Scientific Papers of Josiah Willard Gibbs, Volume 2/Chapter XVI

by Josiah Willard Gibbs

XVI.

REVIEWS OF NEWCOMB AND MICHELSON'S "VELOCITY OF LIGHT IN AIR AND REFRACTING MEDIA" AND OF KETTELER'S "THEORETISCHE OPTIK."

[American Journal of Science, ser. 3, vol. xxxi. pp. 62–67, Jan. 1886.]

Velocity of Light in Air and Refracting Media.

Astronomical Papers prepared for the use of the American Ephemeris and Nautical Almanac, vol. ii. parts 3 and 4,^[1] Washington, 1885.

Professor Newcomb obtains as the final result of his experiments at Washington 299,860 ± 30 kilometers per second for the velocity of light in vacuo. Professor Michelson's entirely independent experiments at Cleveland give substantially the same result (299,853 ± 60). His former experiments at the Naval Academy, after correction of two small errors which he now reports, give 299,910 ± 60. All these experiments were made with the revolving mirror, but the arrangements of the two experimenters were in other respects radically different. The first of these values of the velocity of light with Nyrén's value of the constant of aberration (20".492) gives 149.60 for the distance of the sun in millions of kilometers. On acount of the recent announcement by Messrs. Young and Forbes of a difference of about two per cent, in the velocities of red and blue light, especial attention was paid to this point by both experimenters, without finding the least indication of any difference. In Professor Newcomb's experiments, a difference of only one thousandth in these velocities would have produced a well-marked iridescence on the edges of the return image of the slit formed by reflection from the revolving mirror. No trace of such iridescence could ever be seen. Professor Michelson made an experiment, in which a red glass covered one-half the slit. The two halves of the image—the upper white, the lower red—were exactly in line.

Since Maxwell's electromagnetic theoiy of light makes the velocity of light in air equal to the ratio of the electromagnetic and electrostatic units of electricity, it will be interesting to compare some recent determinations of this ratio. These we give in the following table. Since the determinations are affected by any error in the standard of resistance, we have corrected the results, first, on the supposition that the B. A. ohm = .987 true ohms (Lord Rayleigh's result), and secondly, on the supposition that the B. A. ohm = .989 true ohms, which is essentially assuming that the legal ohm represents the true value.

Ratio of Electromagnetic and Electrostatic units of Electricity in millions of meters and seconds.

	Date.	As Published.	B. A. ohm = .987	B. A. ohm = .989.
Ayrton & Perry,[2]	1878	298.0	296.1	296.4
Hockin,[3]	1879	298.8	296.9	297.2
Shida,[4]	1880	299.5	295.6	296.2
Exner,[5]	1882	301.1 (?)	291.7 (?)	292.3 (?)
J. J. Thomson,[6]	1883	296.3	296.3	296.9
Klemenčič,[7]	1884	301.88 (?)	301.88 (?)	302-48 (?)

These numbers are to be compared with the velocity of light in air, in millions of meters per second, for which Professor Newcomb gives 299.778. Of the electrical determinations, that of J. J. Thomson appears by far the most worthy of confidence. That of Klemenčič—the only one as great as the velocity of light—was obtained by the use of a condenser with glass,—a method which would presumably give too great a ratio. Exner's value is obtained from the mean of three determinations, one of which differed from the others by about three per cent. If we reject this discordant determination, the mean of the other two would give when corrected for resistance 294.4 and 295.0. If we set aside the determinations of Exner and Klemenčič, the remaining four, which represent three different methods, are very accordant, the mean being nearly identical with the result of J. J. Thomson, and about one per cent, less than the velocity of light.

Professor Michelson's experiments on the velocity of light in carbon disulphide afford an interesting illustration of the difference between the velocity of waves and the velocity of groups of waves—a subject which is treated at length in an appendix to the second volume of Lord Rayleigh's Theory of Sound. If we write ${\text{V}}$ for the velocity of waves, ${\text{U}}$ for that of a group of waves, ${\text{L}}$ for the wave-length, and ${\text{T}}$ for the period of vibration,

${\text{V}}={\frac {\text{L}}{\text{T}}},$${\text{U}}={\frac {d({\text{T}}^{-1})}{d({\text{L}}^{-1})}}\cdot$

For purposes of numerical calculation, it will be convenient to transform these formulæ by the use of $\lambda$ for the wave-length in vacuo, $n$ for the index of refraction of the medium considered, and $k$ for the velocity of light in vacuo, which we shall regard as constant, in accordance with general usage. By substitution of these letters we easily obtain

${\frac {k}{\text{V}}}=n,$${\frac {k}{\text{U}}}={\frac {d(n\lambda ^{-1})}{d(\lambda ^{-1})}}\cdot$

The data for the calculation of these quantities for carbon disulphide are given by Verdet (Annales de Chimie et de Physique, (3), vol. lxix, p. 470). They give

${\text{for the line D, }}k/{\text{V}}=1.624,\,k/{\text{U}}=1.722,$ ${\text{for the line E, }}k/{\text{V}}=1.637,\,k/{\text{U}}=1.767.$

The quotient of the velocity in vacuo divided by the velocity in carbon disulphide, according to Professor Michelson's experiments with the light of an arc lamp, is 1.76 ± .02, which agrees very well with $k/{\text{U}}.$ Another theory, which would make the velocity observed in such experiments ${\text{V}}^{2}/{\text{U}}$ (Nature, vol. xxv, p. 52), receives no countenance from these experiments. The value of $k{\text{U}}/{\text{V}}^{2}$ would be about 1.53. Some may think that the experiments on water point in a different direction. Taking our data from Beer's Einleitung in die höhere Optik, 1853, p. 411, we get

${\text{for D}},\,k/{\text{V}}=1.334,\,k/{\text{U}}=1.352,\,k{\text{U}}/{\text{V}}^{2}=1.316,$ ${\text{for E}},\,k/{\text{V}}=1.336,\,k/{\text{U}}=1.359,\,k{\text{U}}/{\text{V}}^{2}=1.313.$

The number obtained by experiment was 1.330, which agrees better with $k/{\text{V}},$ or even with $k{\text{U}}/{\text{V}}^{2},$ than with $k/{\text{U}},$ but the differences are here too small to have much significance.

Theoretische Optik, gegründet auf das Bessel-Sellmeier'sche Princip, zugleich mit den experimentellen Belegen.

Von Dr. E. Ketteler, Professor an der Universität in Bonn. Viewig und Sohn. Braunschweig, 1885.

The principle of Sellmeier, here referred to, relates to vibrations of ponderable particles excited by the etherial vibrations of light, and to the reaction of the former upon the latter. The name of Bessel is added on account of his previous solution of a somewhat analogous problem relating to the pendulum. The object of this work is "to treat theoretical optics in a complete and uniform manner on the new foundation of the simultaneous vibration of etherial and ponderable particles, and to substitute a consistent and systematic new structure for the present conglomerate of more or less disconnected principles." Such a work demands a critical examination, which should not be undertaken from any narrow point of view. Any faults of detail will be readily forgiven, if the author shall give the theory of optics the πού στώ which it has sought so long in vain. We may add that if this effort shall not be judged successful by the scientific world, the author will at least have the satisfaction of being associated in his failure with many of the most distinguished names in mathematical physics.

We have sought to test the proposed theory with respect to that law of optics which seems most conspicuous in its definite mathematical form, and in the rigor of the experimental verifications to which it has been subjected, as well as in the magnificent developments to which it has given rise : the law of double refraction due to Huyghens and Fresnel, and geometrically illustrated by the wave-surface of the latter. We cannot find that the law of Fresnel is proved at all in this treatise. We find on the contrary, that a law is deduced which is different from Fresnel's, and inconsistent with it. We do not refer to anything relating to the direction of vibration of the rays in a crystal, which is a point not touched by the experimental verifications to which we have alluded. We shall confine our comparison to those equations from which the direction of vibration has been eliminated, and which therefore represent relations subject to experimental control. For this purpose equation (13) on page 299 is suitable. It reads

${\frac {u^{2}}{n_{x}^{2}-n^{2}}}+{\frac {v^{2}}{n_{y}^{2}-n^{2}}}+{\frac {w^{2}}{n_{z}^{2}-n^{2}}}=0,$

$n_{x},n_{y},n_{z}$ being the principal indices of refraction. This the author calls the equation of the wave-surface or surface of ray-velocities. It has the form of the equation of FresneUs wave-surface, expressed in terms of the direction-cosines and reciprocal of the radius vector, and if $u,v,w$ are the direction-cosines of the ray, and $n$ the velocity of light in vacuo divided by the so-called ray-velocity in the crystal the equation will express Fresnel's law. But it is impossible to give these meanings to $u,v,w$ and $n.$ They are introduced into the discussion in the expression for the vibrations (p. 295), viz..

$\rho ={\mathfrak {A}}\cos 2\pi \left({\frac {t}{\text{T}}}-{\frac {n(ux+vy+wz)}{\lambda }}\right)\cdot$

The form of this equation shows that $u,v,w$ are proportional to the direction-cosines of the wave-normal, and as the relation $u^{2}+v^{2}+w^{2}=1$ is afterwards used, they must be the direction-cosines of the wave-normal. They cannot possibly denote the direction-cosines of the ray, except in the particular case in which the ray and wave-normal coincide. Again, from the form of this equation, $\lambda /n$ must be the wave-length in the crystal, and if $\lambda$ here as elsewhere in the book (see p. 26) denotes the wave-length in vacuo of light of the period considered, which we doubt not is the intention of the author, $n$ must be the wave-length in vacuo divided by the wave-length in the crystal, i.e., the velocity of light in vacuo divided by the wave-velocity in the crystal With these definitions of $u,v,w,$ and $n,$ equation (13) expresses a law which is different from Fresnel's. Applied to the simple case of a uniaxial crystal, it makes the relation between the wave-velocity of the extraordinary ray and the angle of the wave-normal with the principal axis the same as that of the radius vector and the angle in an ellipse. The law of Huyghens and Fresnel makes the reciprocal of the wave-velocity stand in this relation.

The law which our author has deduced has come up again and again in the history of theoretical optics. Professor Stokes (Report of the British Assoc., 1862, part i, p. 269) and Lord Rayleigh (Phil. Mag., (4), vol. xli, p. 525) have both raised the question whether Huyghens and Fresnel might not have been wrong, and it might not be the wave-velocity and not its reciprocal which is represented by the radius vector in an ellipse. The difference is not very great, for if we lay off on the radii vectores of an ellipse distances inversely proportional to their lengths, the resultant figure will have an oval form approaching that of an ellipse when the eccentricity of the original ellipse is small. Rankine appears to have thought that the difference might be neglected (see Phil. Mag., (4), vol. i, pp. 444, 445) at least he claims that his theory leads to Fresnel's law, while really it would give the same law which our author has found. (Concerning Rankine's "splendid failure," and the whole history of the subject, see Sir Wm. Thomson's Lectures on Molecular Dynamics at the Johns Hopkins University, chap, xx.) Professor Stokes undertook experiments to decide the question. His result, corroborated by Glazebrook (Pro. Roy. Soc., voL xx, p. 443; Phil. Trans., voL clxxi, p. 421), was that Huyghens and Fresnel were right and that the other law was wrong.

To return to our author, we have no doubt from the context that he regards $u,v,w,$ and $n$ as relating to the ray and not to the wave-normal. We suppose that that is the meaning of his remark that the expression for the vibrations (quoted above) is to be referred to the direction of the ray. It seems rather hard not to allow a writer the privilege of defining his own terms. Tet the reader will admit that when the vibrations have been expressed in the above form an inexorable necessity fixes the significance of the direction determined by $u,v,w,$ and leaves nothing in that respect to the choice of the author.

The historical sketches of the development of ideas in the theory of optics, enriched by very numerous references, will be useful to the student. An exception, however, must be made with respect to the statements concerning the electromagnetic theory of light. We are told (p. 450) that the English theory, founded by Maxwell and represented by Glazebrook and Fitzgerald, makes the plane of polarization coincide with the plane of vibration, while Lorentz, on the basis of Helmholtz's equations comes to the conclusion that these planes are at right angles. Since all these writers make the electrical displacement perpendicular to the plane of polarization, we can only attribute this statement to some confusion between the electrical displacement and the magnetic force or "displacement" at right angles to it. We are also told that Glazebrook's "surface-conditions" which determine the intensity of reflected and refracted light are different from those of Lorentz,—a singular error in view of the fact that Mr. Glazebrook (Proc. Camb. Phil. Soc., vol. iv, p. 166) expressly states that his results are the same as those of Lorentz, Fitzgerald, and J. J. Thomson. We have spent much fruitless labor in trying to discover where and how the expressions were obtained which are attributed to Glazebrook, but in which the notation has been altered. They ought to come from Glazebrook's equations (24)–(27) (loc. cit.), but these appear identical with Lorentz's equations (58)–(61) (Zeitschrift f. Math. u. Phys., vol. xxii, p. 27). They might be obtained by interchanging the expressions for vibrations in the plane of incidence and at right angles to it, with two changes of sign.

The reader must be especially cautioned concerning the statements and implications of what has not been done in the electromagnetic theory. These are such as to suggest the question whether the author has taken the trouble to read the titles of the papers which have been published. We refer especially to what is said on pages 248, 249 concerning absorption, dispersion, and the magnetic rotation of the plane of polarization.

In the Experimental Part, with which the treatise closes, we have a comparison of formulæ with the results of experiments by the author and others. The author has been particularly successful in the formula for dispersion. Li the case of quartz (p. 545), the formula (with four constants) represents the results of experiment in a manner entirely satisfactory through the entire range of wave-length from 2.14 to 0.214. Those who may not agree with the author's theoretical views will nevertheless be glad to see the results of experiment brought together, and, so far as may be, represented by formulæ.

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1. [Part 3, "Measures of the Velocity of Light," S. Newcomb; part 4, "Supplementary Measures of the Velocities of white and colored light in air, water, and carbon disulphide," A. A. Michelson.]
2. Phil. Mag., (5), vol. vii, p. 277
3. Report Brit. Assoc., 1879, p. 285.
4. Phil. Mag., (5), vol. x, p. 431
5. Sitzungsberichte Wien. Akad., vol. lxxxvi, p. 106.
6. Phil. Trans., vol. clxxiv, p. 707
7. Sitzungsberichte Wien. Akad., vol. lxxxix, p. 298.