Inclined plane or wedge guiding a vertical rod. Wedge applied to presses. Rods, winch-handles, &c. Disposition of drums or pulleys in the same plane or in different planes; geometrical problem on this subject. Fixed and movable pulleys. Blocks to pulleys. Simple and differential wheel and axle moved by cords. Transmission through a liquid. Ratios of velocities in these different organs. _Direct Transformation of circular progressive motion into progressive and intermittent rectilinear motion._ Rod conducted between guides: 1º, by the simple contact of a wheel; 2º, by cross-straps or chains; 3º, by a projecting cam; 4º, by means of a helicoidal groove set upon the cylindrical axis of the wheel. To-and-fro movement, and heart-shaped or continuous cam, waves, and eccentrics. Simple screw and nut. Left and right handed screws; differential screw of Prony, called the micrometric screw. Ratio of the velocities in these different organs. The example of the cam and pile-driver will be particularly insisted upon; 1º, in the case where this cam and the extremity of the rod have any continuous form given by a simple geometrical drawing; 2º, in the case where this form is defined geometrically by the condition, that the velocity is to be transmitted in an invariable ratio, as takes place for cams in the form of epicycloids or involutes of circles. _Transformation of a circular progressive motion into another similar to the first._ 1º, by contact of cylinders or cones, the two axes being situated in the same plane; 2º, by straps, cords, or endless chains, the axes being in the same situation; 3º, by cams, teeth, and grooves, at very slight intervals; 4º, by the Dutch or universal joint. Case, where the axes are not situated in the same plane; use of an intermediate axis with beveled wheels or a train of pulleys; idea of White or Hooke’s joint in its improved form. Endless screw specially employed in the case of two axes at right angles to one another. Combinations or groupings of wheels. Idea of differential wheels. Relations of velocities in the most important of these systems of transmission. _Transformation of circular progressive Motion into rectilinear or alternating circular motion._ Ordinary circular eccentric. Eccentrics with closed waves or cams. Examples and graphical exercises in the class-rooms relative to the alternate action of the traveling frames of saw-mills, of the slides or entrance valves of steam-engines. Cams for working hammers and bellows. _Transformation of alternating circular motion into alternating rectilinear motion, or into intermittent and progressive circular motion._ Pump rods with or without circular sectors, &c. Examples taken from large exhausting pumps, fire-engines, and common pumps. Suggestions as to the best arrangement of the parts. Lagarousse’s lever, &c. Application of the principle relative to the instantaneous center of rotation to give the relations of the velocities in certain simple cases.