The plane-directer of the surface is also so to all the paraboloids of “raccordement.” Construction of the tangent planes and curves of contact of the circumscribed cones and cylinders.
The line of striction of the surface is its curve of contact with a circumscribed cylinder perpendicular to the directer-plane. Determination of the nature of the plane s.
The lines of striction of the scalene paraboloid are parabolas; those of the isosceles paraboloid are straight lines.
Construction of the tangent plane parallel to a given plane.
Conoid: discussion of the curves of contact of the circumscribed cones and cylinders.
Right conoid. Conoid whose inter with a torus of the same height, whose axis is its rectilinear directrix, has for its projection upon the directer-plane two arcs of Archimedes’ spiral. Construction of the tangents to this curve of inter.
Lessons 23–25. Ruled Surfaces which have not a Directer-Plane. Hyperboloid. Surface of the “biais passe.”
Directer-cone: its advantages for constructing the tangent plane parallel to a given plane, and for determining the nature of the plane s. The tangent planes to the points of the surface, situated at infinity, are respectively parallel to the tangent plane of the directer-cone. Developable surface which is the envelope of these tangent planes at infinity. Construction of a paraboloid of raccordement to a ruled surface defined by two directrices and a directrix cone.
Hyperboloid; double mode of generation by straight lines; center; assymptotic cone.
Scalene hyperboloid; hyperboloid of revolution. Identity of the hyperboloid with one of the five surfaces of the second degree studied in analytical geometry.