Gravity is the only natural force that we are practically concerned with in Rational Statics: and we see, by this, how backward this science is in regard to universal gravi- tation. As for the exterior general circumstances, such as friction, resistance of media, and the like, which are alto- gether excluded in the establishment of the rational laws of Mechanics, we can only say that we are absolutely igno- rant of the way to introduce them into the fundamental relations afforded by analytical Mechanics, because we have nothing to rely on, in working them, but precarious and inaccurate hypotheses, unfit for scientific use.
. As for the theory of equilibrium in regard of^fluid"^^"^ ^^ fluid bodies, — the application which it remains for us to notice, — those bodies must be regarded as either liquid or gaseous.
-rj 1,,. Hydrostatics may be treated in two ways.
We may seek tlie uiws of the equinbrium 01 flaids, according to statical considerations pro])er to that class of bodies: or we may look for them among the laws which relate to sohds, allowing for the new characteristics residting from fluidity.
The first method, being the easiest, was in early times the only one employed. Till a rather recent time, all geometers employed themselves in proposing statical prin- ciples peculiar to fluids; and especially with regard to the grand question of the figure of the earth, on the supposi- tion that it was once fluid. Huyghens first endeavoured to resolve it, taking for his principle of equilibrium the necessary perpendicularity of weight at the free surface of the fluid. Newton's principle was the necessary equality of weight between the two fluid columns going from the centre, — the one to the pole, tlie other to some point of the equator. Bouguer showed that both methods were bad, EQUILIBRIUM OF FLUIDS. 135 because, though each was iucoutestable, the two failed, in many cases, to give the same form to the fluid mass in equilibrium. But he, in his turn, was wrong in believing that the union of the two principles, when they agreed in indicating the same form, was sufficient for equilibrium. It was Clairaut who, in his treatise on the form of the earth, first discovered the true laws of the case, setting out from the evident consideration of the isolated equilibrium of any infinitely small canal; and, tried by this criterion, he showed that the combination required by Bouguer might take place without equilibrium happening. Several great geometers, proceeding on Clairaut' s foundation, have carried on the theory of the equilibrium of fluids a great way. Maclaurin was one of those to whom we owe much; but it was Euler who brought up the subject to its present point, by founding the theory on the principle of equal presstire in all directions. Observation of the statical constitution of fluids indicates this as a general law; and it furnishes the requisite equations with extreme facility.
It was inevitable that the mathematical theory of the equilibrium of fluids should, in the first place, be founded, as we have seen that it was, on statical principles peculiar to this kind of bodies: for, in early days, the y...characteristic differences between solids and fluids must have appeared too great for any geometer to think of applying to the one the general principles appro- priated to the other. But, when the funda- Cases mental laws of hydrostatics were at length obtained, and men's minds were at leisure to estimate the real diversity between the theories of fluids and of solids, they could not but endeavour to attach them to the same general piinciples, and perceive the necessary applicability of the fundamental rules of Statics to the equilibrium of fluids, making allowance for the attendant variability of form. But, before hydrostatics could be comprehended under Statics, it was necessary that the abstract theory of equilibrium should be made so general as to apply directly to fluids as well as solids. This was accomplished when Lagrange supplied, as the basis of the whole of Rational Mechanics, the single principle of Virtual Velocities. One of its most valuable properties is its being as directly 136 POSITIVE PHILOSOPHY.
applicable to fluids as to solids. From that time, Hydro- statics, ceasing to be a natui'al branch of science, lias taken its place as a secondary division of Statics. This arrange- ment has not yet been familiarly admitted; but it must soon become so.
To see how the principle of Virtual Velocities may lead to the fundamental equations of the equilibrium of fluids, we have to consider that all that such an ajjplication requires, is to introduce amoug the forces of the system under notice one new force, — the pressure exerted upon each molecule, which will introduce one term more into the general equation. Proceeding thus, the three general equations of the equilibrium of fluids, employed when hydrostatics was treated as a separate branch, will be immediately reached. If the fluid be a liquid, we must have regard to the condition of incompressibility, — of change of form without change of volume. If the fluid be gaseous, we must substitute for the incompressibility that condition which sul)]ects the volume of the fluid to vary according to a determinate function of the pressiire; for instance, in the inverse ratio of the pressure, according to the physical law on which Mariotte has founded the whole Mechanics of the gases. We know but too little yet of these gaseous conditions; for Mariotte's law can at present be regarded only as an approximation, — sufficiently exact for average circumstances, but not to be rigorously applied in any case whatever.
Some confirmation of the philosophical character of this method of treating hydrostatics arises from its enabling us to pass, almost insensibly, from the order of bodies of invariable form to that of the most variable of all, through intermediate classes, — as flexible and elastic bodies, — whereby we obtain, in an analytical view, a natural filiation of subjects.
We have seen how the department of Statics has been raised to that high degree of speculative perfection which transforms its questions into simj^le problems of Mathema- tical Analysis. We must now take a similar review of the other department of general Mechanics, — that more ex- tended and more complicated study which relates to the laws of Motion.
THEORY OF RECTILINEAR MOTION. 137 SECTION II.
DYNAMICS.
The object of Dynamics is the study of the ^1 •, varied motions produced by continuous forces. The Dynamics of varied motions or continuous forces includes two departments, — the motion of a point, and that of a body. From the positive point of view, this means that, in certain cases, all the parts of the body in question have the same motion, so that the determination of one particle serves for the whole; while in the more general case, each particle of the body, or each body of the system, assuming a distinct motion, it is necessary to examine these different effects, and the action upon them of the re- lations belonging to the system under notice. The second theory being more complicated than the first, the first is the one to begin with, even if both are deduced from the same principles.
With regard to the motion of a point, the question is to determine the circumstances of the compound curvilinear motion, resulting from the simidtaueous action of different continuous forces, it being known what would be the recti- linear motion of the body if influenced by any one of these forces. Like evei-y other, this problem admits of a converse solution.
But here intervenes a preliminary theory, „. j.
which must be noticed before either of the ^^jjj^gj^. ju^^^^j^j^ two departments can be entered upon. This theory is popularly called the tlieory of rectilinear motion, produced by a single continuous force acting indefinitely in the same direction. It may be asked why we want this, after having said that the effect of each separate force is supposed to be known, and the eff'ect of their union the thing to be sought. The answer to this is, that the varied motion produced by each continuous force may be defined in several ways, which depend on each other, and which could never be given simultaneously, though each may be separately the most suitable; whence results the necessity of being able to pass from any one of them to all the rest. The preliminary theory of varied motion relates to these 138 POSITIVE PHILOSOPHY.
transformations, and is therefore inaptly termed the study of the a.ction of a single force. These different equivalent defini- tions of the same varied motions result from the simultaneous consider'ation of the three distinct but co related functions which are presented by it, — space, velocity, and force, con- ceived as dependent on time elapsed. Taking the most extended view, we may say that the definition of a varied motion may be given by any equation containing at once these four variables, of which only one is independent, — time, space, velocity, and force. The problem will consist in deducing from this equation the distinct determination of the three characteristic laws relating to space, velocity, and force, as a function of time, and, consequently, in mutual correlation. This general problem is always re- ducible to a purely analytical research, by the help of the two dynamical formulas which express, as a function of time, velocity and force, when the law of space is supposed to be known. The infinitesimal method leads to these formulas with the utmost ease, the motion being considered uniform during an infinitely small interval of time, and as uniformly accelerated during two consecutive intervals. Thence the velocity, supposed to be constant at the instant, according to the first consideration, will be naturally expressed by the diiferential of the space, divided by that of the time; and, in the same way, the continuous force, according to the second consideration, will evidently be measured by the relation between the infinitely small incre- ment of the velocity, and the time employed in producing this increment.
Lagrange's conception of transcendental analysis ex- cluding him from this use of the infiuitesinaal method for the establisment of the two foregoing dynamic formulas, he was led to present this theory under another point of view, more important than seems to be generally sup- posed. In his Theory of Analytical Functions, he has shown that this dynamic consideration really consists in conceiving any varied motion as compounded, each moment, of a certain uniform motion and another motion uniformly varied, — likening it to the vertical motion of a heavy body under a first impulsion. Lagrange has not given its due advantage to this conception, by developing it as he might THE TWO CASES OF RATIONAL DYNAMICS. 139 have done. In fact, it supplies a complete theory of the assimilation of motions, exactly like the theory of the con- tacts of curves and surfaces, in the department of geometry. Like that theory, it removes the limits within which we sup- posed ourselves to be confined, by disclosing to us, in an abstract way, a much more perfect measure of all varied motion than we obtain by the ordinary theory, though reasons of convenience compel us to abide by the method originally adopted.
The first case or department of rational,r.• ^ T • xij.i?xii- c -J. Motion of a dynamics, — that or the motion ot a point, or point of a body which has all its points or portions affected by the same force, — relates to the study of the curvilinear motion produced by the simultaneous action of any different continuous forces. This case divides itself again into two, — according as the mobile point is free, or as it is compelled to move in a single curve, or on a given surface. The fundamental theory of curvilinear motion may be established in either case, in a different way; each being susceptible of direct treatment, and of being con- nected with the other. In the first case, in order to deduce the second, we have only to regard the active or passive re- sistance of the prescribed curve or surface as a new force to be added to the others proposed. In the other way, we have only to consider the moving point as compelled to describe the curve which it must traverse; and this is enough to afford the fundamental equations, though this curve may then be primitively unknown.
The other, more real and more difiicult., ^.. J.T, £,\.- £, n Motion 01 a case, is tiiat ot the motion or a system ot sv^teni bodies in any way connected, whose proper motions are altered by the conditions of their connection. There is a new elementary conception about the measure- ment of forces which some geometers declare to be logically deducible from antecedent considerations, and to which they would assign the place and title of a fourth law of motion. For the sake of convenience we may make it into a fourth law of motion; but such is not its philosophical character. The idea is, that forces which impress the same velocity on different masses are to each other exactly as those masses; or, in other words, that the 140 POSITIVE PHILOSOPHY.
forces are proportional to the masses, as we have seen them, under the third law of motion, to be proportional to the velocities. All phenomena, such as the communication of motion by collision, or in any other way, have tended to confirm the supposition of this new kind of proportion. It evidently results from this, that when we have to com- pare forces which impress different velocities on unequal masses, each must be measured according to the product of the mass upon which it acts by the corresponding velocity. This product is called by geometers qua7itity of motion; and it determines the percussion of a body, and also the pressure that a body may exercise against any fixed obstacle to its motion.
Proceeding to the second dynamical case, we see that the characteristic difticulty of this order of questions con- sists in the way of estimating the connection of the different bodies of the system, in virtue of which their mutual reactions will necessarily affect the motions which each would take if alone; and we can have no a priori knowledge of what the alterations will be. In the case of the pendulum, for instance, the particles nearest the point of suspension, and those furthest from it, must react on each other by their connection, — the one moving faster and the other slower than if they had been free; and no established dynamic principle exists revealing the law which determines these reactions. Geometers naturally began by laying down a pi'inciple for each particular case; and many were the principles thus offered, which tui-ned out to be only remarkable theorems furnished simul- taneously by fundamental dynamic equations. Lagrange has given us, in his " Analytical Mechanics," the general history of this series of labours: and very interesting it is, as a study of the progressive march of the human intellect. , This method of proceeding continued till mincinle ^ ^^^^ time of D'Alembert, who put an end to all these isolated researches by seeing how to compute the reactions of the liodies of a system in virtue of their connection, and establishing the fundamental ecj[uations of the motion of any system. By the aid of the great principle which bears his name, he made questions of motion merge in simple questions of equilibrium. The principle is simply this. lu the case supposed, the natural motion clearly divides itself into two, — the one which subsists and the one which has been destroyed. By D'Alembert's view, all these last, or, in other words, all the motions that have been lost or gained by the different bodies of the system by their reaction, necessarily balance each other, under the conditions of the connection which characterizes the pro- posed system. James Bernouilli saw this with regard to the particular case of the pendulum; and he was led by it to form an equation adapted to determine the centre of oscillation of the most simple system of weight. But he extended the resource no furtlier; and what he did detracts nothing from the credit of D'Alembert's conception, the excellence of which consists in its entire generality.
In D'Alembert's hands the principle seemed to have a purely logical character. But its germ may be recognized in the second law of motion, established by Newton, under the name of the equality of reaction and action. They are, in fact, the same, with regard to two bodies only acting upon each other in the line which connects them. The one is the greatest possible generalization of the other; and this way of regarding it brings out its true nature, by giving it the physical character which D'Alembert did not impress upon it. Henceforth therefore we recognize in it the second law of motion, extended to any number of bodies connected in any manner.
We see how every dynamical question is thus convertible into one of Statics, by forming, in each case, equations of equilibrium between the destroyed motions. But then comes the difBculty of making out what the destroyed motions are. In endeavouring to get rid of the embarrass- ing consideration of the quantities of motion lost or gained, Euler, above others, has supplied us with the method most suitable for use, — that of attributing to each body a quantity of motion equal and contrary to that which it exhibits, it being evident that if such equal ana contrary motion could be imposed upon it, equilibrium would l)e the result. This method contemplates only the primitive and the actual motions which are the true elements of the dynamic problem,— the given and the unknown; and it is under this method that D'Alembert's principle is habitually 142 POSITIVE PHILOSOPHY.
conceived of. Questions of motion being thus reduced to questions of equilibrium, tlie next step is to combine D'Alembert's principle with that of virtual velocities. This is the combination proposed by Lagrange, and de- veloped in his "Analytical Mechanics," which has carried up the science of abstract Mechanics to the highest degree of logical perfection, — that is, to a rigorous unity. All questions that it can comprehend are brought under a single principle, through which the solution of an}' jDroblem whatever offers only analytical difficulties.
D'Alembert immediately applied his principle to the case of fluids — liquid and gaseous, which evidently admit of its use as well as solids, their peculiar conditions being considered. The result was our obtaining general equa- tions of the motion of fluids, wholly unknown before. The j^rincipal of virtual velocities rendered this perfectly easy, and again left nothing to be desired, in regard to concrete considerations, and presented none but analytical diffi- culties. We must admit however that our actual know- ledge obtained under this theory is extremely imperfect, owing to insurmountable difficulties in the integrations required. If it was so in questions of pure Statics, much more must it be so in the more complex dynamical ques- tions. The problem of the flow of a gravitating liquid through a given orifice, simple as it appears, has never yet been resolved. To simplify as far as they could, geometers have had recourse to Daniel Bernouilli's hypothesis of the parallelism of sections, which admits of our considering motion in regard to horizontal laminse instead of particle by particle. But this method of considering each horizontal lamina of a liquid as moving altogether, and taking the place of the following, is evidently contrary to the fact in almost all cases. The lateral motions are wholly abstracted, and their sensible existence imposes on us the necessity of studying the motion of each particle. We must then con- sider the science of hydrodynamics as being still in its infancy, even with regard to liquids, and much more with regard to gases. Yet, as the fundamental equations of the motions of fluids are irreversibly established, it is clear that what remains to be accomplished is in the direction of mathematical analysis alone.
KESULTS OF RATIONAL MECHANICS. 143 Such is the Method of Rational Mechanics. -p, As for the great theoi-etical results of the science, — the principal general properties of equilibrium and motion thus far discovered, — thej were at first taken for real principles, each being destined to furnish the solution of a certain order of new problems in Mechanics. As the systematic chai'acter of the science has come out however, these supposed principles have shown themselves to be mere theorems, — +i.^ ^^^ necessary results of the fundamental theories of abstract Statics and Dynamics.
Of these theorems, two belong to Statics. ^ ^. The most remarkable is that discovered by p i' - Torricelli with regard to the equilibrium of heavy bodies. It consists in this; that when any system of heavy bodies. is in a situation of ec[uilibrium, its centre of gravity is. necessarily placed at the lowest or highest possil)le point, in comparison with all the positions it might take under any other situation of the system. — Maupertuis afterwards, by his working out of his Laiv of Repose, gave a large generalization to this theorem of Torricelli's, which at once became a mere particular case under that law; Torricelli's applying merely to cases of terrestrial gravitation; while that of Maupertuis extends throughout the whole sphere of the great natural attractive forces.
The other general property relating to Stability and ec[uilibrium may be regarded as a necessary instability of complement of the former. It consists in equihbrium. the fundamental distinction between the cases of stability and instability of equilibrium. There being no such thing- in nature as abstract repose, the term is applied here to that state of stable equilibrium which exists where the centre of gravity is placed as low as possible; while un- stable equilibrium is that which is poi^ularly called equi- librium; and it exists when the centre of gravity is placed as high as possible. Maupertuis's theorem consisted in this, — that the sitviation of equilibrium of any system is always that in which the sum of vii-es vivce (active forces) is a maximum or a minimum; and the one under notice, developed by Lagrange, consists in this, — that in any system equilibrium is stable or unstable according as 144 POSITIVE PHILOSOPHY.
the sum of vires vivce is a maximum, or a Biiuimum. Observation teaches the facts in the most simple cases; but it requires a large theory to exhibit to geometers that the distinction is equally applicable to the most compound systems.
Proceeding to the theorems relative to dynamics, the most direct vray of establishing them is that used by Lagrange, — exhibiting them as immediate consequences . of the general equation of dynamics, deduced +]i'l!!vr,i!c. from the combination of D'Alembert's priuciple With the prmcqjle or virtual velocities. The first theorem is that of the conservation of the motion of the centre of gravity, discovered by Nevrton. Newton showed that the mutual action of the bodies of any system. Conservation vv^hether of attraction, impulsion, or any of the motion other, — regard being had to the constant of the centre equality between action and reaction, — can- gravity. ^^^ ^^^ ^^-^-^ ^^^ affect the state of the centre of gravity; so that if there were no accelerating forces besides, and if the exterior forces of the system were reduced to instantaneous forces, the centre of gravity would remain immovable, or would move uniformly in a right line. D'Alembert generalized this property, and ex- hibited it in such a form that every case in which the motion of the centre of gravity has to be considered may be treated as that of a single molecule. It is seldom that we foi'm an idea af the entire theoretical generality of such great results as those of rational Mechanics. We think of them as relating to inorganic bodies, or as otherwise cir- cumscribed; but we cannot too carefully remember that they aj^ply to all phenomena whatever; and in virtue of this universality alone are the basis of all real science.
The second general theorem of dynamics is area"''' ^ ° ^^^^ principle of areas, the first perception of which is attributable to Kepler. In its sim- plest form it is this; that if the accelerating force of any molecule tends constantly towards a fixed point, the vector radius of the moving body describes equal areas in equal times round the fixed point; so that the area described at the end of any time increases in proportion to the time: and the reciprocal fact is clear, — that the evidence of the DYNAMICAL THEOREMS. 145 areas and the times proves the action upon the body of a force directed towards the fixed point. This discovery of Kepler's is the more remarkable for having been made before dynamics had been really created by Galileo. Its importance in astronomy we shall see hereafter. But though, in its simplest form, it is one of the bases of celes- tial Mechanics, it is, in fact, only the simj^lest particular case of the great general theorem of areas, exhibited in the middle of the last century by D'Arcy, Daniel Bernouilli, and Euler. Kepler's discovery related only to the motion of a point, while the later one refers to the motion of any system of bodies, acting on each other in any manner what- ever; which constitutes a case, not only more complex, but different, on account of the mutual actions involved. It yields proof, however, that though the area described by the vector radius of each molecule may be altered by recij)- rocal actions, the sum of the areas described will remain invariable in a given time, and will increase therefore in proportion to the time. As the theorem of the centre of gravity determines all that relates to motions of transla- tion, this determines all that relates to motions of rotation: