Elimination of the arbitrary functions which enter into an equation by means of partial derivatives. Integration of an equation of partial differences with two independent variables, in the case where it is linear in respect to the derivatives of the unknown function. The general integral contains an arbitrary function. Indication of the geometrical problem, of which the partial differential equation expresses analytically the enunciation. Integration of the partial differential equations to cylindrical, conical, conoidal surfaces of revolution. Determination of the arbitrary functions. Integration of the equation _d^{2}u/dy^{2} = a^{2}d^{2}u/dx^{2}_. The general integral contains two arbitrary functions. Determination of these functions. LESSONS 17-23. _Applications to Mechanics._ Equation to the catenary. Vertical motion of a heavy particle, taking into account the variation of gravity according to the distance from the center of the earth. Vertical motion of a heavy point in a resisting medium, the resistance being supposed proportional to the square of the velocity. Motion of a heavy point compelled to remain in a circle or cycloid. Simple pendulum. Indication of the analytical problem to which we are led in investigating the motion of a free point. Motion of projectiles in a vacuum. Calculation of the longitudinal and transversal vibrations of cords. Longitudinal vibrations of elastic rods. Vibration of gases in cylindical tubes. LESSONS 24-26. _Applications to Astronomy._ Calculation of the force which attracts the planets, deduced from Kepler’s laws. Numerical data of the question. Calculation of the relative motion of two points attracting one another, according to the inverse square of the distance. Determination of the masses of the earth and of the planets accompanied by satellites. Numerical applications. LESSONS 27-30. Elements of the calculus of probabilities and social arithmetic. General principles of the calculus of chances. Simple probability, compound probability, partial probability, total probability. Repeated trials. Enunciation of Bernouilli’s theorem (without proof.) Mathematical expectation. Applications to various cases, and especially to lotteries. Tables of population and mortality. Mean life annuities, life interests, assurances, &c. LESSONS 31-32. _Revision._ General reflections on the subjects comprised in the course. II. DESCRIPTIVE GEOMETRY AND STEREOTOMY. _General Arrangements._ The pupils take in the lecture-room notes and sketches upon sheets, which are presented to the professor and the “répétiteurs” at each interrogation. The care with which these notes are taken is determined by “marks,” of which account is taken in arranging the pupils in order of merit.