VELOCITY OF.PROPAGATION.

33

in the one direction, or forwards, shall correspond with the movements of the second in the opposite direction, or backwards, they will become mutually neutralised, provided they are of equal intensity, and the molecule of the ethereous fluid will continue in repose. This result always takes place, whatever in other respects may be the direction of the oscillatory movement, in reference to the direction in which the waves are propagated, provided that it be equal in the two systems of waves. Thus, for example, in the undulations which are formed on the surface of a liquid, the oscillation is vertical, while the waves are propagated horizontally, and of consequence, in a direction perpendicular to the first. In the waves of sound or light, on the contrary, the oscillatory movement is parallel to the direction of the propagation. And these waves, like the liquid ones, are subject to the above stated law of interference.

We have described in a general manner the waves which may be formed in the interior of a fluid mass. To acquire a precise idea of their mode of propagation, we must consider, that when the fluid has every way the same density and the same elasticity, the vibration produced in one point, must be propagated on all sides with the same velocity; for this velocity of propagation (which must not be confounded with the absolute progressive velocity of the molecules) depends entirely on the density and elasticity of the fluid. It thence follows that all the points that have been made to vibrate at the same instant, will be distributed over a spherical

G

surface, having for its centre the origin of the disturbance. Thus these waves are spherical, whilst those observed at the surface of a liquid are simply circular.

The straight lines drawn from the vibration to the different points of this spherical surface are popularly called rays. These denote merely the directions in which the movement is propagated. This is what is meant both by sonorous rays or rays of sound in acoustics; and by luminous rays, in the system which ascribes the production of light to the vibrations of an universal fluid, styled ether.

The nature of the different elementary movements of which each wave is composed, depends on the nature of the different movements which compose the primary vibration. The most simple hypothesis we can make concerning the formation of luminous waves, is that the small oscillations of the particles of bodies which produce them, are analogous to those of a pendulum, slightly removed from its point of repose; for we are to conceive the particles of bodies, not as immoveably fixed in the positions which they occupy, but as suspended between forces which mutually balance each other in all directions. Now, whatever be the nature of the equal forces, which maintain the particles in this situation, so long as these particles are removed from their position of equilibrium, only by a very small quantity, relative to the sphere of activity of these forces, the accelerative power which tends to bring them back to it, and which thereby makes them oscillate backwards and forwards from the

ADVANCING AND RECEDING PULSES.

85

point of equilibrium, may be regarded as sensibly proportional to the recession. This phenomenon falls precisely under the law of the minute oscillations of the pendulum, and in general of all kinds of small oscillations.

Let us imagine, hung up in the elastic fluid, a little solid plane, which having been removed from its primitive position is brought back to it, by a force proportional to the displacement. At the beginning of its movement, the accelerative force can impress upon it only an infinitely small velocity; but its action continuing, the effects accumulate, and the velocity of the solid plane progressively increases, till the moment of its arrival at the position of equilibrium, where it would remain were it not for its acquired velocity; but in virtue of this velocity, it goes beyond the point of equilibrium. The same force which tends to recall it, and which now acts in a direction contrary to the acquired movement, diminishes continually the velocity till it be reduced to zero. Its elastic power continuing to act produces a velocity in a contrary direction, which restores the particle to its position of equilibrium. This velocity nearly null at the beginning of the return, grows greater by the same degrees as it had diminished, till the instant when the particle arrives at the point of equilibrium, which it passes in consequence of the movement acquired. On departing, however, from this point, the movement diminishes incessantly by the effect of the force which tends to restore the particle to it; and its velocity is once more reduced to zero, as soon

as it reaches its point of departure. It then recommences, with the same stages, the movements just described, and would thus continue to oscillate indefinitely like a pendulum, but for the resistance of the surrounding fluid, whose inertia progressively lessens the amplitude of its oscillations, and eventually extinguishes them altogether, at the expiration of a longer or a shorter time.

To fix our ideas, let us assume the instant when the solid plane has returned to its point of departure, after having executed two oscillations in opposite directions. Then the velocity which it had at the first moment, and which was sensibly null, is at the instant under consideration, transmitted to a section of the fluid distant from the centre of disturbance, by a quantity which we shall represent by d. Immediately, thereafter, the velocity of the solid plane, which has augmented a little, is communicated to the section in contact; from this, it passes successively through all the following sections, and at the moment when the first disturbance arrives at the section situated at the distance d, the second arrives at the immediately preceding section. Continuing thus to divide mentally, the duration of the two oscillations of the solid plane, into an infinity of small intervals of time, and the fluid comprised in the length d, into a like number of corresponding sections infinitely thin, it is easy to see, by the same reasoning, that the dif ferent velocities of the moveable plane at each of these instants is now distributed in the corresponding sections; and that thus, for example, the velo

VIBRATORY DIFFUSION OF LIGHT.

37

city with which the solid plane was animated in the middle of the first oscillation, (equal, tod from the starting point,) must have arrived at the instant under consideration at the distanced. It is therefore the stratum placed at this distance, which is now actuated with the maximum of velocity in advance. In like manner when the plane has arrived at the limit of its first oscillation, its velocity was null, and this absence of motion must coexist in the section situated at the distanced, d being the whole breadth of the waves percurred during a forward and backward oscillation of the above imaginary plane. By its second oscillation, the plane retracing its steps, must give to the section of fluid in contact, and successively to the others, movements contrary to those of the first oscillation; for when the plane retires, the section in contact pressed against this plane by the elasticity or the expansive force of the fluid, necessarily follows it, and fills up the void which its retrograde motion tends to produce. For the same reason, the succeeding section, is drawn towards the first, the third towards the second, and thus in succession. This is the manner in which the retrograde movement is communicated from point to point, till it reaches the most distant sections. Its propa

gation takes place according to the same law, as that of the forward movement. There is no difference except in the direction, or in mathematical language, in the sign of the velocities which they impress on the molecules of the fluid. The extent of fluid, moved by two oscillations of the solid