Re-statement of the notions acquired relative to the composition of accelerating forces; the resulting acceleration, the normal and tangential acceleration animating a point in motion on a curve. The total acceleration of a point upon an axis or plane is the projection upon this axis or plane of the acceleration of the moving body in space. In uniform curvilinear motion the total or resultant acceleration becomes normal to the curve. Particular case of the circle; value of the normal acceleration in terms of the velocity of revolution or the angular velocity of the radius vector. Case of any curve whatever; geometrical expression of the total or resultant acceleration. _Accelerated Compound and Relative Motions._ Geometrical investigation of the simple and compound accelerations arising out of the hypothesis in which the motion of any system of points whatever is referred to another system of invariable form, but also in motion. Geometrical and elementary explanations of the results obtained by means of the transformation of co-ordinates. _Examples or Exercises chosen from among the following Questions:_-- Projection of circular and uniform motion upon a fixed straight line or plane; motion of a circle which rolls uniformly on a straight line; comparison of the motions of the planets relatively to each other, treating them as circular and uniform: comparison of the accelerating force on the moon with that of bodies which fall to the earth. GEOMETRICAL THEORY AND APPLICATION OF MECHANISMS OR CONTRIVANCES FOR THE TRANSFORMATION OF MOTION. LESSONS 10-19. Succinct notions on the classification of elementary motions and organs for transmission of motion in machines after Monge and Hachette, Lanz and Bétancourt. The most essential details upon this subject are set forth in the following order, and made clear by outline drawings previously distributed among the pupils. _Organs fitted to regulate the direction of the circular or rectilinear motion of certain pieces_. Axle; trunnions, gudgeons; pivots and bearings; couplings of axes; adjustment of wheels and of their arms. Joints with hinges, &c.; sheaves and pulleys; chains, ropes, and straps; means of securing them to the necks. Grooves and tongue-pieces. Eyelet-holes sliding along rectilinear or curvilinear rods. Advantages and disadvantages of these different systems of guides under the point of view of accuracy. Rapid indication of some of their applications to drawbridges and to the movable frames or wagons of saw-works and railways. _Transmission at a Distance of Rectilinear Motion in a determinate Direction and Ratio._