao bi 
Pr Pp Pe Pi 
Eliminating p, we get 
if 1 
ae 
Let us also consider the effect of S’ and S,’ on a pencil of 
rays through O, any point on/. The pencil through O un- 
dergoes one-dimensional parabolic transformations due to S’ 
and S,’ which are expressed, Art. 145, by the equations 
cot», — cote =dt and cot, — coto, = d,t,, 
where d=0A and d,=0A,. Eliminating cot, from the 
equations we have 
cot p, — cotg =dt+d,t, = d,t,, 
which shows that the resultant one-dimensional transforma- 
tion in the pencil through O is also parabolic. Hence the 
resultant of S’ and S,’ is an elation S,’(A,/) having its vertex 
at some point A, on/. The characteristic constant and vertex 
of S,’(A,1) are given by the equations 
t, =t+t, and d,t,=dt-+ d,t,. (64) 
The first group property is therefore established for the set 
of * elations having a common axis/. This set is made up 
of «1 one-parameter groups H,'(Al), one for each point on 
l. Since the inverse of every collineation in one of these 
groups is likewise in thesame group, it is also in the set made 
up of these groups. Hence the set of «° elations having the 
same axis is a group of two-parameters, t and d, designated 
byl,’ (L)). 
272. Analytic Proof of H,'(l). Let the axis of the elation 
1 be taken as the x-axis and let the origin be some point O on 
1. The equations of S’ are found by putting B= 0c = 1c’ =0 
