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Fig. 3.

lie near some surface G G' which moves forward with the group velocity. I have shown, in a paper communicated to the recent meeting of the British Association, that the shape of the impulse changes periodically and alternately passes through its original shape and one exactly equal but opposite in direction. If now the impulse has passed through the prism and a wave-front for a homogeneous wave of length would lie in the direction RS, the "impulses " will be confined to a region immediately surrounding a plane HS, the position of which may be calculated by the ordinary law of refraction, substituting the group velocity for the wave velocity. But on HS the impulsive motion is not uniform, but alters periodically from the original type to that which is equal and opposite to it. Hence if the emergent beam be received by a lens, the disturbance at the focus.

of the lens consists of a periodic motion which is the more homogeneous the greater the resolving power of the prism. It will be noticed that this explanation of the modus operandi of a prism differs materially from that given by Dr. J. Larmor (Ether and Matter,' p. 248); but as we may imagine continuous media of such elastic properties as to give dispersion, the true explanation must be independent of the sympathetic vibrations which Dr. Larmor calls to his aid. To calculate the angle between RS and H S, we note that HR is equal to the space passed over in air in the time equal to the difference between that necessary to traverse the thickness t of the prism when the velocity is that of a homogeneous wave and when it is that of the group. Hence U being the group velocity, V the wave velocity in vacuo, and V the wave velocity in the prism,

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We may put with sufficient accuracy V'U in this expression. To observe Talbot's bands, the retarding plate must be brought in on the side of the thin edge of the prism and the best thickness, according to § 1, is that in which that half of the beam which is nearest the thin end of the prism is retarded through half the distance RH. The appropriate λέ αμ thickness is therefore in accordance with the results of 2 αλ

the previous paragraph.

5. The previous investigation gives the retardation which the plate should produce if the best effect is to be observed. If we wish to determine in an actual case the best thickness of plate, we must remember that as we have been dealing with impulses the group velocity comes into play. Hence the usual expression (u-1)e for the retardation, the thickness being e, is not quite accurate.

If U be the group velocity, and V' the velocity of light in racuo, the retardation in time is e

to a distance in air of

U

V'

;

this corresponds

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or if V be the velocity in the substance of the plate, the retardation is

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u' being the refractive index of the material of the retarding plate.

We obtain the right result by adding to (-1)e the distance through which the group has fallen behind the wave; this corresponds to the quantity RH calculated as above, it for the thickness of the prism we substitute e and write μ'

for the refractive index of the plate. This gives for the retardation

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and for the best thickness of the plate this must be equal to

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produce the spectrum.

6. We are so much accustomed to regard the homogeneous wave as the simplest element into which all wave-motions may be resolved that we sometimes overlook the fact that the phenomena of white light may all be reproduced by a single disturbance of short duration. There are cases, and the phenomenon of Talbot's bands may serve as a conspicuous example, where the consideration of the combined group yields to a simpler treatment than the resolution into homogeneous waves. I have shown in my paper on "Interference Phenomena how group velocities may be used to determine the conditions of achromatism. Considerations similar to those used in that paper may perhaps be usefully employed to simplify the treatment of achromatized interference-bands.

II. On the Variation of Entropy as treated by Prof. Willard Gibbs. By H. A. BUMSTEAD, Ph.D., Assistant Professor of Physics, Yale University *.

IN

N the August number of the Philosophical Magazine Mr. S. H. Burbury has discussed certain difficulties which present themselves in Chapter XII. of the "Principles of Statistical Mechanics," by the late Prof. J. Willard Gibbs. Unless I have misunderstood Mr. Burbury's statement, I believe these difficulties may be surmounted, and shall endeavour to give, as briefly as possible, my reasons for this belief. For the sake of brevity I shall assume that the reader has Mr. Burbury's paper before him, and shall refrain from quoting from it unless it seems necessary for clearness of statement.

The first difficulty (which may be more conveniently discussed in terms of the hydrodynamical analogue than in terms of the ensemble of systems) is in regard to Prof. Gibbs's statement that "one may perhaps be allowed to say that a finite amount of stirring will not affect the mean square of the density of the colouring matter, but an infinite amount of stirring may be regarded as producing a condition

* Communicated by the Author.

in which the mean square of the density has its minimum value, and the density is uniform." It seems to me that this statement may be justified as follows:-If, after a certain amount of stirring, we should determine the density of the colouring matter in the liquid, using finite elements of volume sufficiently large, we should find the density sensibly constant in the different elements; but if the elements were chosen small enough (but still finite) some of them would be entirely within the coloured portions and some in the uncoloured portions, and the density in such an estimate would no longer appear to be uniform. If now the stirring were continued, a time would come when these smaller elements would all have the same average density, and so on indefinitely; and no system of finite elements, however small, could be assigned in advance in which the average density could not be made the same for all by a sufficient amount of stirring. In other words, if we are allowed to stir as long as we please, we may use elements (in the estimate of density) as small as we please. That this is what Prof. Gibbs means by his somewhat guarded statement about an infinite amount of stirring, seems plain in the light of the preceding paragraphs in which he discusses the effect of altering the order in which limits are taken. This latter consideration was one of which he not infrequently made use; I recall that he once employed it to reconcile conflicting views as to the interpretation of Fourier's series in a discussion which arose in the columns of Nature' (vol. lix. p. 200).

Although it seems to me that the statement can be thus justified, I nevertheless must agree with Mr. Burbury that the other way of escaping the difficulty, viz. by defining the density by finite elements of volume (or of extension-inphase) is preferable. If I understand the matter correctly, this is not because there is anything in the structure of the ensemble of systems corresponding to a molecular structure in the liquid, for a system of n degrees of freedom occupies no finite extension in the 2n-fold space in which its possible phases lie; but it is because we are unable, owing to the finiteness of our perceptions, to recognize very small differences of phase, just as we are unable to recognize very small differences of position in the analogous case of the liquid. And it is certainly nearer the truth to base the doctrine of the increase of entropy upon the finiteness of our perceptions rather than upon the infiniteness of time. That this was also Prof. Gibbs' opinion I believe is evidenced by the sentence following the one quoted (" one may perhaps be allowed," &c.) in which he says, "We may certainly say that a sensibly

uniform density of the coloured component may be produced by stirring." And it is really this latter form which he uses when he comes to apply the principle, as in the third paragraph on page 154, in which the qualification, "if very small differences of phase are neglected," is of course equivalent to taking finite elements of extension-in-phase. The infinite time idea was, I believe, introduced merely as an alternative (and not a preferable) way of regarding the subject. Admitting then the possibility of variation of the densityin-phase D in the finite elements through which a moving system passes, Mr. Burbury finds a difficulty in the definition. of 7 in the expression D=Ne". He says, "so long as n remains constant for the same system, we may define ʼn to be the entropy which that system has. . . . But 7 being now supposed to be variable for the same system, we require a definition." Here I think the whole trend and spirit of Prof. Gibbs method has been misapprehended; unless I have mistaken his position, Prof. Gibbs would not have admitted that the for a single system, although exactly determined, corresponds to what we call entropy in bodies met with in nature. So far as he applies his results to thermodynamics, he regards the bodies of nature as corresponding, not to a definite system, but to a system chosen at random out of a properly distributed ensemble; so that it is certain average properties of the ensemble which we observe experimentally, and not the properties of a single system. The average value of 7 for the whole ensemble (taken with the negative sign) corresponds to the entropy of any body which the ensemble is capable of representing, and we are no more concerned about the 7 of a single system (except in so far as a knowledge of it may be necessary to get a correct average) than we are concerned with the exact configuration of the system. But it is evident that we may get a sufficiently close value of the average for the whole ensemble by adding, not the η for every system, but the mean values of n for each one of a set of finite elements of extension-in-phase taken sufficiently small. Thus all we are concerned to know is the mean value of 7 in an element, and hence the equation. D=Ne may still serve as the definition of 7, for all necessary purposes, even though D is no longer the exact density at a point but is the mean density throughout an element. Prof. Gibbs' statement in a succeeding paragraph (p. 148) that "is an arbitrary function of the phase," which Mr. Burbury takes to be a new definition, is, I think, not a definition at all, but the statement of an assumed initial condition in the particular problem which he is then considering.

n

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