It is clear that the time required for the body's descent. from A to B is equal to that necessary for the ascent from B to A, so that the whole time required to complete the ascent and descent is equal to T, and is represented by the formula

Now this expression does not contain S,, i. e. the distance from which the body falls to B; the time T is the same therefore, whatever that distance may be.

THE SIMPLE PENDULUM.

98. If a heavy particle P be imagined to be suspended from a point C by a thread without weight, and allowed to oscillate freely, but so as to deviate but little on either side of the vertical, then will its oscillations, so long as they are thus small, be performed in the same time whatever their amplitudes may be.

For let the inclination PCB of CP to the vertical be represented by, and let the weight w of the particle P, which acts in the direction of the vertical VP, be resolved into two others, one of which is in the direction CP, and the other perpendicular to that direction; the former will be wholly counteracted

by the tension of the thread CP, and the latter will be represented by w sin. VPC-w sin. ; and, acting in the direction in which the particle P moves, this will be the whole impressed moving force upon it (Art. 92.). Now so long as the arc is small, this arc does not differ sensibly from its sine, so that for small oscillations the impressed moving force

upon P is represented by we, or by w(), or by w, if/repre

sent the length CP of the suspending thread, and S the length

H

of the arc BP. Now in this expression w and I are constant throughout the oscillation, the moving force varies therefore as S. Hence by the last proposition, the small oscillations on either side of CB are isochronous, since so long as they are thus small, the impressed moving force in the direction of the motion varies as the length of the path BP from the lowest point B. Since in the last proposition the moving force was assumed equal to cS, and that here it is represented by

A single particle thus suspended by a thread without weight, is that which is meant by a SIMPLE PENDULUM. It is evident that the time of oscillation increases with the length I of the pendulum.

IMPULSIVE FORCE.

99. If any number of different moving forces be applied to as many equal bodies, the velocities communicated to them in the same exceedingly small interval of time, will be to one another as the moving forces. For let P1, P2, represent the moving forces, and fi, f, the additional velocities they would communicate per second if each moving force remained continually of the same magnitude (Art. 93.), then would tf,, tf, be the whole velocities communicated on this supposition in t seconds; let these be represented by V1, V2; therefore by Art. 94.

2

2

The proposition is therefore true on the supposition that P, and P, remain constant during the interval of time t; but if t be exceedingly small, then whatever the pressures P, and P, may be, they may be considered to remain the same during

2

that time. Therefore the proposition is true generally, when, as above, the moving forces are supposed to act on equal bodies, or successively on the same body, through equal exceedingly small intervals of time.

Moving forces thus acting through exceedingly small intervals of time only, are called IMPULSIVE FORCES.

THE PARALLELOGRAM OF MOTION.

100. If two impulsive forces P1, P2, whose directions are AB and AC, be impressed at the same time upon a

body at A, which if made to act upon it separately would cause it to move through AB and AC in the same given time, then will the body be made, by the simultaneous action of these impulsive forces, to describe in that time the diagonal AD of the parallelogram, of which AB and AC are adjacent sides.

For the moving forces P, and P, acting separately upon the same body through equal infinitely small times, communicate to it velocities which are (Art. 99.) as those forces, therefore the spaces AB and AC described with these velocities in any given time are also as those forces. Since then AB and AC are to one another as the pressures P1 and P2, therefore by the principle (Art. 2.) of the parallelogram of pressures, the resultant R of P, and P, is in the direction. of the diagonal AD, and bears the same proportion to P1 and P, that AD does to AB and AC.

2

2

Therefore the velocity which the resultant R of P1 and P2 would communicate to the body in any exceedingly small time is to the velocities which P, and P, would separately communicate to it in the same time as AD to AB and AC (Art. 99.), and therefore the spaces which the body would de- scribe uniformly with these three velocities in any equal times are in the ratio of these three lines. But AB and AC are the spaces actually described in the equal times by reason of the impulses of P, and P. Therefore AD is the space de

scribed in that time by reason of the impulse of R, that is, by reason of the simultaneous impulses of P, and P2.

101. THE INDEPENDENCE OF SIMULTANEOUS MOTIONS.

It is evident that if the body starting from A had been made to describe AB in a given time, and then had been made in an equal time to describe BD, it would have arrived precisely at the same point D to which the simultaneous motions AC and AB have brought it, so that the body is made to move by these simultaneous motions precisely to the same point to which it would have been brought by those motions, communicated to it successively, but in half the time. The following may be taken as an illustration of this principle of the independence of simultaneous motions. Let a canal-boat be imagined to

extend across the whole width of the

canal, and let it be supposed that a person standing on the one bank at A is desirous to pass to a point D on the opposite bank, and that for this purpose, as the boat passes him, he steps into it, and walks across it in the direction AB, arriving at the point B in the boat precisely at the instant when the motion of the boat has carried it through BD; it is clear that he will be brought, by the joint effect of his own motion across the boat and the boat's motion along the canal, to the point D (having in reality described the diagonal AD), which point he would have reached in double the time if he had walked across a bridge from A to B in the same time that it took him to walk across the boat, and had then in an equal time walked from B to D along the opposite side.

THE POLYGON OF MOTION.

102. Let any number of impulses be communicated simul

taneously to a body at O, one of which would cause it to move from A to O in a given time, another from B to O in the same time, a third from C to O in that time, and a fourth from D to O. Complete the parallelogram of which AO and BO are adjacent sides; then the impulses AO and BO would simultaneously cause the body to move from E to through the diagonal EO in the time spoken of. Complete the parallelogram EOCF, and draw its diagonal OF, then would the impulses EO and CO, acting simultaneously, cause the body to move through FO in the given time: but the impulse EO produces the same effect on the body as the impulses AO and BO; therefore the impulses AO, BO, and CO, will together cause the body to move through FO in the given time. In the same manner it may be shown that the impulses AO, BO, CO, and DO, will together cause the body to move through GO in a time equal to that occupied by the body's motion through any one of these lines.

It will be observed that GD is the side which completes the polygon OAEFG, whose other sides OA, AE, EF, FG, are respectively equal and parallel to the directions OA, OB, OC, and OD, of the simultaneous impulses.

Instead of the impulses AO, &c. taking place simultaneously, if they had been received successively, the body moving first from O to A in a given time; then through AE, which is equal and parallel to OB, in an equal time; then through EF, which is equal and parallel to OC, in that time; and lastly through FG, which is equal and parallel to OD, in that time, it would have arrived at the same point G, to which these impulses have brought it simultaneously, but after a period as many times greater as there are motions, so that the principle of the independence of simultaneous motions obtains, however great may be the number of such motions.

THE PRINCIPLE OF D'ALEMBERT.

103. Let W1, W2, W3, &c. represent the weights of any