Third solution: helicoidal. Determination of the angular elevation at which the surfaces of the beds become normal to the head planes; construction in the orthogonal and helicoidal _appareil_ of the curves of junction upon the heads, and the angles which they form with the curves of intrados. Cutting of the stones in these different constructions. Broken helicoidal _appareil_, for very long skew arches. Helicoidal _trompes_ at the angles of straight arches; _voussures_ or widenings, which it is necessary to substitute near the heads at the intrados of an arch with a considerable skew; case where the skew is not the same for the two heads. Orthogonal trajectories of the converging sections of a cylinder. LESSONS 23-25. _Conical Intrados--Intrados of Revolution._ Skew _trompe_ in the angle. Suggestions on the general problem of conical skew vaulted roofs. Spherical domes, &c. LESSONS 26-27. _Intrados, a Ruled Surface._ Winding staircases, &c., &c. LESSON 28. _Helicodial Intrados._ Staircase on the Saint-Giles screw. LESSONS 29-31. _Composite Vaulted Roofs._ Various descriptions of vaults. Suggestions on vaulted roofs with polygonal edges and with ogival edges. LESSON 32. _Revision._ Spirit and method of stereotomy. Degree of exactness necessary. Approximate solutions. Case where it is proper to employ calculation in aid of graphical constructions. Review and comparison of different _appareils_. MECHANICS AND MACHINES. GENERAL ARRANGEMENTS. The pupils execute during the two years of study:-- 1. Various drawings or plans of models in relief representing the essential and internal organs of machines, such as articulations of connecting rods, winch-handles and fly-wheels, grease-boxes, eccentrics worked by cams or circles giving motion to rods; the play of slides, &c.; cylinders of steam-engines, condenser, pistons, and various suckers; Archimedes’ screw, and other parts of machines. The sketches of the plan drawings are traced by hand and figured. The drawings in their finished state are washed and colored according to the table of conventional tints; they all carry a scale suitably divided. 2. A drawing of wheel-work by the method of development, and tracing the curves of teeth by arcs of circles from which they are developed. This drawing represents, of the natural size, or on any other scale of size considered suitable to show the nature of the partial actions only, a small number of teeth either in development or projection; the entire wheel-work is represented by the usual method of projection, where in drawings on a small scale the teeth are replaced by truncated pyramids with a trapezoidal base. 3. Finally, numerical exercises concerning the loss of work due to the proejudicial resistances in various machines, the gauging of holes, orifices, &c.