mate equations obtained by neglecting the squares of small quantities. It may be observed, however, that it appears from what has been proved, that it is a property of every plane wave which is the limit of a spherical wave, to have its mean condensation equal to zero; although there is no absurdity in conceiving a plane wave in which that is not the case as already existing, and inquiring in what manner such a wave will be propagated.

There is another way of putting the apparent contradiction arrived at in the case of spherical waves, which Professor Challis has mentioned to me, and has given me permission to publish. Conceive an elastic spherical envelope to exist in an infinite mass of air which is at rest, and conceive it to expand for a certain time, and then to come to rest again, preserving its spherical form and the position of its centre during expansion. We should apparently have a wave consisting of condensation only, without rarefaction, travelling outwards, in which case the conclusion would follow, that the quantity of matter altered with the time.

Now in this or any similar case we have a perfectly definite problem, and our equations are competent to lead to the complete solution, and so make known whether or not a wave will be propagated outwards leaving the fluid about the envelope at rest, and if such a wave be formed, whether it will consist of condensation only, or of condensation accompanied by rarefaction: that condensation will on the whole prevail is evident beforehand, because a certain portion of space which was occupied by the fluid is now occupied by the envelope.

In order to simplify as much as possible the analysis, instead of an expanding envelope, suppose that we have a sphere, of a constant radius b, at the surface of which fluid is supplied in such a manner as to produce a constant velocity V from the centre outwards, the supply lasting from the time 0 to the time T, and then ceasing. This problem is evidently just as good as the former for the purpose intended, and it has the advantage of leading to a result which may be more easily worked out. On account of the length to which the present article has already run, I am unwilling to go into the detail of the solution; I will merely indicate the process, and state the nature of the result.

Since we have no reason to suspect the existence of a function of the form F (r+at) in the value of which belongs to the

present case, we need not burden our equations with this function, but we may assume as the expression for

For we can always, if need be, fall back on the complete integral of (3); and if we find that the particular integral (7) enables us to satisfy all the conditions of the problem, we are certain that we should have arrived at the same result had we used the complete integral all along. These conditions are

...(8);

$=0 when t = 0, from r=b to r = ∞..... for must be equal to a constant, since there is neither condensation nor velocity, and that constant we are at liberty to suppose equal to zero;

(8) determines f(z) from z = b to z = ∞; (9) determines f (z) from z=b_to_z = b − aτ; and (10) determines ƒ(z) from z = b-aτ to 2=- ∞, and thus the motion is completely determined.

It appears from the result that if we consider any particular value of r there is no condensation till at = r — b, when it suddenly commences. The condensation lasts during the time T, when it is suddenly exchanged for rarefaction, which decreases indefinitely, tending to O as its limit as t tends to ∞. The sudden commencement of the condensation, and its sudden change into rarefaction, depend of course on the sudden commencement and cessation of the supply of fluid at the surface of the sphere, and have nothing to do with the object for which the problem was investigated. Since there is no isolated wave of condensation travelling outwards, the complete solution of the problem leads to no contradiction, as might have been confidently anticipated.

[From the Cambridge and Dublin Mathematical Journal, Vol. Iv. p. 1, (February, 1849.)]

ON THE PERFECT BLACKNESS OF THE CENTRAL SPOT IN NEWTON'S RINGS, AND ON THE VERIFICATION OF FRESNEL'S FORMULÆ FOR THE INTENSITIES OF REFLECTED AND REFRACTED RAYS.

WHEN Newton's rings are formed between two glasses of the same kind, the central spot in the reflected rings is observed to be perfectly black. This result is completely at variance with the theory of emissions, according to which the central spot ought to be half as bright as the brightest part of the bright rings, supposing the incident light homogeneous. On the theory of undulations, the intensity of the light reflected at the middle point depends entirely on the proportions in which light is reflected and refracted at the two surfaces of the plate of air, or other interposed medium, whatever it may be. The perfect blackness of the central spot was first explained by Poisson, in the case of a perpendicular incidence, who shewed that when the infinite series of reflections and refractions is taken into account, the expression for the intensity at the centre vanishes, the formula for the intensity of light reflected at a perpendicular incidence first given by Dr Young being assumed. Fresnel extended this conclusion to all incidences by means of a law discovered experimentally by M. Arago, that light is reflected in the same proportions at the first and second surfaces of a transparent plate*. I have thought of a very simple mode of obtaining M. Arago's law from theory, and at the same

* See Dr Lloyd's Report on Physical Optics.-Reports of the British Association, Vol. III. p. 344.

time establishing theoretically the loss of half an undulation in internal, or else in external reflection.

This method rests on what may be called the principle of reversion, a principle which may be enunciated as follows.

If any material system, in which the forces acting depend only on the positions of the particles, be in motion, if at any instant the velocities of the particles be reversed, the previous motion will be repeated in a reverse order. In other words, whatever were the positions of the particles at the time t before the instant of reversion, the same will they be at an equal interval of time t after reversion; from whence it follows that the velocities of the particles in the two cases will be equal in magnitude and opposite in direction.

Let S be the surface of separation of two media which are both transparent, homogeneous, and uncrystallized. For the present purpose S may be supposed a plane. Let A be a point in the surface S where a ray is incident along IA in the first medium. Let AR, AF be the directions of the reflected and refracted rays, . AR' the direction of the reflected ray for a ray incident along FA, and therefore also the direction of the refracted ray for a ray incident along RA. Suppose the vibrations in the incident ray to be either parallel or perpendicular to the plane of incidence. Then the vibrations in the reflected and refracted rays will be in the first case parallel and in the second case perpendicular to the plane of incidence, since everything is symmetrical with respect to that plane. The direction of vibration being determined, it remains to determine the alteration of the coefficient of vibration. Let the maximum vibration in the incident light be taken for unity, and, according to the notation employed in Airy's Tract, let the coefficient of vibration be multiplied by b for reflection and by c for refraction at the surface S, and by e for reflection and ƒ for refraction at a parallel surface separating the second medium from a third, of the same nature as the first.

Let x be measured from A negatively backwards along AI, and positively forwards along AR or AF, and let it denote the distance from A of the particle considered multiplied by the refractive index of the medium in which the particle is situated, so that it expresses an equivalent length of path in vacuum. Let A be the

length of a wave, and v the velocity of propagation in vacuum ; and for shortness sake let

Then sin X, b sin X, c sin X may be taken to represent respectively the incident, reflected, and refracted rays; and it follows from the principle of reversion, if we suppose it applicable to light, that the reflected and refracted rays reversed will produce the incident ray reversed. Now if in the reversed rays we measure x positively along AI or AR', and negatively along AR or AF, the reflected ray reversed will give rise to the rays represented by

b sin X, reflected along AI;

bc sin X, refracted along AR';

and the refracted ray reversed will give rise to

cf sin X, refracted along AI;

ce sin X, reflected along AR'.

*

The two rays along AR' superposed must destroy each other, and the two along AI must give a ray represented by sin X. We have therefore

bc+ce = 0, b2+cf=1;

* It does not at once appear whether on reversing a ray we ought or ought not to change the sign of the coefficient; but the following considerations will shew that we must leave the sign unaltered. Let the portion of a wave, in which the displacement of the ether is in the direction which is considered positive, be called the positive portion, and the remaining part the negative portion; and let the points of separation be called nodes. There are evidently two sorts of nodes: the nodes of one sort, which may be called positive nodes, being situated in front of the positive portions of the waves, and the nodes of the other sort, which may be called negative nodes, being situated behind the positive portions or in front of the negative, the terms in front and behind referring to the direction of propagation. Now when the angle X vanishes, the particle considered is in a node; and since, at the same time, the expression for the velocity of the particle is positive, the coefficient of sin X being supposed positive, the node in question is a positive node. When a ray is reversed, we must in the first instance change the sign of the coefficient, since the velocity is reversed; but since the nodes which in the direct ray were positive are negative in the reversed ray, and vice versâ, we must moreover add to the phase, which comes to the same thing as changing the sign back again. Thus we must take b2 sin X, as in the text, and not-b9 sin X, to represent the ray reflected along AI, and so in other cases.