Application of the same principles to the motion of a heavy body upon a spherical surface. Determination of the time of the oscillations of the moving body. Very small oscillations are isochronal, [67'"— 86] §11
Investigation of the curve which is rigorously isochronal, in a resisting medium ; and particularly if the resistance be proportional to the two first powers of the velocity, [86" — 106]. . . § 12
CHAPTER in. ON THE EaUILIBRIUM OP A SYSTEM OF BODIES 71
Conditions of the equilibrium of two systems of points, which impinge against each other, with directly opposite velocities. Definition of the terms, quantity of motion of a body and similar material points, [106' — 106'"] § 13
On the reciprocal action of material points. Reaction is always equal and contrary to action. Equation of the equilibrium of a system of bodies, from which we may deduce the principle of virtual velocities, [114']. Method of finding the pressures, exerted by bodies, upon the surfaces and curves upon which they are forced to move, [117] §14
Application of these principles, to the case where all the points of the system are rigidly united together, [119] ; conditions of the equilibrium of such a system. On the centre of gravity : method of finding its position ; first, with respect to three fixed rectangular planes, [127] ; second, with respect to three points given in position, [129] § 15
Conditions of equilibrium of a solid body of any figure, [130] §16
CHAPTER IV. ON THE EQUILIBRIUM OF FLUIDS 90
General equations of this equilibrium, [133]. Application to the equilibrium of a homogeneous fluid mass, whose external surface is free, and which covers a fixed solid nucleus, of any figure, [138] §17
CHAPTER V. GENERAL PRINCIPLES OF THE MOTION OF A SYSTEM OF BODIES QQ
General equation of this motion, [142] § 18
Development of the principles comprised in this equation. On the principle of the living force, [144]. It takes place only when the motions of the bodies change by insensible degrees, [145]. Method of estimating the alteration which takes place in the living force, by any sudden change in the motions of the system, [149] § 19
On the principle of the preservation of the motion of the centre of gravity, [155']. It takes place even in those cases, in which the bodies of the system exert on each other, a finite action, in an instant, [159"]. §20
On the principle of the preservation of areas, [167]. It takes place also, like the preceding principle, in the case of a sudden change in the motion of the system, [167'"]. Determination of the system of co-ordinates, in which the sum of the areas described by the projections of the radii vedores, upon two of the rectangular planes, formed by the axes of the co-ordinates, is nothing. This sum is a maximum upon the third rectangular plane ; it is nothing upon every other plane, perpendicular to this third plane, [181"] § 21
The principles of the preservation of the living forces and of the areas take place also, when the origin of the co-ordinates is supposed to have a rectilineal and uniform motion, [182]. In this case, the plane passing constantly through this origin, and upon which the sum of the areas
