STABLE AND UNSTABLE ; STABILITY. A system is said to be stable when a slight disturbance of its actual condition would net produce a continually increasing effect, but one which finally ceases to increase, diminishes, becomes an effect of a contrary character, and so on, in an oscillatory manner. The ordinary vibration of a pendulum is an instance ; the oscillation takes place about a stable position of equilibrium. We can give no instance of an unstable position ; for by definition, such a thing is a mathematical fiction. Any disturbance, however slight, produces upon an unstable system an effect which con tinually increases : no unstable equilibrium therefore can exist a moment, for no system made by human hands can be placed with mathematical exactness in a given position. The pendulum has a position of equilibrium exactly opposite to that about which it can oscillate, but no nicety of adjustment will retain it in that position : it may appear to rest for a moment, but will almost instantly begiu to fall.
The following curves or lines are all such that, supposing them to be rigid matter, a molecule placed ate would rest :— In the first, a displacement to the right or left would produce nothing but oscillation, and the equilibrum is stable ; in the second, neither displacement would be followed by any tendency to restora tion, and the equilibrium is unstable ; in the third, displacement would only be a removal to another position of rest, and the equili brium is called indifferent. In the fourth, displacement to the right would be followed by restoration, but the velocity acquired in restora tion would carry the molecule to the left, on which side there is no tendency to restoration : the equilibrium would then be permanently disturbed, and practically unstable ; though it might be convenient to say that it is stable as to the disturbances to the right, and unstable as to those to the left. In the fifth, the equilibrium at A is unstable, but if a push, however slight, were given to the molecule, it would obviously, by reason of the two contiguous stable positions, oscillate about a, as if A were itself a stable position : and in the same manner a stable position, with an unstable one near to it, might, for a disturbance of sufficient magnitude, present the phenomena of an un stable position.
Now, suppose that the point A, instead of being a single molecule, is the centre of gravity of a system acted on by its own weight only ; and let the curve drawn be the path of the centre of gravity, which, owing to the connection of the parts of the system with its supports, that centre is obliged to take. The phenomena bf the single point still remain true : there is in every case a position of equilibrium when the system is placed in such a position that its centre of gravity is at e. In (1) the equilibrum is stable ; in (2), unstable ; in (3), indifferent; in (4), stable or unstable, according to the direction of disturbance ; in (5), unstable, with results like those of stability. It is au error to state, as is frequently done, that there is no equilibrium in such a system except when its centre of gravity is highest or lowest ; as is obvious from (3) and (4). The general proposition which is true is this—that a system acted on by its own weight is in equili brium then, and then only, when its centre of gravity is placed at that point of its path which has its tangent parallel to the horizon, or per pendicular to the direction of gravity.
When a system is supported on three or more points, it is well known that there is no equilibrium unless the vertical passing through the centre of gravity cuts the polygon formed by joining these points. This must not be confounded. as is sometimes done, with a case of distinction between stable and unstable equilibrium ; for it is a case of equilibrium or no equilibrium, according as the central vertical cuts or does not cut the base of the figure. Of course it is in the power of any one to say that stability means equilibration and instability non-equilibration ; but such is not the technical use of these words in mechanics : stability and instability refer to equili brium, stable equilibrium being that which would only be converted into oscillation by a disturbance, and unstable equilibrium that which would not be so converted.