"The equation of the tangent plane at the point (a, 0, 0) of the conicoid \frac{x^2}{a^2} \pm \frac{y^2}{b^2} \pm \frac{z^2}{c^2} = 1 is x = a; this meets the surface in straight lines whose projection on the plane x = 0 are given by the equation \pm \frac{y^2}{b^2} \pm \frac{z^2}{c^2} = 0. These lines are clearly real when the surface is an hyperboloid of one sheet, and imaginary when the surface is an , or an hyperboloid of two sheets. Hence the hyperboloid of one sheet is a . The hyperbolic paraboloid is a particular case of the hyperboloid of one sheet; hence the hyperbolic paraboloid is also a ruled surface. This can be proved at once from the equation of the paraboloid. For, the tangent plane at the origin is z = 0, and this meets the paraboloid ax^2 + by^2 + 2z = 0 in the straight lines given by the equations ax^2 = by^2 = 0, z = 0; the lines are clearly real when a and b have different signs, and are imaginary when a and b have the same sign. Hence an hyperboloid of one sheet (including an hyperbolic paraboloid as a particular case) is the only ruled conicoid in addition to a cone, a cylinder, and a pair of planes."