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Finding the two values of p, the one above and the other below the angle of minimum deviation for this value of a, and then deducing those corresponding for p and 6, I find
Applying these to the above expression, we find u' – u =.0036783r,
and that the second maximum of the red may occur at the place of the first violet, we must have, if X be the interval of the luminiferous surfaces for red light,
which does not differ greatly from Dr Young's result #. for I have taken a = 1°. 46, and he has taken it = 2".
We see also that if r were very small, as in mist and in ordi
inch, then the primary red would extend far beyond the violet's place, and so likewise with the other colours, and we ought to expect a bow with colour scarcely perceptible, and such is recorded as the fact.
nary clouds not producing rain, probably much less than of an
In the case calculated the primary purple mixing with the second red would give a reddish purple, which agrees with an observation I made on a very splendid display on the 5th of June 1834, immediately after a heavy thunder-shower. I saw three sets of purples at the summit of the bow, and I have this memorandum : ‘but the purple of the principal bow was evidently mingled with the red of the second, and I believed the purple of the second also to be mixed with the red, and perhaps the orange also of the third bow. The three bows thus considered were also of decreasing breadths as fringes in diffraction.’
But an observation by Dr Langwith, quoted by Dr Young, proceeds much more into details than the one I made as above, and the display must have been still more splendid. He says, “You see we had here four orders of colours, and perhaps the beginning of a fifth : for I make no question but that what I call the purple, is a mixture of the purple of each of the upper series with the red of the next below it and the green a mixture of the intermediate colours.”
Again he has this important and philosophical remark: “There are two things which well deserve to be taken notice of, as they may perhaps direct us, in some measure, to the solution of this curious phenomenon. The first is, that the breadth of the first series so far exceeded that of any of the rest, that as near as I can judge, it was equal to them all taken together. The second is that I have never observed these inner orders of colours in the lower parts of the rainbow, though they have often been incomparably more vivid than the upper parts, under which the colours have appeared. I have taken notice of this so very often, that I can hardly look upon it to be accidental; and if it should prove true in general, it will bring the disquisition into a narrow compass; for it will shew that this effect depends upon some property which the drops retain, whilst they are in the upper part of the air, but lose as they come lower, and are more mixed with one
The first question, as to the decreasing breadths, is answered by Dr Young in the previous quotation, and is an effect familiar to those who have studied Physical Optics. The second also receives a complete solution from considering the expression we have obtained, and the state of rain in falling from a cloud. For though the drops were of small size on leaving the cloud, and such as to produce the supernumerary bows, yet as they fall down, having different velocities from
the higher and lower parts of the cloud, they must come in contact, and gradually form large drops, and thus their diameters become at length too great to give an appearance of supernumerary bows. There are other points still, however, which theory will guide us to look for in future; thus if the drops are larger, the second maximum of the red may happen in the green's place, and thus the green be diluted with white light whilst the orange and yellow would be brilliant, but the second maxima of these latter falling in the blue and purple, these colours would again be diluted. In such bows the red, orange and yellow, would form the most striking part. I am not aware that there are any recorded observations relating to this or similar effects.
If we can judge by observation where the second series of maxima commence, we shall be able to calculate the size of the drops forming the bow.
There are observations on record of supernumerary bows attending the secondary rainbow : their solution is perfectly similar to the one given for the primary one.
The comparison of the results of interference with the common explanation of the rainbow, required that the plan followed should be in accordance with the undulatory theory of light. If the effects were considered to be those due to a difference of , an interval in the paths of the rays at the cusp, the results would be similar, only modified a little in quantity.
I have also taken, as ordinarily is done, that u = } for red rays,
although Fraunhofer's observations shew it to belong to the letter D nearly, and the middle of the orange.
Again, I have taken the interval A, as given from Sir Isaac Newton's measures, although unpublished measures of my own confirm those of M. Fresnel in shewing that they are somewhat too small.
VI. On the Dispersion of Light, as eaglained by the Hypothesis of Finite Intervals. By P. KELLAND, Esq., B.A. Fellow of Queens' College.
[Read Feb. 22, 1836.]
THERE is no phenomenon in Optics more familiar and prominent than that a beam of solar light is composed of differently coloured rays, each endued with its own peculiar properties.
It was first satisfactorily proved by Newton, that the parts are distinct from each other, and are susceptible of separation and recomposition, so that any particular colour can be examined apart from the rest. At a very recent period Wollaston and Fraunhofer have examined more intimately the constitution of a beam of ordinary white light, and from the accurate measures of the latter, we are put in possession of a series of data by which, in a variety of substances, the position of each particular portion of the beam is accurately defined. Having then before us such observations, we are in a state to proceed to an explanation not merely of facts broadly and generally stated, but of the minutest details, and most trivial deviations from the rough outline.
It might perhaps be more easy to proceed on the hypothesis which Newton himself advanced, as it would be a matter of little difficulty to assign such forces or inertia to the particles of light, combined with the constant attractive or repulsive forces of the material particles comVol. VI. PART I. U
posing a refracting substance, as should lead to results in unison with those of observation. There are, however, a variety of complex phenomena, to which scarcely any modification of Newton's hypothesis will apply, whilst that of undulations accounts for them in the clearest and most satisfactory manner. The phenomena of dispersion for a considerable time stood almost alone in the way of this theory, and appeared incompatible with its principles. It was assumed, and with good reason, that colour was dependent on the lengths of a wave, whilst the velocity of transmission determined the refractive index of the medium. It became then evident that the theory was at fault, unless the velocity of transmission within refracting media could be shewn to depend on the length of a wave. What was still worse, from the appearance of the stars we were forced to allow, that light of all colours was transmitted uniformly through vacuum.
Several suggestions were made, which, if they did not remove the difficulty, tended at least to clear the theory from suspicion of incapability, and to shift the ground of attack from the principles themselves to our power in applying them. Thus Mr Airy, reasoning from analogy, observes: “We have every reason to think that a part of the velocity of sound depends on the circumstance that the law of elasticity of the air is altered by the instantaneous developement of latent heat on compression, or the contrary effect on expansion. Now if this heat required time for its developement, the quantity of heat developed would depend on the time during which the particles remained in nearly the same relative state; that is, on the time of vibration. Consequently the law of elasticity would be different for different times of vibration, or for different lengths of waves: and therefore the velocity of transmission would be different for waves of different lengths. If we suppose some cause, which is put in action by the vibration of the particles, to affect in a similar manner the elasticity of the medium of light, and if we conclude the degree of developement of that cause to depend on time, we shall have a sufficient explanation of the unequal refrangibility of differently coloured rays.”
These observations are important, inasmuch as they remove from the Undulatory Theory the imputation of being inadequate to account