§ 102. Let l1 be the line S1S2 as a line in the pencil S1. To it corresponds a line l2 in S2. At each of the centres two corresponding lines meet. The two axial pencils with l1 and l2 as axes are projective, and, as, their axes meet at S2, the intersections of corresponding planes form a cone of the second order (§ 58), with S2 as centre. If π1 and π2 be corresponding planes, then their intersection will be a line p2 which passes through S2. Corresponding to it in S1 will be a line p1 which lies in the plane π1, and which therefore meets p2 at some point P. Conversely, if p2 be any line in S2 which meets its corresponding line p1 at a point P, then to the plane l2p2 will correspond the plane l1p1, that is, the plane S1S2P. These planes intersect in p2, so that p2 is a line on the quadric cone generated by the axial pencils l1 and l2. Hence:—
All lines in one pencil which meet their corresponding lines in the other form a cone of the second order which has its centre at the centre of the first pencil, and passes through the centre of the second.
From this follows that the points in which corresponding rays meet lie on two cones of the second order which have the ray joining their centres in common, and form therefore, together with the line S1S2 or l1, the intersection of these cones. Any plane cuts each of the cones in a conic. These two conics have necessarily that point in common in which it cuts the line l1, and therefore besides either one or three other points. It follows that the curve is of the third order as a plane may cut it in three, but not in more than three, points. Hence:—
The locus of points in which corresponding lines on two projective pencils meet is a curve of the third order or a “twisted cubic” k, which passes through the centres of the pencils, and which appears as the intersection of two cones of the second order, which have one line in common.
A line belonging to the congruence determined by the pencils is a secant of the cubic; it has two, or one, or no points in common with this cubic, and is called accordingly a secant proper, a tangent, or a secant improper of the cubic. A secant improper may be considered, to use the language of coordinate geometry, as a secant with imaginary points of intersection.
§ 103. If a1 and a2 be any two corresponding lines in the two pencils, then corresponding planes in the axial pencils having a1 and a2 as axes generate a ruled quadric surface. If P be any point on the cubic k, and if p1, p2 be the corresponding rays in S1 and S2 which meet at P, then to the plane a1p1 in S1 corresponds a2p2 in S2. These therefore meet in a line through P.
This may be stated thus:—
Those secants of the cubic which cut a ray a1, drawn through the centre S1 of one pencil, form a ruled quadric surface which passes through both centres, and which contains the twisted cubic k. Of such surfaces an infinite number exists. Every ray through S1 or S2 which is not a secant determines one of them.
If, however, the rays a1 and a2 are secants meeting at A, then the ruled quadric surface becomes a cone of the second order, having A as centre. Or all lines of the congruence which pass through a point on the twisted cubic k form a cone of the second order. In other words, the projection of a twisted cubic from any point in the curve on to any plane is a conic.
If a1 is not a secant, but made to pass through any point Q in space, the ruled quadric surface determined by a1 will pass through Q. There will therefore be one line of the congruence passing through Q, and only one. For if two such lines pass through Q, then the lines S1Q and S2Q will be corresponding lines; hence Q will be a point on the cubic k, and an infinite number of secants will pass through it. Hence:—