Models in relief or drawings on a large scale, of the machines or elements of the machines mentioned in the course, assist in explaining the lessons. They are brought back, as often as found necessary, under the eyes of the students. When possible, lithographic sketches of the machines, or the elements of the machines, which ought to enter into the course, are distributed among the pupils. The pupils, divided into sections, pay their first visit to the engine factories towards the end of their first year of study; they make one or more additional visits at the end of the second year. _FIRST YEAR._ PART I. KINEMATICS.--PRELIMINARY ELEMENTARY MOVEMENTS OF INVARIABLE POINTS AND SYSTEMS. LESSONS 1-2. Object of kinematics, under the geometrical and experimental point of view. Its principal divisions. Re-statement of the notions relative to the motion of a point, its geometrical representation, and more especially the determination of its velocity. _Simultaneous Velocities of a Point and the Increments of its Velocities._ Ratio of the elementary displacement and the velocity of a point to the displacement, and velocity of its projection upon a straight line or plane. Use of infinitesimals to determine these ratios. Example:--Oscillatory motion of the projection upon a fixed axis of a point moving uniformly upon the circumference of a circle. Analogous considerations for polar co-ordinates. Relations of the velocity of a point, of its velocity of revolution and its angular velocity about a fixed pole; of its velocity in the direction of the radius vector; of the velocity of increase of the area which this radius describes. _Simple Motions of Solids, or Rigid Systems._ 1. Motion of rectilinear or curvilinear translation; simultaneous displacements, and velocities of its different points. 2. Motion of rotation about a fixed axis; relation of the velocities of different points to the angular velocity. Geometrical notions and theorems relative to the _instantaneous center_ of rotation of a body of invariable figure and movable in one plane, or to the _instantaneous axis_ of rotation of a rigid system situated in space, and movable parallel to a fixed plane. Relation of the velocities of different points to their common angular velocity. Use of the instantaneous center of rotation for tracing tangents; examples--and amongst others--that of the plane curve described by a point in a straight line of given length, whose extremities slide upon two fixed lines. Rolling of a curve upon another fixed curve in a plane. Descartes’ theorems upon the intersection of the normals at the successive points of contact: cycloids, epicycloids, involutes, and evolutes. Extension of the preceding motions to the instantaneous axis of rotation of a rigid system movable about a fixed point. COMPOSITION OF MOTIONS. LESSONS 3-6. _Composition of the Velocities of a Point._