PERSPECTIVE GROUPS. 229 
tions with different invariant points. Their resultant is 
(Chap. I, Art. 22) usually loxodromic with two invariant 
points, say B and C; it may however be parabolic with one 
invariant point, B; or it may be identical leaving all points 
on g invariant. In the first case if there are two invariant 
points B and C on g, then the resultant of S’ and S,’ is of 
type I, leaving invariant the triangle (OBC) ; if in the second 
case there is only one invariant point B on g, then the result- 
ant of S’ and S,’ is of type II, leaving invariant the figure 
(OBgq); if in the third case every point in g is invariant, the 
resultant of S’ and S,’ is of type IV, having O for its vertex 
and g for its axis. Hence the resultant of two elations in the 
most general position is of type I, type II or type IV. 
275. Two Elations Whose Resultant is of Type III. In 
the above configuration let O, the intersection of J and l,, co- 
incide with A, the vertex of S,’, Fig. 27(d). The resultant of 
S’ and S,/ now leaves invariant the point A, and the line /. 
Along the line / the two component one-dimensional trans- 
formations are respectively parabolic, with invariant point at 
A,, and identical ; the resultant along / is therefore parabolic 
with invariant point at A,. The two component one-dimen- 
sional transformations of the pencil through A, are respec- 
tively parabolic and identical; their resultant is therefore 
parabolic, having / as the invariant line. The resultant of S’ 
and S,’ leaves invariant the lineal element A,! and produces 
one-dimensional parabolic transformations along / and through 
A, and is therefore of type III. 
276. Anulytic Proofs of Articles (273-275). The results 
of the last three articles may be deduced analytically. We 
shall only outline the proof, leaving the details to the reader. 
Starting with the configuration of Art. 273 let 0 be the origin 
and l and I, the w- and y-axis respectively. The normal forms 
of S’ and S,’ reduce to the following: 
a+Aty 
Se ran a — 
1+ty (66 ) 
y 
1+ty ’? 
