It is natural to suppose, on account of the prodigious rapidity of the vibrations of light, that the luminous particles may perform a great number of regular oscillations in each of the different mechanical situations in which they are placed during the combustion or the incandescence of the luminous body, although these circumstances may still succeed each other in extremely short periods; for the millionth part of a second is sufficient to exhibit, for example, 545 millions of undulations of yellow light; so that the mechanical disturbances, which derange the regular succession of the vibrations of the luminous particles, or which even change their nature, might be repeated a million times in a second without preventing the regular succession of more than 500 millions of consecutive undulations in each state of the particle. We shall soon have occasion to apply this observation to the determination of the circumstances in which the interference of luminous waves is capable of producing sensible effects.
We have seen that each undulation produced by an oscillatory motion was composed of two semiundulations, which [p124] occasioned in the particles of the fluids velocities exactly equal in their intensity, though opposite in the direction of the motions. Let us at first suppose that two whole undulations, moving in the same line and in the same direction, differ half an undulation in their progress: they will then be superinduced on each other through one half of their length, or of their breadth, as we should say in speaking of the waves of a liquid: but I here use in preference the term length as applied to the interval between the two points which are similarly affected by the motions of two consecutive undulations. In the supposed case of the coincidence of one half of each of the undulations, the interference will only take place with respect to the parts so coinciding: that is, to the latter half of the first undulation, and the preceding half of the second: and if these two semiundulations are of equal intensity, since they tend to give, to the same points of the ether, impulses directly opposite, they will wholly neutralise each other, and the motion will be destroyed in this part of the fluid, while it will subsist without alteration in the two other halves of the undulations. In such a case, therefore, half of the motion only would be destroyed.
If now we suppose that each of these undulations, differing in their progress by half the whole length of each, is preceded and followed by a great number of other similar undulations; then, instead of the interference of two detached undulations, we must consider the interference of two systems of waves, which may be supposed equal in their number and their intensity. Since, by the hypothesis, they differ half an undulation in their progress, the semiundulations of the one, which tend to cause in the particles of ether a motion in one direction, coincide with the semiundulations of the other, which urge them in the opposite direction, and these two forces hold each other in equilibrium, so that the motion is wholly destroyed in the whole extent of these two systems of waves, except the two extreme semiundulations, which escape from the interference. But these semiundulations will always constitute a very small part of the whole series to be considered.
This reasoning is obviously applicable to such systems only [p125] as are composed of undulations of the same length; for if the waves were longer one than the other, however small their difference might be, it would happen at last that their relative position would not be the same throughout the extent of the groups; and while the first destroyed each other almost completely, the following ones would be less in opposition, and would ultimately agree completely with each other: hence there would arise a succession of weak and strong vibrations analogous to the beatings which are produced by the coincidence of two sounds differing but little from each other in their tone; but these alternations of weaker and stronger light, succeeding each other with prodigious rapidity, would produce in the eye a continuous sensation only.
It is very probable that the impulse of a single luminous semiundulation, or even of an entire undulation, would be too weak to agitate the particles of the optic nerve, as we find that a single undulation of sound is incapable of causing motion in a body susceptible of a sympathetic vibration. It is the succession of the impulse, which, by the accumulation of the single effects, at last causes the sonorous body to oscillate in a sensible manner; in the same manner as the regular succession of the single efforts of a ringer is at last capable of raising the heaviest church bell into full swing. Applying this mechanical idea to vision, supported as it is by so many analogies, we may easily conceive that it is impossible for the two remaining semiundulations, which have been mentioned, to produce any sensible effect on the retina; and that the result of such a combination of the two systems must be the production of total darkness.
If again we suppose the second system of undulations to be again retarded half an undulation more, so as to make the difference of the progress an entire undulation, the coincidence in the motions of the two groups will be again restored, and the velocities of oscillation will conspire and be augmented in the points of superposition; the intensity of the light being then at its maximum.
Adding another semiundulation to the difference in the progress of the two systems, so as to make it an interval and [p126] a half, it is obvious that the semiundulations, superinduced on each other, will now possess opposite qualities, as in the case of the half interval first supposed: and that all the undulations must in this manner be neutralised, except the extreme three semiundulations on each side, which will be free from interference. Thus almost the whole of the motion will again be destroyed, and the combination of the two pencils of light must produce darkness, as in the case first considered.
Continuing to increase the supposed difference by the length of a semiundulation at each step, we shall have alternately complete darkness and a maximum of light, accordingly as the difference amounts to an odd or an even number of semiundulations: that is, supposing always that the systems of undulations are of equal intensity: for if the one series were less vivid than the other, they would be incapable of destroying them altogether: the velocities of the one series would be subtracted from those of the other, since they would tend to move the particles of the ether in contrary directions, but the remainders would still constitute light, though feebler than that of the strongest single pencil. Thus the second pencil would still occasion a diminution of the light: but the diminution would be the less sensible as the pencil is supposed to be weaker.
Such are the consequences of the principle of the interference of undulations, which agree perfectly, as we have seen, with the law of the mutual influence of the luminous rays which is deduced from experiment: for the results are expressed precisely in the same words, if we give the name of length of undulation to the difference of routes which had been represented by the symbol d. Admitting, therefore, as there is every reason to believe, that light consists in the undulations of a subtile fluid, the period d, after which the same effects of interference are repeated, must be the length of an undulation.
It appears from the table already given for the seven principal kinds of coloured rays, that this period d, or the length of the undulation, varies greatly, according to the [p127] colour of the light, and that for the extreme red rays, for example, it is [more than] half as great again as for the violet rays situated at the other extremity of the spectrum.