58 PENCILS OF LINES. 
73. Parabolic Transformations. The parabolic case re- 
quires special attention, for the characteristic constant t of 
a parabolic transformation is not subject to a dualistic inter- 
pretation. Leta transformation T be determined by a para- 
bola K passing through O and O’ (Fig. 8). The single ray 
Fic. 8. 
left invariant by the transformation is parallel to the axis of 
the parabola K. Let r and 7’ be a pair of corresponding rays 
in the pencils through O and O’; and let them meet the conic 
K in the point @. We make use of the following theorem for 
a parabola: If from any point @ on a parabola chords be 
drawn to O and O’, two fixed points on the parabola, the dif- 
ference of the cotangents of the angles which these chords 
make with the axis of the parabola is constant. Thus 
cot, — cot? = cot», 6 and 6,, being the angles which the rays 
r and 7’ make with the invariant ray, and ¢ being the angle 
made with the invariant ray by the ray r” which is trans- 
formed into the perpendicular to the invariant ray. 
The relation cot#,=cot#+t, for a parabolic transforma- 
tion is readily deduced as a limiting case of the cross-ratio 
formula of a hyperbolic transformation. (See art. 37.) 
74. Projective Transformations of a Pencil of Planes. The 
theory of the real transformations of a pencil of planes is so 
