Re-statement without demonstration of the properties of this surface found by analysis, principally as regards its generation by the conic sections. LESSONS 19-20. _General Properties of Warped or Ruled Surfaces._ Principal modes of generation of warped surfaces. When two warped surfaces touch in three points of a common generatrix, they touch each other in every point of this straight line. Every plane passing through a generatrix touches the surface at one point in this line. The tangent plane at infinity is the plane-directer to all the paraboloids of “raccordement.” Construction of the tangent planes and curves of contact of circumscribed cones and cylinders. When two infinitely near generatrices of a warped surface are in the same plane, all the curves of contact of the circumscribed cones and cylinders pass through their point of concourse. The normals to a warped surface along a generatrix form an isosceles paraboloid. The name of central point of a generatrix is given to the point where it is met by the straight line upon which is measured its shortest distance from the adjoining generatrix. The locus of these points forms the line of striction of the surface. The vertex of the normal paraboloid along a generating line is situated at the central point. If the point of contact of a plane touching a warped surface moves along a generatrix, beginning from the central point, the tangent of the angle which the tangent plane makes with its primitive position is proportional to the length described by the point of contact. The tangent plane at the central point is perpendicular to the tangent plane at infinity upon the same generatrix. Construction of the line of striction by aid of this property. LESSONS 21-22. _Ruled Surfaces with plane-divecters Conoids._ 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 sections. 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 intersection 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 intersection. LESSONS 23-25. _Ruled Surfaces which have not a Directer-Plane. Hyperboloid. Surface of the “biais passe.”_