The present work is, in the main, a reprint of the article "Mechanics," which I wrote for the last edition of the Encyclopedia Britannica. It was published in 1883, and I introduced it at once as one of the text-books in my advanced class. I there found it so useful that I have employed it ever since. The publishers have suggested that I should at last make a separate volume of it, instead of applying to them annually for a number of "pulls" from the stereotype plates. This course enables me to make use of the mass of corrections, modifications of portions which have been found to present difficulties, etc., which have been suggested to me from time to time by a long series of vitally interested and highly intelligent (albeit wholly unprofessional) critics.

The article was written at the request of Dr. Eobertson Smith, who was at the time the acting editor of the Encyclopaedia; and its arrangement and scope were, in great part, suggested by him, the main object being to give, in necessarily moderate compass, a tolerably complete elementary view of the subject, in so far as that was not already supplied by other articles in that great work. Some years earlier Dr. Smith had been my official assistant, and was thoroughly acquainted with the wants of students.

In this reprint I have of course had to complete the article by filling up the gaps already alluded to. For one of these I have reprinted my article on Waves. Thus the portions of this book which deal with such subjects as Attraction, Hydrostatics, Hydrokinetics, etc., are wholly additions to the original article. I have endeavoured to assimilate them, as far as I could, to the rest of the text, drawing my materials mainly from notes made from time to time on my interleaved lecture copy.

The general plan, - acted up to as far as possible, though by no means uniformly, - has been to lay fully the foundations of each branch of this extensive subject, and to give enough of the mathematical development to show the usual modes of attack, keeping by preference to examples whose main difficulty is physical rather than merely mathematical.

While using the article for teaching purposes, I have found the special advantage of having the formulae ready before the students, in type; so that I could profitably employ their time and mine upon the reasons for each successive step, instead of making my lecture a mere display of dexterity in " writing out " with chalk on the black-board. After more than forty years' practice at such work one comes, however reluctantly, to the conclusion that it is, in the true as well as in the usual sense of the word, vanity. The real source of the mischief, so far as the student is concerned, is his spending in mere unintellectual copying or transcribing the time which ought to be spent in sedulous attention to the explanations which the teacher gives, or at least ought to give, while manipulating the chalk.

One obvious objection may be made to many parts of this book - undue brevity. It was inevitable, when much had to be compressed into moderate space; yet, at the worst, is not brevity, if it but convey its message, transcendently preferable to prolixity ? Still, in teaching, I have found it advantageous to supplement the work at each stage by additional examples of the processes given in the text ; as well as by references to special books in which particular questions are examined with greater detail. Thus Tait and Steele's Dynamics of a Particle will supply much of what is wanted, both in the way of numerous additional examples and in that of minute detail in some important problems.

I am particularly indebted to Messrs. A. D. Kussell and A. C. Smith, at present workers in my Laboratory, for the care which they have bestowed on the correction of the proof-sheets.

The law of energy, in abstract dynamics, may be expressed as follows: the whole work done in any time, on any limited material system, by applied forces, is equal to the whole effect in the forms of potential and kinetic energy produced in the system, together with the work lost in friction. This principle may be regarded as comprehending the whole of abstract dynamics, because the conditions of equilibrium and of motion, in every possible case, may be derived from it.

A material system, whose relative motions are unresisted by friction, is in equilibrium in any configuration if, and is not in equilibrium, unless, the rate at which the applied forces perform work at the instant of passing through it is equal to that at which potential energy is gained, in every possible motion through that configuration. This is the celebrated principle of "virtual velocities," which Lagrange made the basis of his
Mecanique Analytique.

To prove it, we have first to remark that the system cannot possibly move away from any particular configuration except by work being done upon it by the forces to which it is subject ; it is therefore in equilibrium if the stated condition is fulfilled.

To ascertain that nothing less than this condition can secure the equilibrium, let us first consider a system having only one degree of freedom to move. Whatever forces act on the whole system, we may always hold it in equilibrium by a single force applied to any one point of the system in its line of motion, opposite to the direction in which it tends to move, and of such magnitude that, in any infinitely small motion in either direction, it shall resist or shall do as much work as the other forces, whether applied or internal, altogether do or resist. Now, by the principle of superposition of forces in equilibrium, we might, without altering their effect, apply to any one point of the system such a force as we have just seen would hold the system in equilibrium, and another force equal and opposite to it. All the other forces being balanced by one of these two, they and it might again, by the principle of superposition of forces in equilibrium, be removed ; and therefore the whole set of given forces would produce the same effect, whether for equilibrium or for motion, as the single force which is left acting alone. This single force, since it is in a line in which the point of its application is free to move, must move the system. Hence the given forces, to which the single force has been proved equivalent, cannot possibly be 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 freedom 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, constraint 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 application of constraints limiting motion cannot possibly disturb 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.

If a material system, under the influence of internal and applied forces, varying according to some 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 (§ 230), approximately equivalent to a single force through this point.

But if, when displaced infinitely little in any 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 thickness 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.

If, on the other hand, the system can be displaced in any way from a position of equilibrium, so that when 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 realised in practice, cases of unstable equilibrium.

When, as in many cases, the nature of the equilibrium 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 perpendicular to its plane, is practically unstable. A sphere resting 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.