722 MECHANICS in equilibrium unless their whole work for an infinitely small motion is nothing, in which case the single equivalent force is reduced to nothing. But whatever amount of free dom to move the whole system may have, we may always, by the application of frictionless constraint, limit it to one degree of freedom only; and this may be freedom to execute any particular motion whatever, possible under the given conditions of the system. If, therefore, in any such infinitely small motion there is variation of potential energy uncompensated by work of the applied forces, con straint limiting the freedom of the system to only this motion will bring us to the case in which we have just demonstrated there cannot be equilibrium. But the appli cation of constraints limiting motion cannot possibly dis turb equilibrium, and therefore the given system under the actual conditions cannot be in equilibrium in any particular configuration if the rate of doing work is greater than that at which potential energy is stored up in any possible motion through that configuration. Neutral 194. If a material system, under the influence of equi- internal and applied forces, varying according to some mm. definite law, is balanced by them in any position in which it may be placed, its equilibrium is said to be neutral. This is the case with any spherical body of uniform material resting on a horizontal plane. A right cylinder or cone, bounded by plane ends perpendicular to the axis, is also in neutral equilibrium on a horizontal plane. Practically, any mass of moderate dimensions is in neutral equilibrium when its centre of inertia only is fixed, since, when its longest dimension is small in comparison with the earth s radius, the action of gravity is, as we shall see ( 222), ap proximately equivalent to a single force through this point. Stable 195. But if, when displaced infinitely little in any ^ u !" direction from a particular position of equilibrium, and left to itself, it commences and continues vibrating, without ever experiencing more than infinitely small deviation, in any one of its parts, from the position of equilibrium, the equilibrium in this position is said to be stable. A weight suspended by a string, a uniform sphere in a hollow bowl, a loaded sphere resting on a horizontal plane with the loaded side lowest, an oblate body resting with one end of its shortest diameter on a horizontal plane, a plank, whose thick ness is small compared with its length and breadth, floating on water, are all cases of stable equilibrium, if we neglect the motions of rotation about a vertical axis in the second, third, and fourth cases, and horizontal motion in general in the fifth, for all of which the equilibrium is neutral. Unstable 196. If, on the other hand, the system can be displaced equi- j n anv wav f rorn a position of equilibrium, so that when mi> left to itself it will not vibrate within infinitely small limits about the position of equilibrium, but will move .farther and farther away from it, the equilibrium in this position is said to be unstable. Thus a loaded sphere resting on a horizontal plane with its load as high as possible, an egg- shaped body standing on one end, a board floating edgewise in water, would present, if they could be realized in practice, cases of unstable equilibrium. 197. When, as in many cases, the nature of the equili brium varies with the direction of displacement, if unstable for any possible displacement it is practically unstable on the whole. Thus a circular disk standing on its edge, though in neutral equilibrium for displacements in its plane, yet being in unstable equilibrium for those perpen dicular to its plane, is practically unstable. A sphere rest ing in equilibrium on a saddle presents a case in which there is stable, neutral, or unstable equilibrium according to the direction in which it may be displaced by rolling ; but practically it is unstable. 198. The theory of energy shows a very clear and simple test for discriminating these characters, or deter mining whether the equilibrium is neutral, stable, or Energy. unstable, in any case. If there is just as much potential test . of energy stored up as there is work performed by the applied f.i u !" and internal forces in any possible displacement, the equili brium is neutral, but not unless. If in every possible infinitely small displacement from a position of equilibrium there is more potential energy stored up than work done, the equilibrium is thoroughly stable, and not unless. If in any or in every infinitely small displacement from a position of equilibrium there is more work done than energy stored up, the equilibrium is unstable. It follows that if the system is influenced only by internal forces, or if the applied forces follow the law of doing always the same amount of work upon the system while passing from one configuration to another by all possible paths, the whole potential energy must be constant in all positions for neutral equilibrium, must be a minimum for positions of thoroughly stable equilibrium, and must be either a maximum for all displacements or a maximum for some displacements and a minimum for others when there is unstable equilibrium. 199. We have seen that, according to D Alembert s Forma- principle, as explained above, forces acting on the different tion of points of a material system, and their reactions against the | e ^ 0] accelerations which they actually experience in any case of O f ino y, motion, are in equilibrium with one another. Hence, in any actual case of motion, not only is the actual work by the forces equal to the kinetic energy produced in any infinitely small time, in virtue of the actual accelerations, but so also is the work which would be done by the forces, in any infinitely small time, if the velocities of the points constituting the system were at any instant changed to any possible infinitely small velocities, and the accelerations unchanged. This statement, when put into the concise language of mathematical analysis, constitutes Lagrange s application of the " principle of virtual velocities " to express the conditions of D Alembert s equilibrium between the forces acting and the resistances of the masses to the acceleration. It comprehends, as we have seen, every possible condition of every case of motion. The " equa tions of motion " in any particular case are, as Lagrange has shown, deduced from it with great ease. Commencing again with the equations of motion of a particle let us introduce quantities Sx, &c., consistent with, the conditions, otherwise perfectly arbitrary, and we have the general equation S,m(xSx + . . . ) = 2(X5a; + . . . ), in which, by D Alembert s principle, the forces X , Y , Z , due to the constraints, do not appear. If the system be conservative the right-hand member of this is, of course, equal to the loss of potential energy, so that and therefore, quite generally in such a system, 2m(x$x + . . . )= -5V ...... (1). In the actual motion of any system we have, for each particle, 5x = x8t, &c., so that we have This is the complete statement of Newton s scholium, 2 above. The right-hand member is the expression of the algebraic sum of the actioncs agentium and of the rcactioncs resisteniium, so far as these depend upon gravity, friction, &c., and the left-hand member that of the reactioncs due to the accelerations of the several particles. If the system be conservative, this becomes dV whose integral is of Bourse, the general statement of the conservation of energy. In Lagrange s general equation above, as we have stated, the variations Sx, &c., are not usually independent. "We must take
account of the various constraints imposed 011 the system. If these