Every plane section of a quadric surface is a conic or a line-pair.
Instead of a circle or sphere we may take any conic or quadric.
Now from a quadric equation we derive, in like manner, the notion of a complex or imaginary number such as is spoken of above.
Evidently the method gives for L a quadric equation, which is the “resolvent” equation in this particular case.
Each ray cuts its corresponding plane in a point, the locus of these points is a quadric surface.
The surface itself is therefore called a quadric surface, or a surface of the second order.
If through one point on a quadric surface no line on the surface can be drawn, then the surface contains no lines.
On a quadric surface the points are all hyperbolic, or all parabolic, or all elliptic.
This plane is called the polar plane of the point P, with regard to the quadric surface.
Every line which has three points in common with a quadric surface lies on the surface.