PENCILS OF LINES. BiG 
points. Let O and O’ (Fig. 7) be any two points in the plane. 
Pass any conic K through O and O’ and draw a pencil of lines 
through O. Join the points where these lines cut the conic K 
to O’. We thereby construct a pencil through O’ projective 
a 
Fig. 7. 
with the given pencil through O. Corresponding rays of 
these pencils meet on K. If the whole plane be translated 
along the line OO’ without rotation until O’ is carried to O, 
we then have two pencils through O which are projectively 
related. These operations construct a projective transforma- 
tion of the pencil through O. 
72. Invariant Rays. The transformation thus constructed 
usually leaves two rays of the pencil invariant ; these are the 
rays parallel to the asymptotes of the conic K. If the conic 
is a hyperbola, the two invariant rays are real and the trans- 
formation is hyperbolic. If the conic is an ellipse, the inva- 
riant rays are conjugate imaginary and the transformation is 
elliptic. If the conic is a parabola, there is only one inva- 
riant ray, and the transformation is parabolic. 
In the hyperbolic and elliptic transformations, the cross- 
ratio of the invariant rays and a pair of corresponding rays is 
constant for all pairs of corresponding rays. Thus cross-ratio 
in the first case is real; in the second case, of the form e’”’. 
No further developments are necessary in these cases. 
