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Let us assume log a' = 2:0000000,
With these logarithms, proceeding exactly as in the calculation Art. 23, we find a = 7, and hence, as in Art. 21, y = 9, z = 10.
We might find a otherwise from formula vii. Art. 18, which may
and it is a remarkable property of this expression that it is exactly similar to the equation
where #. : and “. take the places of y, x, and a.
V. Mathematical Considerations on the Problem of the Rainbow, shewing it to belong to Physical Optics. By R. Potter, Esq. of Queens' College.
[Read Dec. 14, 1835.]
HAVING lately, in the course of my academical studies, had occasion to read more carefully the theory of the Rainbow, I was convinced of the inadequacy of the popular mode in which it is treated in elementary books (this is Sir Isaac Newton's explanation, see his Optics, p. 147). The reason which is given, why the various prismatic colours are seen, each, so brilliantly in the rainbow, is that the intensity of any colour fades away so rapidly from its maximum, that it does not prevent, in any great degree, the other colours being seen as such.
This is clearly an arbitrary assumption, which will not bear examination, for within the primary bow, the grayish light is of very considerable intensity when the display is a fine one, and in a like proportion in fainter displays. So that we should naturally expect the brilliant colours yellow and green (even laying aside the red and orange) to have still an intensity at the places of the indigo and violet, sufficient to drown the effect of colour in those weak shades.
We find that this is the fact in certain cases, as in Fog bows and similar appearances, when the size of the aqueous spheres is very small, but in other cases the violet especially is very bright. The popular theory, however, offers no reason why the size of the spherical drops should influence in any degree the colours of the bows, and it is only in referring the problem to Physical Optics, and considering the interference of the light which arises, that we understand how the size of the drops varying and determining the angular positions of the bright and dark fringes for any colour, causes the appearances to vary, by modifying the extent of the overlapping of the various colours.
The existence of the supernumerary bows furnishes a still more weighty objection to the common explanation; which supposes only one maximum of intensity for each colour, at the angle at which two consecutive rays emerge from the drop parallel to each other. But these supernumerary bows shew that the true explanation must furnish reasons for a succession of maxima and minima for each colour, and this the principle of interference does in a manner perfectly in accordance with recorded observations of the phenomena.
It was not until after I had finished the mathematical investigations, that I learned from Mr Whewell, that Dr Young had previously applied the principle of interferences to explain the supernumerary rainbows. If I had known this earlier, I probably should never have entered further into the subject; as it is, it will be found that I have shewn a method of obtaining the result, which I am not aware that he has any where given; and at any rate I shall have awakened the attention of mathematicians to this interesting phenomenon.
Dr Young's account is very concise, and insufficient as a mathematical explanation; he, however, notices the brilliancy of some of the colours being assisted by the interference of the others. At page 470 of his Lectures on Natural Philosophy, he says, “We have already seen that the light producing the ordinary rainbow is double, its intensity being only greatest at its termination where the common bow appears, while the whole light is extended much more widely. The two portions concerned in its production must divide this light into fringes; but unless almost all the drops of a shower happen to be of the same magnitude, the effects of these fringes must be confounded and destroyed; in general however they must at least co-operate more or less in producing one dark fringe, which must cut off the common rainbow much more abruptly than it would otherwise have been terminated, and consequently assist the distinctness of the colours. The magnitudes of the drops of rain, required for producing such of these rainbows as are usually observed, is between the 50" and 100" of an inch; they (i.e. the supernumerary bows) become gradually narrower as they are more remote from the common rainbows, nearly in the same proportions as the external fringes of a shadow, or the rings seen in a con
At page 643. Vol. II. in a reprint of a paper of his in the Phil. Trans. for 1803, he goes further into particulars, and says, “In order to understand the phenomenon, we have only to attend to the two portions of light which are exhibited in the common diagrams explanatory of the rainbow, regularly reflected from the posterior surface of the drop, and crossing each other in various directions, till, at the angle of greatest deviation, they coincide with each other, so as to produce, by the greater intensity of this redoubled light, the common rainbow of 41 degrees. Other parts of these two portions will quit the drop in directions parallel to each other; and these would exhibit a continued diffusion of fainter light for 25" within the bright termination which forms the rainbow, but for the general law of interference, which, as in other similar cases divides the light into concentric rings; the magnitude of these rings depending on that of the drop, according to the difference of time occupied in the passage of the two portions, which thus proceed in parallel directions to the spectator's eye, after having been differently refracted and reflected within the drop. This difference varies at first, nearly as the square of the angular distance from the primitive rainbow: and if the first additional red be at the distance of 2" from the red of the rainbow, so as to interfere a little with the primitive violet the fourth additional red will be at the distance of nearly 2" more, and the intermediate colours will occupy a space nearly equal to the original rainbow. In order to produce this effect the drops must be about ; of an inch or .013 in diameter: it would be sufficient if they were be
Dr Young does not explain the method by which he found the diameters of the rain-drops should be # of an inch.
I regret my inability to deduce results perfectly rigorous by the method which I have followed on account of the complicated and transcendental nature of the relations between the quantities to be expressed, but I have pushed the mathematical part of the investigation to as close an approximation as the general discussion of the problem may require.
I have adopted the method, of first finding the caustic; because this and a very numerous class of interferences is produced, not by a separation of the original luminiferous surface into two separate surfaces, as in the cases ordinarily considered, but by a reduplication of the surface upon itself after reflection, or refraction, or both. In these cases, as I have shewn in a paper read before the Physical Section, at the meeting of the British Association at Cambridge, and published at Brussels in M. Quetelet’s ‘Correspondance Mathematique et Physique, there is an arête de rebroussement of the luminiferous surface at the caustic surface, or, in the usual sections a cusp in the section of the luminiferous surface at the caustic, and the former curve is always an involute of the latter. Having once found the caustic, this consideration enables us to proceed to the calculation of these complicated effects with a close approximation to the accurate result.
Proceeding first to find the expressions for the caustic when parallel rays have been twice refracted and once reflected in a transparent sphere, as in the primary rainbow, and using the ordinary mode of determining the caustic by considering it the locus of the intersections of consecutive rays. Let q = z of incidence,