In order to complete the bases of the general theory of diffraction, it remains for us to consider the principle of Huygens, which appears to be a rigorous consequence of the system of undulations.
The principle may be thus expressed: The vibrations of a luminous undulation, in each of its points, may be regarded as the result of the elementary motions which would be transmitted to that point, at the same instant, from all the points of the undulation, considered separately, as they existed in any one of its earlier situations.
It is a consequence of the principle of the co-existence of small motions, that the vibrations, produced at any point of an elastic fluid, by several agitations, are represented by the result of all the velocities belonging to that point at the same instant, as derived from the different centres of the undulations, combined according to the laws of motion, whatever may be the number and situation of the centres, and whatever the periods and nature of the undulations. This general principle is applicable to every particular case. We may suppose the agitations infinite in number, of the same kind, simultaneous, and taking place in contiguous points of a plane or a spherical surface: it will also be convenient to suppose the motions of the particles to take place in the same direction, perpendicular to the surface, their velocity being proportional to the condensation of the medium, and none of them retrograde in their direction. In this manner a derivative undulation will be produced by the union of these agitations, and the principle of Huygens may be truly applied to such a propagation. [This may be called a rigorous consequence of the system, but it can scarcely be considered as a proposition mathematically demonstrated: and the fundamental law of Huygens must perhaps be assumed as an axiom or a phenomenon. TR.]
The intensity of the primitive undulation being uniform throughout the surface, it results from this “theoretical” [p435] consideration, as well as from other reasoning, that the uniformity will be preserved throughout the progress of the undulation, unless any part of it be intercepted or retarded; because the result of the elementary motions, which have been mentioned, will be the same for all the points. But if a portion of the undulation be intercepted by the interposition of an opaque body, then the intensity of each part will vary according to the distance from the margin of the shadow, and these variations will be particularly sensible in the neighbourhood of the tangent rays.
Let C be the luminous point, AG the screen, and AME the wave, arrived at A, and partly intercepted by the opaque body. We may suppose it to be divided into an infinite number of small arcs, Am′, m′m, mM, Mn″, n″n′, n′n, and so forth. In order to find its intensity at the point P, belonging to any subsequent situation of the undulation, BPD, we must find the result of all the elementary agitations which each of these portions of the primitive undulation would produce there if they acted separately.
The impulse, which has been given to every part of the [p436] primitive undulation, being perpendicular to its surface, the motions of the particles of ether in this direction must be more considerable than in any other; and the rays depending on these motions, if separately considered, would be so much the weaker as they deviated the more from this direction.
The investigation of the law by which their intensity would be governed, according to their direction, as derived from any separate centre of agitation, would certainly be of very difficult investigation: but happily we are not obliged to determine this law, for it is easy to see that when the inclination to the perpendicular is considerable, the effects of the different rays must very nearly destroy each other: so that these rays, which sensibly affect the quantity of light received at each point P, may safely be regarded as being equal in intensity.
When the centre of agitation has undergone a condensation, the expansive force tends to urge the molecules in every direction; and if they do not perform a retrograde motion, it is only because their initial velocities forwards destroy those which the expansion of the condensed fluid would otherwise generate backwards: but it does not follow from this that the agitation can only be propagated in the direction of the initial velocities; for the expansive force in a perpendicular direction, for example, will combine with the primitive impulse without any diminution of its effects. It is obvious that the intensity of the undulation thus produced may vary much at the different points of its circumference, not only from the nature of the initial impulse, but also because the condensations are not subject to the same law on every side of the centre of the agitated part[?]. But the variations of the intensity of the derivative undulation must necessarily be subjected to a law of continuity, and may consequently be considered as insensible in a very small angular interval, especially in the neighbourhood of the perpendicular to the surface of the primitive undulation; for the initial velocities of the molecules, referred to any given direction, being proportional to the cosines of the angles made by that direction with the perpendicular, these results vary much [p437] more slowly than the angles themselves, while they remain inconsiderable.
If, in fact, we consider rays sensibly inclined to each other, such as EP, FP, IP, meeting in the point P, which we may suppose at the distance of a great number of breadths from the undulation EA: and if we take two arcs, EF and FI, of such a length that the differences EP−FP and FP−IP may be equal to half an undulation: on account of the marked obliquity of the rays, and of the smallness of a semiundulation, in proportion to their length, these two arcs will be almost equal, and the rays which come from them to the point P will be nearly parallel; so that on account of the difference of a semiundulation between the corresponding rays of the two arcs, their effects will mutually destroy each other.