From this same principle, his son, D. Bernouilli, deduced the law of the motion of fluids in vases, Which he explains in the Berlin Memoirs for 17A3: a subject before but little understood.
The advantage of this principle consists in its affording immediately an equation between the velocities of the bodies and the %ariable quantities which determine their position in space so that when by the nature of the problem these variable quantities are reduced to one, the equation is of itself sufficient for its solution, as in the instance of the problem relating to the centre of oscillation. In general, the conservation of the vis viva gives a first integral of the several differential equations of each problem, which is often of great utility.
The second principle above alluded to, conservation of the motiun of the centre of yramity, is given by Sir Isaac Newton in his Principia, as au elementary proposition ; where he demonstrates, that the state of repose or of motion of the centre of gravity of several bodies, is not altered by the recip rocal action of these bodies, in any manner whatever : so that the centre of gravity of bodies acting upon each other, either by means of cords or of levers, or by the laws of attraction, remains always in repose, or move uniformly in a direct line, unless disturbed by some exterior action or obstacle. This theorem has been extended by D'Alembert, who has demon strated, that if every body in the system be solicited by a eonstant accelerating force, either acting in parallel lines, or directed towards a fixed point, but varying with the distance, the centre of gravity will describe a similar curve to w hat it would have done, had the bodies heel, free. And, it might be added, the motion of this centre will be the saute as if all the .forces of the bodies were applied to it, each in its proper direction. This principle serves to determine the motion of the centre of gia%ity, independently of the respective motions of the bodies ; and thus it will ever alliaal three finite equations between the co-ordinates of the bodies and the times ; and these equations will be the integrals of the differential equations of the problem.
The third principle, the conservation of equal areas, is more modern than the two former, and appears to have been separately discovered by Euler, D. Bernouilli, and D'Arey, about the same period, though under different forms.
Euler and Bernouilli describe the principle thus : In the motion of several bodies round a fixed centre, the sum of the products of the mass of each body by the velocity of rotation round the centre, and by its distance the saute centre, is always independent of any mutual action exerted by the bodies upon each other, and preserves itself the same as long as there is no exterior action or obstacle. Such is the prin ciple described by 1). Bernouilli in the first volume of the Memoirs of the Berlin Academy, 1'7-16 ; and by D'Alembert, in the same year, in his Opuscula. The Chevalier D'Are?.
also in the same year, sent his Memoir to the Academy of Paris, though it was not printed till 1752, wherein he says, "The sum of the products of the mass of each body by the area traced by its radius vector about a fixed point, is always proportional to the times." This principle, however, is only a generalization of Sir Isaac's theorem of equality of areas described by centripetal forces : and to perceive its analogy, or rather its identity with that of Euler and Bernouilli, it is only requisite to recollect, that the velocity of rotation is expressed by the element of the circular arc divided by that of the time ; and that the first of these elements multiplied by the distance from the centre, gives the element of the area described abont it. It appears then that this latter principle is only the difThrential expression of that of the Chevalier, %vho afterwards gave the same principle in another form, which renders it more similar to the preceding, viz. The sum of the products of the masses by the velocities, and by the perpendiculars drawn from the centre to the direction of the forces, is always a constant quantity. Under this point of view, I. D'Arcy set up a kind of metaphysical principle, which be denominates the conservation of action, in opposition to, or rather as a substitute for, the principle of the least action.