§ 88. There follow several important theorems:
Through four points two, one, or no conics may be drawn which touch any given line, according as the involution determined by the given four-point on the line has real, coincident or imaginary foci.
Two, one, or no conics may be drawn which touch four given lines and pass through a given point, according as the involution determined by the given four-side at the point has real, coincident or imaginary focal rays.
For the conic through four points which touches a given line has its point of contact at a focus of the involution determined by the four-point on the line.
As a special case we get, by taking the line at infinity:
Through four points of which none is at infinity either two or no parabolas may be drawn.
The problem of drawing a conic through four points and touching a given line is solved by determining the points of contact on the line, that is, by determining the foci of the involution in which the line cuts the sides of the four-point. The corresponding remark holds for the problem of drawing the conics which touch four lines and pass through a given point.
Ruled Quadric Surfaces
§ 89. We have considered hitherto projective rows which lie in the same plane, in which case lines joining corresponding points envelop a conic. We shall now consider projective rows whose bases do not meet. In this case, corresponding points will be joined by lines which do not lie in a plane, but on some surface, which like every surface generated by lines is called a ruled surface. This surface clearly contains the bases of the two rows.
If the points in either row be joined to the base of the other, we obtain two axial pencils which are also projective, those planes being corresponding which pass through corresponding points in the given rows. If A′, A be two corresponding points, α, α′ the planes in the axial pencils passing through them, then AA′ will be the line of intersection of the corresponding planes α, α′ and also the line joining corresponding points in the rows.