member my would thus become negative, which is impossible.

1

The motion of the system can then only be continued by the directions of the motions of certain of the elements which compose it being changed, so that the corresponding quantities by which EU, and EU, are respectively increased may change their signs, and the whole quantity U ̧ — £U, which before increased continually may now continually diminish. This being the case, Emv will increase again until, when U, - £U,=0, it becomes again equal to Em V2; that is, until the system acquires again the vis viva with which its disturbance commenced.

Thus, then, it has been shown, that in respect to every one of the supposed positions of the system there is a certain impulse or amount of vis viva, which being communicated to the system when in equilibrium, will just cause it to oscillate as far as that position, remain for an instant at rest in it, then return again towards its position of equili brium, and re-acquire the vis viva with which its displacement commenced. Now this being true of every position of the supposed range of positions, it follows that it is true of every disturbance or impulse which will not carry the system beyond this supposed range of positions; so that, having been displaced by any such disturbance or impulse, the system will constantly return again towards the position of equilibrium from which it set out, and is STABLE in respect to that position.

On the other hand, if the supposed position of equili brium be one in which the vis viva is a minimum, then the aggregate work of the forces which tend to accelerate the motion must, after the system has passed through that position, exceed that of the forces which tend to retard the motion; so that, adopting the same notation as before, EU, must be greater than U,, and the second member of the equation essentially positive. Whatever may have been the original impulse, and the communicated vis viva Σm V2, Emo' must therefore continually increase; so that the whole system can never come to a position of instantaneous repose;† but on the contrary, the motions of its parts must continuously increase, and it must deviate continually farther from its position of equilibrium, in which position it can never

That is, of that range of positions over which the supposed position of equilibrium holds the relation of a position of maximum vis viva.

Within that range of positions over which the supposed position of equilibrium holds the relation of minimum vis viva.

rest. The position is thus one of unstable equilibrium. Therefore, &c.

DYNAMICAL STABILITY.*

If a body be made, by the action of certain disturbing forces, to pass from one position of equilibrium into another, and if in each of the intermediate positions these forces are in excess of the forces opposed to its motion, it is obvious that, by reason of this excess, the motion will be continually accelerated, and that the body will reach its second position with a certain finite velocity, whose effect (measured under the form of vis viva) will be to carry it beyond that position. This however passed, the case will be reversed, the resistances will be in excess of the moving forces, and the body's velocity being continually diminished and eventually destroyed, it will, after resting for an instant, again return towards the position of equilibrium through which it had passed. It will not however finally rest in this position until it has completed other oscillations about it. Now the amplitude of the first oscillation of the body beyond the position in which it is finally to rest, being its greatest amplitude of oscillation, involves practically an important condition of its stability; for it may be an amplitude sufficient to carry the body into its next adjacent position of equilibrium, which being, of necessity, a position of unstable equilibrium, the motion will be yet further continued and the body overturned. Different bodies requiring moreover different amounts of work to be done upon them to produce in all the same amplitude of oscillation, that is (relatively to that amplitude) the most stable which requires the greatest amount of work to be so done upon it. It is this condition of stability, dependent upon dynamical considerations, to which, in the following paper, the name of dynamical stability is given.

I cannot find that the question has before been considered in this point of view, but only in that which determines whether any given position be one of stable, unstable, or mixed equilibrium; or which determines what pressure is necessary to retain the body at any given inclination from such a position.

* Extracted from a paper "On Dynamical Stability, and on the Oscillations of Floating Bodies," by the author of this work, published in the Transactions of the Royal Society, Part. II. for 1850. The remainder of the paper will be found in the Appendix.

1. To the discussion of the conditions of the dynamica. stability of a body the principle of vis viva readily lends itself. That principle, when translated into a language which the labours of M. PONCELET have made familiar to the uses of practical science, may be stated as follows:

"When, being acted upon by given forces, a body or system of bodies has been moved from a state of rest, the difference between the aggregate work of those forces whose tendencies are in the directions in which their points of application have been moved, and that of the forces whose tendencies are in the opposite direction, is equal to one-half the vis viva of the system."

Thus, if zu, be taken to represent the aggregate work of the forces by which a body has been displaced from a position in which it was at rest, and Zu, the aggregate work (during this displacement) of the other forces applied to it; and if the terms which compose Zu, and Zu, be understood to be taken positively or negatively, according as the tendencies of the corresponding forces are in the directions in which their points of application have been made to move or in the opposite directions; then representing the aggregate vis viva of the body by

1

Zwv2.

Now Eu, representing the aggregate work of those forces which acted upon the body in the position from which it has been moved, may be supposed to the known; Eu, may therefore be determined in terms of the vis viva, or conversely.

2. In the extreme position into which the body is made to oscillate and from which it begins to return, it, for an instant, rests. In this position, therefore, its vis viva disappears, and we have

Σu,+£u,=0. . . . . . (2').

This equation, in which Eu, and Eu, are functions of the impressed forces and of the inclination, determines the extreme position into which the body is made to roll by the action of given disturbing forces; or, conversely, it determines the forces by which it may be made to roll into a given extreme position.

* See Art. 129.

3. The position in which it will finally rest is determined by the maximum value of Eu,+Eu, in equation (1); for, by a well-known property, the vis viva of a system* attains a maximum value when it passes through a position of stable, and a minimum, when it passes through a position of unstable equilibrium. The extreme position into which the body oscillates is therefore essentially different from that in which it will finally rest.

4. Different bodies, requiring different amounts of work to be done upon them to bring them to the same given inclination, that is (relatively to that inclination) the most stable which requires the greatest amount of work to be so done upon it, or in respect to which Eu, is the greatest. If, instead of all being brought to the same given inclination, each is brought into a position of unstable equilibrium, the corresponding value of Eu, represents the amount of work which must be done upon it to overthrow it, and may be considered to measure its absolute, as the former value measures its relative dynamical stability. The absolute dynamical stability of a body thus measured I propose to represent by the symbol U, and its relative dynamical stability, as to the inclination, by U().

The measure of the absolute dynamical stability of a body is the maximum value of its relative stability, or U the maximum of U(); for whilst the body is made to incline from its position of stable equilibrium, it continually tends to return to it until it passes through a position of unstable equilibrium, when it tends to recede from it; the aggregate amount of work necessary to produce this inclination must therefore continually increase until it passes through that position and afterwards diminish.

5. The work opposed by the weight of a body to any change in its position is measured by the product of the vertical elevation of its centre of gravity by its weight.+ Representing therefore by W the weight of the body, and by AH the vertical displacement of its centre of gravity when it is made to incline through an angle 4, and observing that the displacement of this point is in a direction opposite to that in which the force applied to it acts, we have Σu,=—W.^II, and by equation (2′),

* Art. 132.

It is obvious that the absolute dynamical stability of a body may be greater than that of another, whilst its stability, relatively to a given inclina. tion, is less; less work being required to incline it than the other at that angle, but more, entirely to overthrow it.

Art. 60.

If therefore no other force than its weight be opposed to a body's being overthrown, its absolute dynamical stability, when resting on a rigid surface, is measured by the product of its weight by the height through which its centre of gravity must be raised to bring it from a stable into an unstable position of equilibrium.

6. The Dynamical Stability of Floating Bodies.-The action of gusts of wind upon a ship, or of blows of the sea, being measured in their effects upon it by their work, that vessel is the most stable under the influence of these, or will roll and pitch the least (other things being the same), which requires the greatest amount of work to be done upon it to bring it to a given inclination; or, in respect to which the relative dynamical stability (8) is the greatest for a given value of . In another sense, that ship may be said to be the most stable which would require the greatest amount of work to be done upon it to bring it into a position from which it would not again right itself, or whose absolute dynamical stability U is the greatest. Subject to the one condition, the ship will roll the least, and subject to the other, it will be the least likely to roll over.

Thus the theory of dynamical stability involves a question of naval construction. It will be found discussed in its application to this question in the Appendix.

FRICTION.

133. It is a matter of constant experience, that a certain resistance is opposed to the motion of one body on the surface of another under any pressure, however smooth may be the surfaces of contact, not only at the first commencement, but at every subsequent period of the motion; so that, not only is the exertion of a certain force necessary to cause the one body to pass at first from a state of rest to a state of motion upon the surface of the other, but that a certain force is further requisite to keep up this state of motion. The resistance thus opposed to the motion of one body on the surface of another when the two are pressed together, is called fric