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 50th and 100th 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 concave plate." 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 2o 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 20 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 between and " &c. 1 70 80 76 Dr Young does not explain the method by which he found the diameters of the rain-drops should be of an inch. 76 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. D= ‹ OTS=supplement of the angle of deviation of any ray (this angle is frequently itself called, incorrectly, the deviation), P and D, the values of and D corresponding to the minimum deviation, or the maximum of its supplement D. Let O be the centre of the sphere and origin of polar co-ordinates, let QOAT be the ray incident perpendicularly on the sphere, qPSPT any other ray emergent at S, OS= radius=r, also OST= < of incidence, =4, by property of a refracting sphere; let also < OPT=y=π−(D+0), then Op=p.sin y=r. sin ; r. sin o Now when the point P is the intersection of consecutive rays, p, and remain constant, whilst and y vary; or_tan (D+ 0) = tan 4d4D ....... But as shewn in elementary treatises on Optics, Again, substituting for sin y its value, we have This (3) with the equations (1) and (2) would suffice to eliminate p terms of and constants, if the transcendental 0 and D, and give p in relations of p, p' and did not prevent it: that expression would be the polar equation of the caustic for the primary bow: we may however trace this curve from equation (3). 4 3 Taking μ = as is usually done for red light, we find two which, as seen in elementary books, is the angle of minimum deviation, the ray at this angle being an asymptote to the caustic. = the intersections of π Фт to ф = 2 = The deviation diminishing from 0 to = consecutive rays are behind the sphere, but from the deviation increases, and the intersections are in front. So that the two branches of the caustic are as at Aa, Bb in the Fig. 2. being perpendicular to the sphere at A, and tangential to it at B, and the line cc being the ray which has the minimum deviation and an asymptote to the two branches. The section of the luminiferous surface at any position will be similar to ege', the branch e'g being an involute of the caustic Bb, and eg of the caustic Aa, or more accurately the part of eg as far as the asymptote is the involute of the virtual branch Aa, and the remainder of the other branch. For want of knowing the equation of the curve ege', I have used this approximation of considering the two branches near g for the small angular distances which we employ as coinciding with their osculating circles, at the given points. Then finding p and p, which correspond to this angular distance, and taking rectangular co-ordinates parallel and perpendicular to cc, we easily get the values of the co-ordinates (a, b) (a', b') of the centres of |