Right-lined sections of the hyperbolic paraboloid.--Through each point of the surface of the hyperbolic paraboloid two right lines may be traced, whence results the generation of the paraboloid by two systems of right lines.--Two right lines of the same system do not meet, but two right lines of different systems always meet.--All the right lines of the same system are parallel to the same plane.--The hyperbolic paraboloid may be generated by the movement of a right line which slides on three fixed right lines which are parallel to the same plane; or by a right line which slides on two fixed right lines, itself remaining always parallel to a given plane. Reciprocally, every surface resulting from one of these two modes of generation is a hyperbolic paraboloid. General equations of conical surfaces and of cylindrical surfaces. VI. DESCRIPTIVE GEOMETRY. The general methods of Descriptive Geometry,--their uses in Stone-cutting and Carpentry, in Linear Perspective, and in the determination of the Shadows of bodies,--constitute one of the most fruitful branches of the applications of mathematics. The course has always been given at the Polytechnic School with particular care, according to the plans traced by the illustrious _Monge_, but no part of the subject has heretofore been required for admission. The time given to it in the school, being however complained of on all sides as insufficient for its great extent and important applications, the general methods of Descriptive Geometry will henceforth be retrenched from the internal course, and be required of all candidates for admission. As to the programme itself, it is needless to say any thing, for it was established by _Monge_, and the extent which he gave to it, as well as the methods which he had created, have thus far been maintained. We merely suppress the construction of the shortest distance between two right-lines, which presents a disagreeable and useless complication. Candidates will have to present to the examiner a collection of their graphical constructions (_épures_) of all the questions of the programme, signed by their teacher. They are farther required to make free-hand sketches of five of their _épures_. PROGRAMME OF DESCRIPTIVE GEOMETRY. _Problems relating to the point, to the straight line, and to the plane._[7] [Footnote 7: The method of the change of the planes of projection will be used for the resolution of these problems.] Through a point given in space, to pass a right line parallel to a given right line, and to find the length of a part of that right line. Through a given point, to pass a plane parallel to a given plane. To construct the plane which passes through three points given in space. Two planes being given, to find the projections of their intersection. A right line and a plane being given, to find the projections of the point in which the right line meets the plane.