We now proceed to some account of the principal mechanical con siderations connected with vibrations. If any system whatsoever be slightly disturbed from a position of stable equilibrium, every particle makes an effort to return to that position; and it can be shown that the force of restitution varies as the distance from the position of equilibrium, so that all the particles perform either simple vibrations, or motions compounded of simple vibrations. Not that it is accurately and geometrically true that the force of restitution always varies as the distance from the position of equilibrium, but only exceedingly near to it. The consequence of the restitutive force is, that the system, in returning to its position of equilibrium, acquires velocity, and the several particles pass through or near to their positions of equilibrium with their several velocities, until the force of restitution, which begins to set in a contrary direction the moment the position of equilibrium is passed, destroys the acquired velocity, and causes the particles to return. The same vibration is then repeated, or rather would be repeated if there were no retarding forces : as it is, the resistance of the air. &c., continually diminish the extent of the vibrations, until at last they become insensible. • But it can be satisfactorily shown that these resistances have no sensible tendency to alter the times of the vibrations; and few persons are aware how much of their comfort depends upon this circumstance. Whenever a sound is produced, a musical note generally accompaulen it; the sound in the consequence of the vibrations excited in the disturbed system, and the permanency of its musical pitch is the consequence of these vibrations being all made in the same time, or very nearly so. The air does not retain the vibrations communicated to it, but poem them on, so to speak ; and it is therefore an agent which communicates the successive vibrations of a disturbed body just as they are communicated to it. If the vibra tions gradually slackened in their times, as they do In their excursions from the 'fleet of the resistances, the consequence would ho that there would be no sustained notes, but every sound would be a sliding chromatic, descent, like the cry of some animals, which are therefore (smoldered very annoying neighbours; and most musical instruments would be rendered unusable.
There is a principle in mechanics which is called that of the coex. islencc of titration*, and sometimes the superposition of vibrations, which teems to be only a particular case of what might be called the coexistence or superposition of small changca of any kind. If a set of small ribrationa be given to any system, solid or fluid, the disturbance of any particle at any one Instant, arising from the united effect of the vibrations, will be the sum or difference of the disturbances arising from the several different vibrations, according as they are In the same or opposite directions. This is not strictly true In any cane, but it is very nearly true when the vibrations are small, and the more nearly so the smaller the vibrations are. For instance, two stones are dropped into water at two different places, and at a certain time, on a certain part of the surface, the resulting waves cross one another. If there bo a particle which is at the same time raised on both waves, a tenth of an inch say, from one only, and three.tentha of an inch from the other, that particle will altogether raised four-tenths of an inch, or insensibly near to it. Thus the effects of the two waves travel without any apparent interference with each other, and the eye can easily follow any one wave, even though a dozen disturbances should have been excited at the same time. A handful of small pebbles thrown into smooth water will show the coexistence of the resulting waves very satisfactorily ; and it is curious to observe bow readily the non interference of the several disturbances is seen when looked for, so readily, that it never is looked for unless the attention be specially directed to it.