General theorem of work and _vis viva_. Part which appertains to the interior forces in this theorem. Particular case called principle of the conservation of _vires vivæ_, where the sum of the elements of work done by the exterior and interior forces is the differential of a function of the co-ordinates of different points of the system. Application of the theorem of work to the stability of the equilibrium of heavy systems. Extension of the preceding theorems to the case of relative motions. Particular case of relative equilibrium. Motion of any material system relative to axes always passing through the center of gravity, and moving parallel to themselves. Invariable plane of Laplace. Relation between the absolute _vis viva_ of a material system, and that which would be due to its motion, referred to the system of movable axes above indicated. _Examples and Applications._ The following examples, amongst others, to be taken as applications or subjects of exercises relative to the general principles which precede. Walking. Recoil of guns. Eolypile. Flight of rockets. Pressure of fluid veins, resistance of mediums, &c. Direct collision of bodies more or less hard, elastic, or penetrable. Exchange of quantities of motion. Loss of _vis viva_ under different hypotheses. Influence of vibrations and permanent molecular displacements. Pile driving; advantage of large rammers. Comparison of effects of the shocks and of simple pressures due to the weight of the construction. Oblique collision, and ricochet. Data furnished by experiment. Oscillations of a vertical elastic prism suspended to a fixed point, and loaded with a weight, neglecting the inertia, and the weight of the material parts of this prism. Case of a sudden blow. What is meant by the “_resistance vive_” of a prism to rupture? Results of experiments. Work developed by powder upon projectiles, estimated according to the _vis viva_ which it impresses on them, as well as upon the gun and the gases upon hypothesis of a mean velocity. SPECIAL DYNAMICS OF SOLID BODIES. LESSONS 9-12. _Simple Rotation of an invariable Solid about its Axis._ In applying to this case the first general rule of dynamics, the theorem of the moments of the quantities of motion, and the theorem of work, we are led to the notion of the moment of inertia; explanation of the origin of this name. The angular acceleration is equal to the sum of the moments of the exterior forces divided by the moment of inertia about the axis of rotation. Sum of the moments of the quantities of motion relative to this axis. _Vis viva_ of a solid simply turning about an axis. What is meant by _radius of gyration_? _Remind_ of the geometrical properties of moments of inertia, of the ellipsoid which represents them, of the principal axes at any point, of those which are referred to the center of gravity.