§ 58. There are theorems about cones of second order and second class in a pencil which are reciprocal to the above, according to § 43. We mention only a few of the more important ones.
The locus of intersections of corresponding planes in two projective axial pencils whose axes meet is a cone of the second order.
The envelope of planes which join corresponding lines in two projective flat pencils, not in the same plane, is a cone of the second class.
Cones of second order and cones of second class are identical.
Every plane cuts a cone of the second order in a conic.
A cone of second order is uniquely determined by five of its edges or by five of its tangent planes, or by four edges and the tangent plane at one of them, &c. &c.
Pascal’s Theorem.—If a solid angle of six faces be inscribed in a cone of the second order, then the intersections of opposite faces are three lines in a plane.
Brianchon’s Theorem.—If a solid angle of six edges be circumscribed about a cone of the second order, then the planes through opposite edges meet in a line.
Each of the other theorems about conics may be stated for cones of the second order.
| Fig. 22. |