will be in equilibria. Ile contents himself with ing this principle to inclined planes ; but it equally applies to all machines.
T)es Cartes deduced the equi'ibrium of difliotent forces from a similar principle; hot he presented it uncler another, and less general point of view, than Galileo had done; for he argues that to lift a given weight to a certain height. pre cisely the same three is requisite that would be suflieient to raise a heavier to a height proportionally less, or a lighter to a height proportionally greater; therefore two unequal weights will be in equilibria, when the perpendicular spaces described by them are reciprocally proportioned to them. In the application of this principle, however. only the spaces described in the first instant of motion are to be considered ; otherwise the accurate law of equilibrium will not be attained.
Another principle, recurred to by some authors in the solution or problems relative to the equilibrium of fitrecc, of the foregoing, viz. When a system of heavy is in equilibria, the centre of gravity is the lowest possible. For the centre of gravity ola body is the lowest, when the differential of its descent is 0, as can be demon strated from the principle de mallInns et minitnns ; that is, when the centre of gravity neithee ascends nor descends by an infinitely small change in the position of the system.
J. Bernonilli first perceived the great utility of generalizing this principle of virtual velocity, and applied it to the solu tion of problems; in which he was followed by Yarignon, who has devoted the whole of the ninth section of his Nouvelle .11eclionique to demonstrate its truth and exemplify its utility in various cases in statics.
in the ..111'inoires de l' Academie fin. 1 7,10, Maupertuis pro posed another principle, originating in the source, under the title ()I' "The. Law of Repose :” which was afterwards extended icy Euler, and explained in the of the Berlin Academy for 1751 : and the principle assumed by Mons Courtivron. in the ...1ilonoires de l' .Ira for is of the same nature: viz., that of all the situations which a system of bodies can successively take, that wherein the system must be placed to remain' in equilibrio, is that in which the Pis ?qv,/ is either it maximum or a minimum, because the vi• viva is the sum of the respective masses composing the system, each multiplied into the square of its velocity.
Of all these methods, that of virtual velocity appears to be most •rent:rally useful ; indeed all the others are derived front it, and are serviceable in as they approach nearer to it. La Grange has given practical examples of the aua ly tical processes tbr determining general form uhe or equations for the equilibrium of any system ; and La Place has demon strated the principle on which the calculus is tbunded.
in the foirroitio! ohservations, force is supposed to be the product of the mass of a material point, by the velocity it \M ilk] receive if entirely free. By confining these CI rations to the case of a single material point, the conditions of equilibrium be tbund to be analogous to those above spoken of, but much simplified.
The most elementary equation to express the state of equi librium of a material point, acted on by any number of forces, is, that every fiffee, multiplied by the clement of its direction. equals 0 : thus, suppose the point to change its position in an infHtely small degree in any direction; then, in the ease of equilibrium, if every Circe bo multiplied by the approached to, or reced.•d front by the point, the force being estiinated in its direction, the product, will be 0.
I Jere the point is supposed to he free ; but if constrained to move on a curved surthee, it will experience a reaction equal and contrary to the pressure which it exerts on such surface, but perpendicular to it, or in the direction of the radius of the curve. This reaction may be considered as a new force, which, multiplied by the elements of its direction, must ho added to the tbriner eluation. But it' the variation of position, instead of being taken arbitrarily, be taken upon the curve, so as not to alter the conditions of the problem, the preceding equation will still hold good. because the elementary variation of the radius is canal to 0, as is evident from inspection. Again, if the magnitude of any ti true. or its intensify, multiplied by the distance of its direetion from any fixed point, be denominated its moment, relatively to such point, it will be found that the sum of the moments of the producing threes is always canal to that ()I• the resulting, force ; and in ease of equilibrium the sum of the moments of all the forces equals 0.