VI. On the Dispersion of Light, as explained by the Hypothesis of Finite Intervals. By P. KELLAND, ESQ., B.A. Fellow of Queens' College.

[Read Feb. 22, 1836.]

PRELIMINARY OBSERVATIONS.

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 com

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

for dispersion, at the same time I think that simple as they may appear at a first glance, and satisfactory as they undoubtedly are to a certain extent, it will be found a difficult task to pursue them into detail, even in the case of sound. We know little or nothing of the laws which regulate the developement of heat, which affect the velocity of light, at least if we adopt the hypothesis of molecular radiation, and have thus only shifted our difficulty without removing it. If on the other hand, we choose to regard heat as an effect consequent on the alteration of the positions of the attractive or repulsive particles within a medium (which seems reasonable from some recent experiments on the Polarization, &c. of Heat), then, by analogy, Mr Airy's hypothesis amounts in fact to supposing the particles endued with attractive or repulsive energies, influenced by the particular positions into which they are thrown, and varying with the change of these positions, to the action of which all the effects are assigned.

The great obstacle to a simple explanation of this subject appears to have arisen from the fact, that theorists generally have not divested themselves of the idea of motion directly velocity has been substituted for force, and wave for change of force.

It occurred to me about two years since, that if we could deduce a simple equation of motion on the supposition that the particles of a medium are at a finite distance from each other, we might arrive at results very different in form from those usually adopted. In fact it appeared probable that the velocity might depend on the positions into which the particles should arrange themselves, and thus might be affected by the length of a wave.

Such a formula I actually obtained, and deduced from it the necessary result, that the square of the velocity is represented by a series sin 02 of terms of which c

(sin

is a type. There was, however, one point

in my analysis which I regarded as fatal to the whole; namely, that having a function involving the distance between two consecutive particles, and the space through which a particle is disturbed, I had expanded it in terms of the ratio of the latter quantity to the former.

U 2

It appeared to me at the time doubtful whether this series might not be a diverging one, and thence it became extremely probable that the existence of the function in form above, was owing to the absence of terms omitted in this expansion. Lately M. Cauchy's Memoir on the same subject has fallen into my hands, and an opportunity has been afforded me of comparing his results with my own. The comparison has shewn me that although in some points we differ, in the essentials of principle, at least, we coincide.

Whatever difficulty may attach itself to my hypothesis as to the sphere of action of the particles, will attach itself equally to his, as they are identical: at least I have reduced mine to the same state as his.

Although there were many points of coincidence in our processes, there were not a few of difference, to many of which the present Memoir is indebted; as I have not scrupled to adopt anything which would tend either to simplify or generalize the results; my object being by no means to regard my formula as a re-discovery of what M. Cauchy had published in 1830, but rather to attempt an improvement on what is already known. I may be allowed to add, that M. Cauchy's equations, owing to his proceeding with great generality at first, and only adding new hypotheses to simplify them when they became perfectly unwieldy, are so buried in symbols, that a person must possess no ordinary sagacity to give to them any interpretation. And further, there are some points in which the result is more general than the hypothesis would render necessary.

The plan which I have pursued is to simplify the equations as I proceed, and not to retain any result which admits of reduction.

SECTION I.

Analytical Investigation.

THE problem about which we are to occupy ourselves, is the motion of any system of material particles, exerting on each other forces

varying according to any function of the distance. It would be a useless generalization, in the present state of analysis, to proceed at once to the solution of this problem without any further restrictions, for even should we succeed in integrating the resulting equations, whether by approximation or any other method, we should at length be obliged to have recourse to particular hypotheses in order to interpret our results.

I

propose then to make the following hypotheses:

1. That the distance between the particles is sufficiently large compared with their sphere of motion, to allow the square of the latter quantity to be omitted compared with that of the former.

2. That the disposition of the particles is a disposition of symmetry. It may serve to fix our ideas, if we consider them symmetrically situated with respect to the three co-ordinate planes; as, for instance, arranged in the angular summits of cubes, whose edges are parallel to the co-ordinate axes, and whose centre is the origin. This is, however, merely stated as something to guide us, since we must suppose, in whatever manner it can be accomplished, that the disposition is perfectly symmetrical. On these two hypotheses, which, virtually at least, are M. Cauchy's hypotheses, I shall now proceed to determine the equations of motion.

Let x, y, ≈ be the co-ordinates of any particle P in its state of rest, the origin being taken at pleasure, and the axes any axes of symmetry. x + dx, y + dy, ≈ + dx those of any other particle Q, which lies within the sphere of sensible attraction to P; r the distance PQ; x +a, y + ß, +y the co-ordinates of P after any time t from the beginning of the motion; x + a + dx + da, y + ß + dy + dß, ≈ + y + x + dy those of Q at the same time; r+p' the corresponding value of the distance PQ. Let the accelerating force of Q on P at the distance PQ be represented by the function (r + p'). p (r + p'). Resolving this attraction parallel to the axis of x, it gives $(r + p'). (dx + da), whence we obtain