Directer-cone: its advantages for constructing the tangent plane parallel to a given plane, and for determining the nature of the plane sections. 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. Re-statement without demonstration of the properties of this surface, found by analysis, principally as to what regards the axis, the vertices, the principal planes, and the generation by conic sections. Hyperboloid of _raccordement_ to a ruled surface along a generatrix; all their centers are in the same plane. Transformation of a hyperboloid of _raccordement_. Surface of the _biais passé_. Construction of a hyperboloid of _raccordement_; its transformation into a paraboloid. Construction of the tangent plane at a given point. LESSONS 26-28. _Curvature of Surfaces. Lines of Curvature._ Re-statement without proof of the formula of Euler given in the course of analysis. There exists an infinity of surfaces of the second degree, which at one of their vertices osculate any surface whatever at a given point. In the tangent plane, at a point of a surface, there exists a conic section, whose diameters are proportional to the square roots of the radii of curvature of the normal sections to which they are tangents. This curve is called the indicatrix. It is defined in form and position, but not in magnitude. The normal sections tangential to the axes of the indicatrix are called the principal sections. The indicatrix an ellipse; convex surfaces; umbilici; line of spherical curvatures. The indicatrix a hyperbola; surfaces with opposite curvatures. The assymplotes of the indicatrix have a contact of the second order with the surface, and of the first order with the section of the surface by its tangent plane. A ruled surface has contrary curvatures at every point. The second assymplotes of the indicatrices of all the points of the same generatrix form a hyperboloid, if the surface has not directer-plane,--a paraboloid, if it have one. Curvature of developable surfaces. There exists upon every surface two systems of orthogonal lines, such that every straight line subject to move by gliding over either of them, and remaining normal to the surface, will engender a developable surface. These lines are called lines of curvature. The two lines of curvature which cross at a point, are tangents to the principal sections of the surface at that point.