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.
83. Determination of the pencil. We now show that it is possible to assign arbitrarily three lines, a, b, and c, of [pg 50] the system (besides the lines u and u'); but if these three lines are chosen, the system is completely determined.
This statement is equivalent to the following:
Given three pairs of corresponding points in two projective point-rows, it is possible to find a point in one which corresponds to any point of the other.
We proceed, then, to the solution of the fundamental
Problem. Given three pairs of points, AA', BB', and CC', of two projective point-rows u and u', to find the point D' of u' which corresponds to any given point D of u.
