THEORY OF PROJECTION. 351 
tains ~* one-parameter elliptic subgroups, one for each pair 
of conjugate imaginary points on the line; it contains ~* one- 
parameter parabolic subgroups, one for each real point on the 
line; it contains ©’ two-parameter groups, G,(A), one for 
each real point on the line. The structure of G, may be rep- 
resented by the formula 
G, = 01G,(A) = ~?hG,(AA’) + ~*%eG,(AA’) + ~'pG,(A). 
$6. Theory of Projection. 
Definitions. We begin with a few definitions of the 
terms which will be frequently used in this section. A set or 
row of points on a line is called a range of points; the line on 
which the points are situated is called the base of the range. 
A set of lines lying in a plane and passing through a fixed 
point is called a pencil of lines; the fixed point is called the 
vertex of the pencil, and each line of the pencil is called a ray. 
44, Perspective Projection. Let arange of points, A, B, C, 
D..., Fig. 1, be given on a line 1; let lines be drawn to A, B, 
C, D...., from a point P not on the line /; these lines form 
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