Through two conics which lie in different planes, but have two points in common, and through one external point always one quadric surface may be drawn.
§ 97. Every plane which cuts a quadric surface in a line-pair is a tangent plane. For every line in this plane through the centre of the line-pair (the point of intersection of the two lines) cuts the surface in two coincident points and is therefore a tangent to the surface, the centre of the line-pair being the point of contact.
If a quadric surface contains a line, then every plane through this line cuts the surface in a line-pair (or in two coincident lines). For this plane cannot cut the surface in a conic. Hence:—
If a quadric surface contains one line p then it contains an infinite number of lines, and through every point Q on the surface, one line q can be drawn which cuts p. For the plane through the point Q and the line p cuts the surface in a line-pair which must pass through Q and of which p is one line.
No two such lines q on the surface can meet. For as both meet p their plane would contain p and therefore cut the surface in a triangle.
Every line which cuts three lines q will be on the surface; for it has three points in common with it.
Hence the quadric surfaces which contain lines are the same as the ruled quadric surfaces considered in §§ 89-93, but with one important exception. In the last investigation we have left out of consideration the possibility of a plane having only one line (two coincident lines) in common with a quadric surface.
§ 98. To investigate this case we suppose first that there is one point A on the surface through which two different lines a, b can be drawn, which lie altogether on the surface.
If P is any other point on the surface which lies neither on a nor b, then the plane through P and a will cut the surface in a second line a′ which passes through P and which cuts a. Similarly there is a line b′ through P which cuts b. These two lines a′ and b′ may coincide, but then they must coincide with PA.
If this happens for one point P, it happens for every other point Q. For if two different lines could be drawn through Q, then by the same reasoning the line PQ would be altogether on the surface, hence two lines would be drawn through P against the assumption. From this follows:—