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 the triangles POA and POC are equal, as also are the triangles P’OB and P’OC.  Therefore the angle POP’ is constant, being equal to half the constant angle AOC + COB.  This being true, if we take any four harmonic points, P_1_, P_2_, P_3_, P_4_, on the line u, they will project to O in four harmonic lines, and the tangents to the circle from these four points will meet u’ in four harmonic points, P’_1_, P’_2_, P’_3_, P’_4_, because the lines from these points to O inclose the same angles as the lines from the points P_1_, P_2_, P_3_, P_4_ on u.  The point-row on u is therefore projective to the point-row on u’.  Thus the tangents to a circle are seen to join corresponding points on two projective point-rows, and so, according to the definition, form a pencil of rays of the second order.

81.  Tangents to a conic. If now this figure be projected to a point outside the plane of the circle, and any section of the resulting cone be made by a plane, we can easily see that the system of rays tangent to any conic section is a pencil of rays of the second order.  The converse is also true, as we shall see later, and a pencil of rays of the second order is also a set of lines tangent to a conic section.

82. The point-rows u and u’ are, themselves, lines of the system, for to the common point of the two point-rows, considered as a point of u, must correspond some point of u’, and the line joining these two corresponding points is clearly u’ itself.  Similarly for the line u.