be less than; if the arc be greater than this = 2

the expansion instead of proceeding in terms of the arc itself must proceed in terms of its supplement, and an analogous rule applies to larger arcs :

The expansion, then, with this restriction, which is perhaps not requisite, as at any rate the principal terms are those which arise from particles near P, proceeds with powers of ; and we obtain

p=A1Σ {p(r) +

q=Β,Σ {φ(r) +

λ

Suppose the function which expresses the mutual action of two particles on each other, some power of the distance, or let rø(r) = 1;

let the distance between two consecutive particles be denoted by e, and for x, y, z, dp respectively write e, en, e, eo: & being the numdx, dy, dz, ber of particles along a line through Q perpendicular to the plane of yx, and similarly of n, and ☛.

Now each part of these expressions, with the exception of the factors A

and is a numerical quantity not dependent on the nature of

the medium, except inasmuch as it requires the medium of symmetry, B

π-2 and is evidently some number: in fact it is equal to A 3 × (100,000)2 by page 28; hence the only possible mode of causing p to become large in vacuum whilst q is small, at the same time that p is not large and q not small, (I speak comparatively) in glass, will be by

-1

3

and that e-1 may be small at the same time, and vice versá, €”-3 must be negative; therefore n > 1< 3, or if n=2, all the conditions are satisVOL. VI. PART I. Ꮓ

fied: this requires the further condition, that e shall be much less in vacuum than in refracting media.

We have, then, clearly been led to the conclusion, not only that the density of the æther is greater in vacuum than in refracting media, but also that its particles act on each other with forces varying inversely as the square of the distance.

But I do not stop here; it is a remarkable fact, and one which demands particular attention, that various phenomena appear to indicate not merely that the motion is in general transversal, but that it is altogether so.

Suppose the

as

Let us consider this point a little more accurately. forces which the particles of the medium exert to be repulsive, those of air, from which arise the phenomena of sound. A series of particles constituting any vertical line being simultaneously impelled in a horizontal direction would, by virtue of their repulsion, cause a similar motion in those immediately in front of them, whilst the latter particles would tend to check the impetus of the former, and thus vibrations in the direction of transmission are simple to conceive, and easy to explain.

On the other hand were the particles attractive, no such motion would be possible, except under peculiar restrictions.

But suppose, notwithstanding, that the forces which the particles exert are attractive-Let the system of particles in a vertical line have a vertical motion, and the slightest consideration will shew us that the immediate consequence is the production of a vertical motion in the particles immediately in advance of them; whilst, as before, the reciprocal action of the latter particles tends to impede the motion of the former. Here, then, we have as clear a case as before, and our general conclusion from the whole is, that repulsive forces allow of direct, attractive, of transversal vibrations only. In the former case I would refer for a simple conception to the wind blowing over a field of corn, and to a rope held between two persons, and jerked by one of them; in the latter, as they illustrate both the mode of vibration and the action of the forces.

I do not mean, however, to consider the above reasoning as applicable to all possible arrangements of particles; it might happen, and very probably is the case, that a differently constituted medium from that which we, merely for the sake of simplicity, have imagined, would lead us to results different from the above. As my object is not now to examine every possible circumstance attending the motion, but merely to illustrate one particular view, I shall proceed on the hypothesis of the arrangement in cubical forms. We will then endeavour to ascertain, from an examination of our mathematical results, what assistance analysis affords us in the investigation of the law of transversality of vibration. I shall here assume that the law of the inverse square of the distance has been proved; and shall adopt the same notation which I applied to the investigation of that property. Suppose, then, to fix the ideas that the wave is transmitted along the axis of

y.

If v, v', v" be the velocities of transmission respectively of vibrations whose motion is parallel to x, y, z, we have

Now, from the symmetry of the medium, we have

But by reference to the preceding part of the investigation it will be found, that the forces have been considered positive when they acted in the direction in which the disturbing particle lay, and vice versá; that is, they have been considered attractive. It appears, then, that such a supposition makes v and v" possible and equal, but v'impossible, and of a different magnitude.

If, on the other hand, we had considered the forces repulsive, the factor A would have been negative, we should also have had v' possible, whilst and v" would have been impossible. Hence attractive forces give rise to transversal vibrations only, repulsive to direct vibrations only. The latter corresponds, both as to forces and vibrations, to the particles of air, the former then may be reasonably supposed true for light; and hence it follows, that from a comparison of our formula with observed facts, the forces are found to be attractive. I must, however, observe, that the equations deduced as I have obtained them will but very imperfectly apply to sound; there seems, however, a great probability that the general form will be an analogous one; and should it be found to be the same, then since all the waves of sound of different lengths travel with an equal velocity, the conclusions which we have deduced as to the forces varying inversely as the square of the distance might hold equally in air, a conclusion to which I hope shortly