moving according to the direction of its axis supposed to be hori zontal; the relative apparent motion of the globe would not be parallel to this axis, in all the positions of the axis relatively to the horizon. Here, then, is a simple method of ascertaining, by experi ence, whether '(ƒ) has a sensible value on the Earth: but the mast exact experiments render imperceptible, in the apparent motion of the globe, any deviation from the direction of the impressed force; whence it follows that, on the Earth, (f) is nearly nothing. Its value, were it ever so little sensible, would be principally manifest in the duration of the oscillations of the pendulum; a duration which would be different according to the position of the plane of its motion, with respect to the direction of the Earth's motion. The most exact observations not discovering any like difference, we must conclude that '(ƒ) is insensible, and probably nothing on the Earth.

If the equation '(ƒ)= obtained, whatever were the force f (f) would be constant, and the velocity would be proportional to the force; again, it would be proportional thereto, if the function () were composed of only one term; since otherwise '(ƒ) would never be nothing, f not being so. It would be necessary, then, if the velocity were not proportional to the force, to suppose that, in nature, the function of the velocity which expresses the force is composed of several terms; which is scarcely probable. It is moreover necessary to suppose, that the velocity of the Earth is exactly that which is conformable to the equation (f), which is against all probability, Besides, the velocity of the Earth varies in different seasons of the year; it is a thirtieth part greater in winter than in summer. This variation is still more considerable, if, as every thing seems to indicate, the solar system be moved in space; for, according as the progressive motion conspires with that of the Earth, or according as it is contrary, there ought thence to result, during the course of the year, great variations in the absolute motion of the earth; which ought to alter the equation in question, and the ratio of the impressed force to the absolute velocity which thence results; if this equation and this ratio were not inde. pendent of the Earth's motion:-but observation discovers no sensible alteration.'

Having thus established the two fundamental propositions, the inertia of matter, and the proportionality of force to velocity, (the only two propositions which mechanics borrow from experience,) the author considers the movement of a material point solicited by forces that appear to act like gravity in a continued manner. The equations for the motion of a point are obtained by employing the principle excogitated by D'Alem bert, and applied in the Dynamique of that author: by this principle, the velocity which a body has at the beginning of a new instant of time is decomposed into two; one, the velocity which the body has in the new instant; and the other must necessarily be such that the body, by virtue of it, must remain in equilibrio.-The equation next deduced is that of the square of the velocity (c+2 ), c being a constant quantity, and

a function, the exact differential of which is Pdx+2dy+Rdã, x, y, and z being three rectangular co-ordinates, and P, Q, R, three forces acting parallel to them.-M. LA PLACE next demonstrates the singular property of the curve described by a body; which property consists in this, that the integral fuds. (v= velocity, ds element of the curve,) comprized between the two extreme points of the described curve, is less than in every other curve if the body be free, or on every other curve subjected to the same surface on which it must move if not entirely free,

From the equations established in the first chapter, is deduced this proposition: that the pressure against a surface is equal to the square of the velocity divided by the radius of curvature; and thence immediately this, that the centrifugal force of a body revolving in a circle is equal to the square of the velocity divided by the radius..

The case of a body moving in a resisting medium, and acted on by a constant force, is next investigated. If, in the equa-, tions deduced, the resistance be puto, three equations are obtained, which contain all the theory of projectiles. The motion of a body on a spherical surface is reduced to three, differential equations of the first order; thence may be had the expression for the square of the velocity, and for the differential of the time, the integral of which can only be obtained in an infinite series. When the oscillations are very small, this integral is nearly r-radius of circle, g=gravity,

semidiameter of circle, radius=1. This expression is equivalent to this; that the oscillations in small circular arcst are isochronous: but, as the isochronism of the pendulums in circular arcs are only approximate, a question naturally presents itself, concerning the curve which a body must des cribe in a resisting medium, so as to arrive in the same time at the point at which its motion ceases, whatever be the arc described from the lowest point. M. LA PLACE deduces the general equation to the tautochronous curve, which equation converts in that of the cyclofd in the two cases of an evanes cent resistance, and of a resistance varying as the velocity, agreeably to what Newton has demonstrated in his Principia. (Vol 2. prop. 26.)

Chapter III. On the Equilibrium of a System of Bodies.-The most simple case of equilibrium is that of two material points, meeting each other with equal and directly contrary velocities; their impenetrability destroys their velocity, and reduces them to a state of repose. From a principle like this, M. D'Alembert proposed to deduce all the cases relative to the impact of bodies; the cases in which the bodies are to each as number to number are easily solved; and the demonstrations are extended

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tended to incommensurable bodies, by the introduction of a proposition built on the method reductio ad absurdum. The method of M. LA PLACE is not essentially different from that of D'Alembert. His first proposition is to investigate the relation between the velocities u and u', pertaining to two systems of material points m and m', in the same right line, but moving in directly contrary directions: this relation is expressed by the equation' mu=m'u'. This proposition is made to include all bodies; all bodies are therefore supposed to be commensurable, or (according to the author's expression) to consist of an assignable number of similar material points; similar material points being those which, meeting with equal and contrary velocities, produce a mutual equilibrium.

After the conditions of the equilibrium of two systems of bodies are stated, the equation of the equilibrium is deduced; whence results the famous principle of virtual velocities invented by M. LAGRANGE, and applied by him in his Nouvelle Mecanique.

The case next considered is that in which the points of a system are invariably united together; the conditions of their equilibrium are given; the centre of gravity is investigated, &c.

In the course of this chapter, is deduced the equation Σ.mv2=c+29; which is the analytical translation of the principle of the conservation of forces vives. As this principle does not subsist in the case of a sudden variation of the motions of the system, M. LA PLACE gives a method of estimating the alteration which the force vive undergoes. The principle of the conservation of the motion of the centre of gravity, of areas, is demonstrated; each of which subsists in the sudden change of the system:-next, the system of co-ordinates, in which the sum of the areas described by the projection of the radii vectores is nothing, on two rectangular planes formed by the axes of these co-ordinates. This sum is a maximum on the third rectangular plane, and nothing on every other plane perpendicular to this.

At the conclusion of the chapter, it is shewn that E. fmvds is a minimum. In this equation, is involved the principle of the least action discovered by Maupertuis, and treated by that author on metaphysical considerations; although the principle is merely a mathematical result of the primordial laws of the equilibrium and motion of matter.

In Chap. IV. are given the general Equations of the Equili brium of Fluids.

Chap. V. On the general Principles of the Motion of a System of Bodies. The laws of the motion of a point are reduced to those of equilibrium, by decomposing its instantaneous motions into two others; one of which subsists, and the other is destroyed by the forces soliciting this point. The equilibrium

between

between these forces and the motion lost by the body gives the differential equation of its motion,-A like method may be used to determine the motion of a system of bodies m, m', m11, &c. If mP, m2, mR, be the forces soliciting m, parallel to the axis of its rectangular co-ordinates, x, y, z, and m'P', m'Q', m1R', &c. forces soliciting m', &c.—and dx, dy, dz, the variations of the directions of the forces, the equations will be of this form:

o=m, dx {

ddx

{des

dt

Chap. VI. On the Laws of Motion of a System of Bodies, in all relations mathematically possible between the Force and Velocity.

M. LA PLACE observes that there are an infinite number of ways of expressing the force by the velocity, which imply no contradiction. The most simple of all is that of the force proportional to the velocity, and observation proves it to be the law of nature. Agreeably to this law, the differential equations of the motion of a system of bodies have been determined: but the analysis employed relates to all laws mathematically possible between the force and velocity. The author designates this relation between the velocity and force, by the equation F=q(v), (v) representing any function of the velocity.

This hypothesis renders the solution of the problems extremely difficult: but from the equations may be deduced principles analogous to those of the conservation of the forces vives, of areas, of the centre of gravity, &c. The conservation of the forces vives has place in all laws mathematically possible between the force and velocity; understanding, by the force vive of a body, the product of its mass by the double of the integral of its velocity, multiplied by the differential of the function of the velocity which expresses the force; -and the principle of the least action has place, when it means that the sum of the integrals of the finite forces of the bodies of a system, respectively multiplied by the elements of their direction, is a minimum.

Chapter VII. On the Motions of a solid Body, of any figure whatever. Here we find the equations for the motions of the translation and rotation of a body; the principal axes of a body; its instantaneous axis of rotation; the oscillations of a body vibrating about one of its principal axes; the state of stability for a system of bodies; or such a state that the system deranged from it, by an infinitely small quantity, only varies from it by an infinitely small quantity, making continual oscillations about this state.

Chapter VIII. On the Motions of Fluids-As the laws of the motion of a system of bodies were deduced from those of its equilibrium, so the laws of the motions of fluids are made to depend from those of their equilibrium. M. LA PLACE deduces the equations of their motions, transforms them, and shews that they are integrable when, the density being any function of the pressure, the sum of the velocities parallel to three rectangular axes, multiplied each by the element of its direction, is an exact variation. He then applies his principles to the motion of a fluid homogenous mass, having an uniform motion of rotation round one of the axes of its co-ordinates; to the determination of the very small oscillations of a fluid homogeneous mass, covering a spheroid that has a motion of rotation; to the motion of the Sea, supposing it deranged from a state of equilibrium by the action of very small forces; to the oscillation of the Earth's atmosphere in a state of motion, regarding only the regular causes that agitate it; and to the variations which these motions produce in the heights of the barometer.

Book II. On the Law of universal Gravitation, and on the Motion of the Centres of Gravity of Heavenly Bodies..

In his Exposition du Système du Monde, M. LA PLACE stated that, in order to arrive at the comprehensive view which is now formed of the system of the World, it was necessary to observe, during a great number of ages, the phænomena of the heavens; to recognize the real motions of the Earth; to ascend to the laws of the planetary motions, and from these laws to the principle of universal gravitation; and finally to descend from these laws to the complete explication of all the heavenly phanomena, even in their minutest details. To descend from the law of gravitation to the calculation of the phænomena, is the business of physical astronomy; of which Newton is the father. Its appearance caused a most memorable epoch in the history of science.

The first book of the Principia, it is known, is conversant in the solution of geometrical and mechanical problems; and in the third book, application is made of these problems to the system of the world. Newton, in his 2d section, proves that, if equal areas in equal times be described by a body round a fixed point, the body is urged by a force tending towards that point: but Kepler, by observation, found that the planets described equal areas round the Sun: the planets, therefore, were urged by forces tending towards the Sun. Again, Newton proved (3d section) that a body, moving in an ellipse round a force in the focus, was urged by a force of which the law of variation was the inverse square of the distance: but Kepler had observed that

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