It' the tortes actin)* on a point, or on a system of points, he not so proportioned as to maintain the system ill a motion must necessarily Like place, the laws of Which may he deduced from an extension of the principles laid down fin: investigating the state of equilibrium ; a method pursued by La Grange, and after him by La Place. The former combines the principle of virtual velocities with that of D'Alembert, Which is very simple, and, though long unob served, may be considered as an axiom. It is as full t Ws If several bodies have a tendency to motion. in directions, and with velocities, which they are constrained to change in consequence of their reciprocal reaction ; the motion so induced may be considered as composed of two others, one of which the bodies act naily assume, and the other such. that had the bodies been only acted upon by it, they would have remained in equilibrium. This theorem is not of itself suffieient to solve a problem, because it is always necessary to derive some condition relative to the equilibrium from other considerations; and the difficulty of determining the forces and the laws of their equilibrium, sometimes renders this application more difficult, and the process more tedious, than ifthe solution were performed upon some principle more complex and more indirect. To obviate this objection, there fore, La Grange attempted to combine the principle of D'Alembert with that of virtual velocity ; in which he was so successful, that he was enabled to deduce the general equations relating to the forees acting on a system of bodies. His description of the method is as follows: To tbrin an accurate conception of the mode in which these principles are applied, it is necessary to recur to the general 1»•inciple or virtual velocity, viz. When a system of material points, solicited by any force, is in equilibrium, if the system receive ever so small an alteration in its position, every point Will naturally and consequently describe a small space ; each of which spaces being multiplied by the slim of each force, according to the direction of such 11)ree, must equal 0.
Now, supposing the system to be in motion, the motion that each point makes in an instant may be considered as composed of two, one of them being that which the point acquires in the 1; illowing instant ; consequently. the other must be destroyed by the reciprocal :teflon of the points or bodies upon each (Alter, as well as of the moving Ibrees by which they are solicited. There therefore be all equilibritnn between those threes and the pressures or resist ances resulting front the motions lost by the bodies front one instant to another. Tlurefore, to extend to the motion of a system of bodies, the flirmuke of its equilibrium, it is only necessary to add the terms due to the last-mentioned forces.
The decrement of the velocities, which every particle has in the direction of three fixed rectangular co-ordinates, represents the motions lost in those directions; and their increment represents such as are lost in the opposite direc tions. Therefore, the resulting pressures or f frees of these
motions destroy eh Ni ill be generally expressed by the mass multiplied into the element if' the velocity, divided by the element of the time ; and their directions will be directly opposite to those 1>f the velocities.
By means terms required may he analytically expressed, and a general Iiirmula obtained for the motion of a sy stein of bodies, which will comprehend the solution of all the problems in dy namics; and a simple extension of it will give the necessary equations flu. cavil problem.
A great advantage derived from this tbrmula is, that it gives directly a number of general equations, wherein are included the principles or theorems, known under the appella ()I' conservation of the vis viva ; conservation of flee motion of the centre of ft rarity ; conservation of equal areas ; and the principle (y" the least action.
Of these, the first. the conservation of the vis viva, was discovered by Huygens, though under a form somewhat different from that which we now give to it. As employed by him, it consisted in the equality between the ascent and descent of the centre of gravity of several weighty bodies, Which descend together, and then ascend separately by the force they had respectively acquired. But by t he known properties of the centre of gravity, the space it describes in any direction is expressed by the sum of the products of the mass of each body by the space such body has described in the same direction, divided by the sum of the masses. Galileo, on the other hand, has shown in his problems, that the vertical space described by a weighty body in its descent is proportilmal to the square of the velocity acquired, and by which it will reaseend to its former elevation. The principle of Huygens is therefore reduced to this; that in the motion of a system of bodies, the sum of the masses by the squares of the velocities is constantly the same, whether the bodies descend conjointly, or whether they freely descend separately through the same vertical channel.
This principle had been considered only as a simple theorem of mechanics, till J. Bernouilli adopted the distinc tion, establish( d between such pressures as set without producing actual motion, and the living forces, as they were termed, which produced motion ; as liken ise the measures of these fin.ces by the products of the masses by the squares of the velocities. Bernouilli saw nothing in this principle but a consequence of the theory of the /is viva, and a general law of nature, in consequence of which, the suet of the vis viva of several bodies preserves itself the same, as long as they continue to act upon each other by simple pres sures, and is always equal to the simple cis viva, resulting front the action of the forces by which the body is really moved. To this principle he gave the mime of conservatio vivium vivarum, and successfully employed it in the solution of several problems that had not befbre been effected.