these osculating circles. Returning thence to polar co-ordinates for the branches eg and e'g to the same pole O, we find the space (pp') between the branches at the required point. The caustic in any observation of the rainbow may be considered at the eye as coinciding with its asymptote, as a spherule of water of inch diameter subtends no sensible angle at 1000 yards distance.

70

substituting u2 - 2 (a. cos a + b. sin a) u+a2 + b2 = R2 ;

.. u = a.cos a + b. sin a ±√R2 - a2 − b2 + (a . cos a + b. sin a)2

· · pp' = u' — u = (a' − a) cos a + (b' -- b) sin a ± (R′ – R + &c.)

To establish the condition that the two branches are in contact at the cusp g, and remembering at the same time that, in an observation, the rain-drop is at a great distance, and that therefore R and R' are large quantities compared with a, b, a', b', we have

substituting u' — u = (a' — a) cos a + (b' - b) sin a- (a'-a) nearly.

Or, since A' lies on the side of the negative ; therefore a' is negative; also b> b', and they are also both negative, in our problem; therefore

u' — u = (a — a') (1 − cos a) − (b - b') sin a

α

= 2 (a - a') sin? (b − b') sin a.

2

To put this into a form for use in calculation, let

The quantities a, a, b, b, are to be calculated from the values found for P and corresponding to any particular value of a, which I have taken, for example, as the angular distance from the red to the purple;

.. a = 1°. 46', we have also D= Dm - a = 40°. 16'.

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 and 0, I find

ρ

and that the second maximum of the red may occur at the place of the first violet, we must have, if λ be the interval of the luminiferous surfaces for red light,

1

76'

which does not differ greatly from Dr Young's result for I have

1 th 1000

We see also that if r were very small, as in mist and in ordinary clouds not producing rain, probably much less than of an 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.

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 another."

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.

4 3

for red rays,

I have also taken, as ordinarily is done, that μ = 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 λ, 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.