Responsionem Dn. Catelani, num. 27. Ephem. Gallic. anni 1684, insertam." Let m, m', denote two equal bodies attached to an inflexible straight line which is capable of motion in a vertical plane about one extremity which is fixed; let r, r', denote the distances of m, m', respectively, from the fixed extremity; v, v', their velocities for any position of the inflexible line in its descent from an assigned position; u, u', the velocities which they would have acquired by descending down the same arcs unconnectedly. Then, in consequence of the connection of the bodies, a velocity uv will be lost by m and a velocity v' - u' gained by m' in their 'descent. Bernoulli proposes it to the consideration of mathematicians whether, according to the statical relation of two forces in equilibrium on a lever, the proportion u-v: v'u'r'r be an accurate expression of the circumstances of the motion. This idea of Bernoulli's, although not free from error, contains however the first germ of the Principle of reducing the determination of the motions of material systems to the solution of statical problems. L'Hôpital, in a letter addressed to Huyghens', correctly observed that instead of considering the velocities acquired in a finite time, he should have considered the infinitesimal velocities acquired in an instant of time, and have compared them with those which gravity tends to impress upon the bodies during the same instant. He takes a complex pendulum, consisting of any two bodies attached to an inflexible straight line, and considers equilibrium to subsist between the quantities of motion lost and gained by these bodies in any instant of time, that is, between the differences of the quantities of motion which the bodies really acquire in this instant, and those which gravity tends to impress on them. He applies this Principle, which agrees with the general Principle of D'Alembert, to the determination of the Centre of Oscillation of a pendulum consisting of two bodies attached to an inflexible straight line oscillating about one extremity. He then extends his theory to a greater number of bodies in a straight line, and determines their Centre of Oscillation on the supposition, the truth of which is not however sufficiently obvious without demonstration, that

1 Histoire des Ouvrages des Sçavans, 1699, Juin, p. 410.

any two of them may be collected at their particular Centre of Oscillation. On the publication of L'Hôpital's letter, James Bernoulli' reverted to the subject of the Centre of Oscillation, and at length succeeded in obtaining a direct and rigorous solution of the problem in the case where all the bodies are in one line, by the application of the principle laid down by L'Hôpital. Bernoulli' afterwards extended his method to the general case of the oscillations of bodies of any figure.

6

An ingenious investigation of the Centre of Oscillation, a problem from the beginning intimately connected with the deve lopment of D'Alembert's Principle, was shortly afterwards given by Brook Taylor3 and John Bernoulli', between whom arose an angry controversy respecting priority of discovery; the method given by these mathematicians, although depending upon the statical principles of the lever, did not however involve, in an explicit form, L'Hôpital's Principle of Equilibrium. Finally, Hermann determined the Centre of Oscillation by the principle of the statical equivalence of the solicitations of gravity, and the vicarious solicitations applied in opposite directions, or, as it is expressed by modern mathematicians, by the equilibrium subsisting between the impressed forces of gravity and the effective forces applied in opposite directions; a method of investigation virtually coincident with that given by James Bernoulli. The idea of L'Hôpital became still more general in the hands of Euler', in a memoir on the determination of the oscillations of flexible strings printed in the year 1740. From the above historical sketch it will be easily seen that in the enunciation of a general Principle of Motion, Fontaine and D'Alembert had little more to do than to express in general language what had been distinctly conceived in the prosecution of particular re

1 Acta Erudit. Lips. 1691, Jul. p. 317, Opera, Tom. 1. p. 460.

2 Mémoires de l'Académie des Sciences de Paris, 1703, 1704.

3 Philosophical Transactions, 1714, May. Methodus Incrementorum.

4 Acta Erudit. Lips. 1714, Jun. p. 257; Mém. Acad. Par. 1714, p. 208. Opera, Tom. II. p. 168.

5 Act. Erudit. Lips. 1716, 1718, 1719, 1721, 1722.

6 Phoronomia; Lib. 1. cap. 5.

7 Comment. Petrop. Tom. VII.

searches by L'Hôpital, James and John Bernoulli, Brook Taylor, Hermann, and Euler. For additional information on the historical development of D'Alembert's Principle, the reader is referred to Lagrange's Mécanique Analytique, Seconde Partie, Section 1; Montucla's Histoire des Mathématiques, part. v. liv. 3, part. IV. liv. 7; and Whewell's History of the Inductive Sciences, Vol. II. In modern treatises on Mechanics, D'Alembert's Principle is expressed under one or other of the following forms:

(1) When any material system is in motion under the action of any forces, the moving forces lost by the different molecules of the system must be in equlibrium.

(2) If the effective moving forces of the several particles of a system be applied to them in directions opposite to those in which they act; they will, conjointly with the impressed moving forces, constitute a system of forces statically disposed.

The former of these enunciations it will be seen is substantially the same as that given by D'Alembert, while the latter is a generalization of the idea developed by Hermann in his investigations on the particular problem of the Centre of Oscillation.

SECT. 1. Motion of a single Particle1.

The object of this section is to apply D'Alembert's Principle to the exemplification of a general method for the determination of the motion of a particle within tubes and between contiguous surfaces, of which either the position, or the form, or both, are made to vary according to any assigned law whatever, the particle being acted on by given forces. Several of the problems of this section have been solved by particular methods in Chapter IV.

I. We will commence with the consideration of the motion of a particle along a tube, and, for the sake of perfect generality, we will suppose the tube to be one of double curvature. The tube is considered in all cases to be indefinitely narrow and

1 The substance of this Section was published in the Cambridge Mathematical Journal, Vol. III. p. 49.

perfectly smooth, and every section at right angles to its axis to be circular.

Let the particle be referred to three fixed rectangular axes, and let x, y, z, be its co-ordinates at any time t; let x, y, z, become x+8x, y+dy, z+dz, when t becomes t+St; St, and consequently dx, Sy, dz, being considered to be indefinitely small. Then the effective accelerating forces on the particle parallel to the three fixed axes will be, at the time t,

Also, let X, Y, Z, represent the impressed accelerating forces on the particle resolved parallel to the axes of x, y, z; and let x+dx, y+dy, z+dz, be the co-ordinates of a point in the tube very near to the point x, y, z, which the particle occupies at the time t. Then, observing that the action of the tube on the particle is always at right angles to its axis at every point and therefore, at the time t, to the line joining the two points x, y, z, and x+dx, y+dy, z+dz, we have, by D'Alembert's Principle, combined with the Principle of Virtual Velocities,

Again, since the form and position of the tube are supposed to vary according to some assigned law, it is clear that, when t is known, the equations to the tube must be known; hence it is evident that, in addition to the equation (A), we shall have, from the particular conditions of each individual problem, a number of equations equivalent to two of the form

$ (x, y, z, t) = 0,

x(x, y, z, t) = 0.........(B);

where and x are symbols of functionality depending upon the law of the variations of the form and position of the tube.

The three equations (A) and (B) involve the four quantities x, y, z, t, and therefore, in any particular case, if the difficulty of the analytical processes be not insuperable, we may ascertain x, y, z, each of them in terms of. t; in which consists the complete solution of the problem.

If the tube remain during the whole of the motion within one.

plane, then, the plane of x, y, being so chosen as to coincide with this plane, the three equations (A) and (B) will evidently be reduced to the two

We proceed to illustrate the general formulæ of the motion by the discussion of a few problems.

(1) A rectilinear tube revolves with a uniform angular velocity about one extremity in a horizontal plane: to find the motion of a particle within the tube.

Let o be the constant angular velocity; r the distance of the particle at any time t from the fixed extremity of the tube; then, the plane of x, y, being taken horizontal, and the origin of co-ordinates at the fixed extremity of the tube, we shall have, supposing the tube initially to coincide with the axis of x,

Substituting in the general formula (C) the values which we

8x Sy

have obtained for da, dy, St

St, we have, since X=0, Y=0,