21, 22 for the paraboloid, 25, 26 for the hyperboloid, and 27-32 for the spheroid).
This theorem has been generalized for any tetrahedron; a sphere can be drawn through the four feet of the perpendiculars, and consequently through the mid-points of the lines from the vertices to the centre of the hyperboloid having these perpendiculars as generators, and through the orthogonal projections of these points on the opposite faces.
He then supposed this cylindrical column of water to be divided into two parts, - the first, which he called the " cataract," being an hyperboloid generated by the revolution of an hyperbola of the fifth degree around the axis of the cylinder which should pass through the orifice, and the second the remainder of the water in the cylindrical vessel.
He considered the horizontal strata of this hyperboloid as always in motion, while the remainder of the water was in a state of rest, and imagined that there was a kind of cataract in the middle of the fluid.
The ntill-lines which are at a given distance r from a point 0 of the central axis will therefore form one system of generators of a hyperboloid of revolution; and by varying r we get a series of such hyperboloids with a common centre and axis.
Sliding Cont?,zct (lateral): Skew-Bevel Wheels.An hyperboloid of revolution is a surface resembling a sheaf or a dice box, generated by the rotation of a straight iAn line round an axis from which it is at a constant distance, - c and to which it is inclined at -~- a constant angle.