That which is called the preservation of living forces is a consequence of the discovery of Huyghens concerning the movement of the centre of gravity in a compound body. For the space described by that centre is expressed by the quotient arising from the sum of the pro ducts of the mass of each body into the distance it passes over, divided by the sum of the masses; and since the spaces descended by bodies when acted on by gravity are proportional to the squares of the velocities, it follows that the sum of the products of the mass of each body into the square of its velocity is constant, whether the move jointly in any manner, or whether they descend freely through equal vertical spaces.
The preservation of the centre of gravity is a principle which contains the discovery of Newton, that the motion of the common centre of gravity of several bodies is not affected by the mutual attractions of the bodies. It was subsequently extended by D'Alembert, who shows that if the bodies are solicited by a constant accelerative force in direc tions either parallel to each other or tending to a fixed point, the centre of gravity must describe the same line as if the bodies were f roe.
The preservation of areas seems to have been discovered simulta neously by Euler, Daniel Bernoulli, and the Chevalier D'Arci, about 1750. According to the latter it is an extension of Newton's theorem
that the radii reetores of revolving bodies describe equal areas in equal times, and it consists in this : that the sum of the products of the muses of revolving bodies inW the areas described by their radii rat ores about a fixed point is proportional to the time. Or the sum of the products of the masses into the velocities and into the perpendicu lars let fall from the fixed point on the lines of direction of the motions is constant.
The principle of least action originally signified, that when bodies act on each other, the sum of the products of the masses into the velocities and spaces described is a minimum. But considered in the most general sense, agreeably to the extension given to it by La Grange, the principle consists in this: that in trajectories described by bodies subject to central forces, the integral of the velocity multiplied by the clement of the orbit is always a inaximum or a minimum.
A general outline of that part of mechanics which relates to the equilibrium of solid bodies is given under the word STATICS ; and the details of the subjects may be seen under LEVER, &c. The part of mechanics which relates to bodies in motion appears under the words referred to in the article Ds:issues.