*79. Pencil of rays of the second order defined.* If the corresponding
points of two projective point-rows be joined by straight lines, a system
of lines is obtained which is called a pencil of rays of the second order.
This name arises from the fact, easily shown (§ 57), that at most two
lines of the system may pass through any arbitrary point in the plane. For
if through any point there should pass three lines of the system, then
this point might be taken as the center of two projective pencils, one
projecting one point-row and the other projecting the other. Since, now,
these pencils have three rays of one coincident with the corresponding
rays of the other, the two are identical and the two point-rows are in
perspective position, which was not supposed.

[Figure 19]

FIG. 19




*80. Tangents to a circle.* To get a clear notion of this system of
lines, we may first show that the tangents to a circle form a system of
this kind. For take any two tangents, _u_ and _u’_, to a circle, and let
_A_ and _B_ be the points of contact (Fig. 19). Let now _t_ be any third
tangent with point of contact at _C_ and meeting _u_ and _u’_ in _P_ and
_P’_ respectively. Join _A_, _B_, _P_, _P’_, and _C_ to _O_, the center of
the circle. Tangents from any point to a circle are equal, and therefore