PERSPECTIVE GROUPS. 233 
All perspective collineations haying a common axis forma, 
three-parameter group. 
279. The Group H,(ll’). We have already seen that the 
cross-ratio of the resultant of any two collineations in H,(/) 
is equal to the product of the cross-ratios of the components; 
thus k,=kk,. We next inquire into the position of the ver- 
tex of the resultant, whose coordinates are (A,, B.). 
From equations (68) we find 
AB, (k —1) + A, Bk (ky — BB, (kk, — 
(lee Ee ef ebealev Lt (GO) 
By (k — 1) + Bk (k; — 1) B, (k — 1) + Bk (ky — 1) 
It is easy to verify the following equation, 
Abas Siteeiet 
eee ane —a()s (70) 
TN aa ee 
Hence the point (A., B,) is collinear with (A, B) and (A,, B,). 
From this fact we infer that all collineations having the same 
axis and whose vertices are collinear form a group. This 
group is designated by H,(ll’), l’ being the line on which 
the vertices lie. 
The group H,(/) contains ©’ two-parameter sub-groups 
H,(ll’), one for each line l’ of the plane ; but two such sub- 
groups contain one one-parameter group H,(A,/) in common, 
A being the point of intersection of /’ and 1,’. 
280. The Groups H,(A) and H,(AA!‘). It may be shown 
in a manner similar to Art. (278) that all perspective collinea- 
tions having the same vertex A form a_ three-parameter 
group H,(A), and that all such collineations having their 
axes concurrent at A’ form a two-parameter group H,(A A’). 
The same thing may be shown geometrically as follows: Let 
S(A,1/) and S,(A,l,) be two perspective collineations having 
the common vertex A. Fig. 28(b). Both leave invariant all 
lines through A; hence their resultant also leaves invariant 
the pencil through A and is also a perspective collineation. 
Both also leave invariant A’, the intersection of / and 1,; 
hence the resultant is of type IV and its axis, l,, goes 
