Remarks upon the lines of curvature of developable surfaces, and surfaces of revolution. Determination of the radii of curvature, and assymplotes of the indicatrix at a point of a surface of revolution. LESSONS 29-30. _Division of Curves of Apparent Contour, and of Separation of Light and Shadow into Real and Virtual Parts._ When a cone is circumscribed about a surface, at any point whatever of the curve of contact, the tangent to this curve and the generatrix of the cone are parallel to two conjugate diameters of the indicatrix. Surfaces, as they are considered in shadows, envelop opaque bodies, and the curve of contact of a circumscribed cone, only forms a separation of light and shadow, for a luminous point at the summit of the cone, when the generatrices of this cone are exterior. This line is thus sometimes real and sometimes virtual. Upon a convex surface, the curve of separation of light and shade is either all real or all virtual. Upon a surface with contrary curvatures, this curve presents generally a succession of real and virtual parts: the curve of shadow cast from the surface upon itself presents a like succession. These curves meet tangentially, and the transition from the real to the virtual parts upon one and the other, take place at their points of contact in such a way that the real part of the curve of shadow continues the real part of the curve of separation of light and shade. The circumscribed cones have edges of regression along the generatrices, which correspond to the points of transition. The lines of visible contour present analogous circumstances. General method of determining the position of the transition points. Special method for a surface of revolution. LESSONS 31-34. _Ruled Helicoidal Surfaces._ Surface of the thread of the triangular screw; generation, representation, sections by planes and conical cylinders. Construction of the tangent plane at a given point, or parallel to a given plane. The axis is the line of striction. Construction of lines of shadow and perspective: their infinite branches, their assymplotes. Determination of the osculating hyperboloid along a generatrix. Representation and shading of the screw with a triangular thread and its nut. Surface of the thread of the square screw; generation, sections by planes and conical cylinders; tangent planes; curve of contact of a circumscribed cone. The curve of contact of a circumscribed cylinder is a helix whose _step_ is half that of the surface. Determination of the osculating paraboloid. At any point whatever of the surface, the absolute lengths of the radii of curvature are equal. Representation and shading of the screw with a square thread, and of its nut. Observations on the general ruled helicoidal surface, and on the surface of intrados of the winding staircase. LESSON 35. _Different Helicoidal Surfaces._