| The lines which join any point in space to the points on a curve of the second order form a cone of the second order. | Every plane section of a cone of the second order is a curve of the second order. |
| The planes which join any point in space to the lines enveloping a curve of the second class envelope themselves a cone of the second class. | Every plane section of a cone of the second class is a curve of the second class. |
By its aid, or by the principle of duality, it will be easy to obtain theorems about them from the theorems about the curves.
We prove the first. A curve of the second order is generated by two projective pencils. These pencils, when joined to the point in space, give rise to two projective axial pencils, which generate the cone in question as the locus of the lines where corresponding planes meet.
§48.
| Theorem.—The curve of second order which is generated by two projective flat pencils passes through the centres of the two pencils. | Theorem.—The envelope of second class which is generated by two projective rows contains the bases of these rows as enveloping lines or tangents. |
| Proof.—If S and S′ are the two pencils, then to the ray SS′ or p′ in the pencil S′ corresponds in the pencil S a ray p, which is different from p′, for the pencils are not perspective. But p and p′ meet at S, so that S is a point on the curve, and similarly S′. | Proof.—If s and s′ are the two rows, then to the point ss′ or P′ as a point in s′ corresponds in s a point P, which is not coincident with P′, for the rows are not perspective. But P and P′ are joined by s, so that s is one of the enveloping lines, and similarly s′. |
It follows that every line in one of the two pencils cuts the curve in two points, viz. once at the centre S of the pencil, and once where it cuts its corresponding ray in the other pencil. These two points, however, coincide, if the line is cut by its corresponding line at S itself. The line p in S, which corresponds to the line SS′ in S′, is therefore the only line through S which has but one point in common with the curve, or which cuts the curve in two coincident points. Such a line is called a tangent to the curve, touching the latter at the point S, which is called the “point of contact.”
In the same manner we get in the reciprocal investigation the result that through every point in one of the rows, say in s, two tangents may be drawn to the curve, the one being s, the other the line joining the point to its corresponding point in s′. There is, however, one point P in s for which these two lines coincide. Such a point in one of the tangents is called the “point of contact” of the tangent. We thus get—
| Theorem.—To the line joining the centres of the projective pencils as a line in one pencil corresponds in the other the tangent at its centre. | Theorem.—To the point of intersection of the bases of two projective rows as a point in one row corresponds in the other the point of contact of its base. |
§ 49. Two projective pencils are determined if three pairs of corresponding lines are given. Hence if a1, b1, c1 are three lines in a pencil S1, and a2, b2, c2 the corresponding lines in a projective pencil S2, the correspondence and therefore the curve of the second order generated by the points of intersection of corresponding rays is determined. Of this curve we know the two centres S1 and S2, and the three points a1a2, b1b2, c1c2, hence five points in all. This and the reciprocal considerations enable us to solve the following two problems: