When we see a system in equilibrium, experience tells us that there are efforts it motion which are counteracted. Remove any one of the forces, or any part of one of them, and motion immediately begins. It is true that friction and other resistances prevent our having so good a perception of this truth as we otherwise might have ; since, when equilibrating forces are removed in whole or in part,friction frequently supplies the place and maintains the equilibrium. A little reflection will however make it apparent that when a system is once iu equi librium, no addition nor subtraction of forces can be made without pro. ducing motion, unless the forces added or withdrawn be such as by themselves would maintain equilibrium.
A system, then, at rest, makes efforts to move, which efforts are counteracted ; and the mathematical conditions of equilibrium, what ever they may be, must express that every force endeavours to produce motion ; must contain, directly or indirectly, a measure of the ening) of that force; and must show that a complete oounteraction of all the efforts at motion takes place. But here arises a question, and one which is of the utmost importance lu the comprehension of our prin. eiple. The number of virtual motions is usually infinite : —Does an3 given system of forces make an effort to produce every one of them, in sonic only f We know that, if the forces do not produce equilibrinm, one of the virtual motions ensues in the first dt following the applica tion of the forces, to the exclusion of all the rest ; it ought not, there fore, to surprise the student if he were told that, for every given set of forces (a given system being always understood), Borne one motion prevented, is every motion prevented. But in point of fact the direct contrary is true, in rigid systems at least : generally speaking. there is but one class of virtual motions which a given set of forces has not a tendency to produce, and any one of the rest may be produced. Our meaning will appear in the followiug explanation : —We have seen that every infinitely small motion of a rigid system mny produced by a scrowlikemotion,* namely, rotation round an axis, accompanied by a slipping up or down that axis. Take any line fur an axis, and suppose n screw, fitted to its receiving screw (the latter im moveably fixed in space), to be described with that axis : suppose also that the system to which the forces are applied is fixed to the screw. Ilere then is every virtual motion prevented, except one; so that if the system begin to move, it must take that one motion. Now apply the given set of forces, and resolve them all in directions parallel to the axis, and in planes perpendicular to it. There must be motion unless the former forces destroy each other, and the latter have a resultant or resultants pasting through the axis. Consequently, with certain ex ceptions (which, though infinite in number, are few compared with the rest), a given set of forces, acting on a given system, will produce any virtual motion, if others be excluded : but when there are various virtual motions not excluded, the system, if the forces do not balance one another, will take one in preference to any of the rest. The pro ceding argument ought to be more developed, but we have not room for such an explanation as would be intelligible to every one: most of the difficulty indeed lies iu the purely geometrical conception of motion, and is foreign to our article.
We are to expect, then, as the condition of equilibrium, a collection of conditions, an infinite number, implying that, of an infinite number of motions, possible d priori, the given system of forces makes each and every one impossible. To make it appear in what the condition may
probably consist, look at the following cases :—If one point of the system be fixed, forces applied at that point are useless, for they only produce a pressure or strain on the fixed point, and neither promote nor retard any virtual motion. If one point be restrained to move upon a given surface or curve, forces applied at that point per pendicular to that surface or curve are useless, for a similar reason. Thus suppose one point must be retained on n given horizontal plane : any weight added to that point has no effect on the equilibrium ; it is merely equivalent to so much weighty laid upon the plane. Generally, then, a force produces no effect in equilibrium unless the point to which it is applied can move in the direction of that force : thus weight produces no effect when applied to a point of which all the virtual motions are horizontal. But let the plane be ever so little inclined to the horizon, a point restricted to move upon it has somewhat of vertical motion : weight applied at that point will have some offeot in equilibrium. It would be natural to conclude (and let it be remem bered that in these it priori views we are only stating strong probabi lities) that the more freely a point may move in the direction of the force which acts upon it, the greater the effect of that force iu pro ducing or disturbing equilibrium. Now since it is sufficiently evident that, ccrtcris paribus, a force has more or leas effect in proportion to its magnitude, for instance, that, under given circumstances, two pounds of pressure produce twice the effect of one pound, it seems that for any given virtual motion, the effect of each force varies jointly as the magnitude of the force, and the length over which, in that virtual motion, the point of application moves in the direction of the force. That is, suppose A to be the point of application of the force, and A to represent its direction and magnitude. In one virtual or possible motion of the system, let A be transferred to e, infinitely near to A. Draw n s perpendicular upon Q A, then A 13 is the space moved over in the direction of the force ; and if the force contain P units of pressure, exAs is the product on the value of which the efficiency of the force seems to have some dependence. Here, however, the motion A s is in the direction of the force, and the force helps to produce that motion ; for it is obviously easier, exteris paribus, that the point A should move in the direction A R, when the force acts in the direction A Q, than it would have been if the same force had acted in the opposite direction A T. But suppose that another virtual motion might bring A to V. Draw v w perpendicular to A T ; then A W is the space moved over in the direction of the force, and P x AW is the product on which the efficiency of the force seems to depend. But here the motion A w is in the direction opposite to that of the force, and it is obviously less easy that the point A should move in the direction AT, when the force acts in the direction AQ, than it would have been if the force had acted in the opposite direction AT. Hence, to what has preceded, we may probably add that the efficiency of a force, in promoting,or preventing one given kind of virtual motion, is to be considered as of one kind or another according, as, for that motion, the virtual motion of the point of application, estimated in the line of action of the force, is with the direction of the force, or opposite to it.