to establish the law in question, afforded strong presumptive evidence in its favour; not confined to the action of the particles of ether, but extending to those of air, and giving normal vibrations in the latter instance as the cause of the phenomena of sound.

All the investigations were, however, confined to a perfectly symmetrical medium, on which account the results were limited to noncrystallized substances.

My object at present is to complete the view I have taken of the subject, by extending analogous artifices of simplification to particles arranged not in a perfectly symmetrical manner, but symmetrical only with respect to three planes at right angles to one another.

In entering on this subject, I must remind you that I take for granted the law of the inverse square of the distance as established; and the novelty which is presented by the present view of the subject arises from the difference in the form of the force corresponding to a disturbance in the normal direction, from that put in play by a disturbance in the transverse direction.

I have limited my operations to one series of particles, from the circumstance that the form is not altered by introducing another series, provided the latter act on the former, and are themselves subject to the action of the former. The results arising from the combination of two sets, I have proved to be the sums or differences of the results arising from each set respectively.

It is true that the action of material particles has been totally omitted, the material particles being supposed to exert on those of ether an influence by which they themselves are not reciprocally affected. My reason for this omission is, that such influence will not affect the motion in a non-crystallized medium, (see Trans. Camb. Phil. Soc. Vol. VI. p. 244.) and, consequently, will not materially affect it in a crystallized one. The charge which has lately been brought against the hypothesis which M. Cauchy and others have adopted, is, that it omits altogether the action of the particles of matter.

Now I conceive that this is by no means a fair charge, for if the material particles themselves vibrate, we have two systems of vibrating particles, the combined motion of which has been considered, and if they do not vibrate, they produce no effect.

As regards the law of force, a Memoir has lately been circulated, in which M. Cauchy arrives at the conclusion that it is the inverse fourth power of the distance. Adopting this law, Professor Lloyd has proved that the vibrations are transversal, in a paper read before the Irish Academy, in November last. In a short abstract of that paper, it is stated that the object of the Author has been simplification, and the mode of accomplishing that object is given. This mode is precisely that which I adopted, and some of the conclusions are apparently the same; as for instance, that the vibrations are transversal. This conclusion is stated as follows: "When this law of force (the inverse fourth power) is substituted in the corresponding relation for the normal vibration, the velocity of propagation is infinite; so that the normal disturbance is propagated instantaneously, and gives rise to no wave.”

I do not think from this statement that the grounds on which the law of the inverse square stands, are less tenable than those which lead to the inverse fourth power, and shall not therefore consider it incumbent on me to change my views with respect to the law.

I have dwelt at considerable length on this point, as it is of essential importance to all my succeeding investigations that the law of the inverse square of the distance be not set aside; and I think it will be allowed, that as far as the above speculations are concerned, that of the inverse fourth power does not appear to be established.

In attempting to offer any investigations connected with the transmission of light through crystals, we are naturally prompted to recur, as to the established theory, to those of M. Fresnel, which stand prominent as an example of clearness of conception and distinctness of explanation. The agreement of the results with those of observation, the remarkable predictions which they have afforded of phenomena which have fully verified those predictions, the simplicity with which

they explain a multitude of various and complex phenomena, have stamped them with a character so firm that it would be presumptuous to attempt to set them aside. Truth however compels me to state, that whilst I feel the highest admiration of M. Fresnel's theory, I am at the same time doubtful whether some of the points on which it rests are not defective, at least as commonly stated. I allude only to the mechanical part of it; nothing can be more complete or more elegant than the geometrical part. I trust I shall not be understood in anything which follows as endeavouring in the slightest to detract from M. Fresnel's fame. I mean far otherwise; but having advanced the opinion that some parts of the mechanical theory are inaccurate, it becomes incumbent on me to explain in what manner this inaccuracy is introduced, and how it happens that from imperfect premises accurate conclusions have been deduced.

It shall be my endeavour then to point out, as clearly as I am able, the circumstances in which the theory labours under a difficulty, and then to shew the cause of this difficulty.

SECTION I.

Remarks on M. Fresnel's Theory.

M. FRESNEL in his Memoir on double refraction, p. 103, states the principle, that "the elasticity put in play by luminous vibrations depends solely on their direction and not on that of the waves." Of this principle he demonstrates, in a very satisfactory manner, the theoretic possibility, and there appears little room to doubt its truth. Taking it for granted then, he proceeds (p. 106.) to an application of it in the following manner.

If we have two displacements corresponding to different waves, we may consider each of them as belonging to a new wave, the front of which is the plane passing through them, and shall, if we wish to combine the two, have only to combine two vibrations in the front of a common wave. Thus far, I think, there can be no ground for the slightest objection. But the statement in p. 107 cannot, I think, lay claim to the same degree of evidence as this.

It would occupy too much space to give here the whole of this statement. It will be quite sufficient to give an abstract of it, which I copy from Professor Airy's Tracts, p. 343.

"If the displacement of a particle considered in any direction be resolved into three displacements in the directions of x, y, z, the variations of force in those directions produced by the alteration of a single particle (and consequently the force produced by the whole system) are the same as if the displacements in those directions had been made independently. From this it easily follows that the sum of any number of displacements causes forces equal to the sum of the forces corresponding to the separate displacements: and then any number of undulations, produced by vibrations in different directions, may coexist without destroying each other." It will be seen that this statement supposes the force put in play to depend only on the

displacement, and not at all on the position of the front of the wave. If, indeed, it could restrict the hypothesis by adding that the force put in play by a displacement in any direction in the front of a wave is independent of the position of that front, or remains constant whilst that front is made to revolve about the line of displacement, it would coincide with what M. Fresnel had established above.

It will be seen then that I object not to the supposition that the force put in play is independent of the position of the plane of the wave, but to the converse, that if a displacement be resolved parallel to x, y, z, the forces put in play will be the same as if the wave was in each individual case perpendicular to y, z, xy respectively.

It is clear that such an hypothesis takes for granted, what I should not think Fresnel could mean, that the force on each particle in any direction is of the same form as if that particle alone were in motion in that direction.

It will not suffice to urge in answer to these objections, that however they might apply to motion in general, in the particular instance of vibrations the nature of the arrangement is such as to render them invalid. On no hypothesis, that I can conceive, would the force due to a displacement in the direction of transmission be the same as that in a perpendicular direction, and when the law of force is that of the inverse square of the distance, far from being identical, they are of a directly opposite character; and the effects which they produce, instead of being analogous, are totally different even in form; the one being an oscillation, the other a progression. But this is not the only objection which I adduce. M. Fresnel determines by his construction the two directions in which a vibration taking place the law of transmission is satisfied. He finds the force in each of these directions, by supposing the whole force put in play to be resolved into two; one in the direction itself, and the other, perpendicular to the front of the wave. The former of these he assumes as the force which produces vibration, the latter he omits altogether.