Notes

¹Here as in the following the German “K¨orperw¨arme” will be rendered simply as “heat.” (Tr.)

²See, e.g., Lobry de Bruyn and L. K. Wolff , Rec. des Trav. Chim. des Pays-Bas 23, p. 155, 1904.

³To restrict the word homogeneous to its absolute sense would mean that it could not be applied to any material substance.

⁴Lord Rayleigh, Phil. Mag., 47, p. 379, 1899.

⁵G. Kirchhoff , Pogg. Ann., 109, p. 275, 1860. Gesammelte Abhandlungen, J. A. Barth, Leipzig, 1882, p. 573. In defining a black body Kirchhoff also assumes that the absorption of incident rays takes place in a layer “infinitely thin.” We do not include this in our definition.

⁶For this point see especially A. Schuster, Astrophysical Journal, 21, p. 1, 1905, who has pointed out that an infinite layer of gas with a black surface need by no means be a black body.

⁷See, e.g., M. Planck, Vorlesungen ¨uber Thermodynamik, Leipzig, Veit and Comp., 1911 (or English Translation, Longmans Green & Co.), Secs. 165 and 189, et seq.

⁸H. v. Helmholtz, Handbuch d. physiologischen Optik 1. Lieferung, Leipzig, Leop. Voss, 1856, p. 169. See also Helmholtz, Vorlesungen ¨uber die Theorie der W¨arme herausgegeben von F. Richarz, Leipzig, J. A. Barth, 1903, p. 161. The restrictions of the law of reciprocity made there do not bear on our problems, since we are concerned with temperature radiation only (Sec. 7).

⁹G. Kirchhoff , Gesammelte Abhandlungen, Leipzig, J. A. Barth, 1882, p. 594. R. Clausius, Pogg. Ann. 121, p. 1, 1864.

¹⁰G. Kirchhoff , Gesammelte Abhandlungen, 1882, p. 574.

¹¹E. Pringsheim, Verhandlungen der Deutschen Physikalischen Gesellschaft, 3, p. 81, 1901.

¹²In this law it is assumed that the quantity $q$ in (24) is the same as in (37). This does not hold for strongly dispersing or absorbing substances. For the generalization applying to such cases see M. Laue, Annalen d. Physik 32, p. 1085, 1910.

¹³W. Wien and O. Lummer, Wied. Annalen, 56, p. 451, 1895.

¹⁴The strength of the emission influences only the time required to establish stationary radiation, but not its character. It is essential, however, that the walls transmit no radiation to the exterior.

¹⁵M. Thiesen, Verhandlungen d. Deutschen Physikal. Gesellschaft, 2, p. 65, 1900.

¹⁶P. Lebedew, Annalen d. Phys. 6, p. 433, 1901. See also E. F. Nichols and G. F. Hull, Annalen d. Phys. 12, p. 225, 1903.

¹⁷J. Stefan, Wien. Berichte, 79, p. 391, 1879.

¹⁸L. Boltzmann, Wied. Annalen, 22, p. 291, 1884.

¹⁹O. Lummer und E. Pringsheim, Wied. Annalen, 63, p. 395, 1897. Annalen d. Physik 3, p. 159, 1900.

²⁰F. Kurlbaum, Wied. Annalen, 65, p. 759, 1898.

²¹F. Kurlbaum, Verhandlungen d. Deutsch. physikal. Gesellschaft, 14, p. 580, 1912.

²²According to private information kindly furnished by my colleague H. Rubens (July, 1912). (These results have since been published. See W. H. Westphal, Verhandlungen d. Deutsch. physikal. Gesellschaft, 14, p. 987, 1912, Tr.)

²³W. Wien, Sitzungsberichte d. Akad. d. Wissensch. Berlin, Febr. 9, 1893, p. 55. Wiedemann’s Annal., 52, p. 132, 1894. See also among others M. Thiesen, Verhandl. d. Deutsch. phys. Gesellsch., 2, p. 65, 1900. H. A. Lorentz, Akad. d. Wissensch. Amsterdam, May 18, 1901, p. 607. M. Abraham, Annal. d. Physik 14, p. 236, 1904.

²⁴The complete solution of the problem of reflection of a pencil from a moving absolutely reflecting surface including the case of an arbitrarily large velocity of the surface may be found in the paper by M. Abraham quoted in Sec. 71. See also the text-book by the same author. Electromagnetische Theorie der Strahlung, 1908 (Leipzig, B. G. Teubner).

²⁵It is clear that the change in intensity of the reflected radiation caused by the motion of the reflector can also be derived from purely electrodynamical considerations, since electrodynamics are consistent with the energy principle. This method is somewhat lengthy, but it affords a deeper insight into the details of the phenomenon of reflection.

²⁶E.g., F. Paschen, Sitzungsber. d. Akad. d. Wissensch. Berlin, pp. 405 and 959, 1899. O. Lummer und E. Pringsheim, Verhandlungen d. Deutschen physikalischen Gesellschaft 1, pp. 23 and 215, 1899. Annal. d. Physik 6, p. 192, 1901.

²⁷O. Lummer und E. Pringsheim, l. c.

²⁸F. Paschen, Annal. d. Physik 6, p. 657, 1901.

²⁹“Independent” in the sense of “non-coherent.” If, e.g., a ray with the principal intensities $K$ and $K\prime$ is elliptically polarized, its entropy is not equal to $L+L\prime$, but equal to the entropy of a plane polarized ray of intensity $K+K\prime$. For an elliptically polarized ray may be transformed at once into a plane polarized one, e.g., by total reflection. For the entropy of a ray with coherent components see below Sec. 104, et seq.

³⁰On the average, since the solar rays of different color do not have exactly the same temperature.

³¹L. Holborn und F. Kurlbaum, Annal. d. Physik 10, p. 229, 1903.

³²M. Laue, Annalen d. Phys. 23, p. 1, 1907.

³³Compare footnote to page 490 (Tr.).

³⁴L. Boltzmann, Vorlesungen ¨uber Gastheorie 1, p. 21, 1896. Wiener Sitzungsberichte 78, Juni, 1878, at the end. Compare also S. H. Burbury, Nature, 51, p. 78, 1894.

³⁵Hereafter Boltzmann’s “Unordnung” will be rendered by chaos, “ungeordnet” by chaotic (Tr.).

³⁶To avoid misunderstanding I must emphasize that the question, whether the hypothesis of elemental chaos is really everywhere satisfied in nature, is not touched upon by the preceding considerations. I intended only to show at this point that, wherever this hypothesis does not hold, the natural processes, if viewed from the thermodynamic (macroscopic) point of view, do not take place unambiguously.

³⁷It is well known that the condition that the two systems be independent of each other is essential for the validity of the expression (163). That it is also a necessary condition for the additive combination of the entropy was proven first by M. Laue in the case of optically coherent rays. Annalen d. Physik 20, p. 365, 1906.

³⁸L. Boltzmann, Vorlesungen ¨uber Gastheorie, 1, Sec. 6.

³⁹Abridged in the sense that factors which in the logarithmic expression (173) would give rise to small additive terms have been omitted at the outset. A brief derivation of equation (173) may be found on p. 473 (Tr.).

⁴⁰See for example E. Czuber, Wahrscheinlichkeitsrechnung (Leipzig, B. G. Teubner) p. 22, 1903; H. Poincaré, Calcul des Probabilités (Paris, Gauthier-Villars), p. 85, 1912.

⁴¹Compare, for example, L. Boltzmann, Gastheorie, 2, p. 62 et seq., 1898, or J. W. Gibbs, Elementary principles in statistical mechanics, Chapter I, 1902.

⁴²L. Boltzmann, Sitzungsber. d. Akad. d. Wissensch. zu Wien (II) 76, p. 373, 1877. Compare also Gastheorie, 1, p. 38, 1896.

⁴³Compare F. Richarz, Wiedemann’s Annal., 67, p. 705, 1899.

⁴⁴E.g., M. Planck, Vorlesungen ¨uber Thermodynamik, Leipzig, Veit und Comp., 1911, Sec. 287, equation 267.

⁴⁵Compare also O. Sackur, Annal. d. Physik 36, p. 958, 1911, Nernst-Festschrift, p. 405, 1912, and H. Tetrode, Annal. d. Physik 38, p. 434, 1912.

⁴⁶M. Planck, l. c., Sec. 288, equation 271.

⁴⁷The quantity from which the principle of least action takes its name. (Tr.)

⁴⁸A. Einstein, Ann. d. Phys. 22, p. 180, 1907. Compare also M. Born und Th. von Kárman, Phys. Zeitschr. 13, p. 297, 1912.

⁴⁹Compare footnote to page 491. See also page 473.

⁵⁰A complete mathematical discussion of the subject of this chapter has been given by H. A. Lorentz. Compare, e.g., Nature, 92, p. 305, Nov. 6, 1913. (Tr.)

⁵¹H. Hertz, Wied. Ann. 36, p. 12, 1889.

⁵²See among others H. Rubens und F. Kurlbaum, Sitz. Ber. d. Akad. d. Wiss. zu Berlin vom 25. Okt., 1900, p. 929. Ann. d. Phys. 4, p. 649, 1901. F. Paschen, Ann. d. Phys. 4, p. 277, 1901. O. Lummer und E. Pringsheim, Ann. d. Phys. 6, p. 210, 1901. T¨atigkeitsbericht der Phys.-Techn. Reichsanstalt vom J. 1911, Zeitschr. f. Instrumentenkunde, 1912, April, p. 134 ff.

⁵³According to private information kindly furnished by the president, Mr. Warburg.

⁵⁴W. Wien, Wied. Ann. 58, p. 662, 1896.

⁵⁵Lord Rayleigh, Phil. Mag. 49, p. 539, 1900.

⁵⁶J. H. Jeans, Phil. Mag. Febr., 1909, p. 229, H. A. Lorentz, Nuovo Cimento V, vol. 16, 1908.

⁵⁷Here as well as later on the value given above (79) has been replaced by $a=7.39\cdot 10^{-15}$, obtained from $\sigma =\frac{ac}{4}=5.54\cdot 10^{-5}$. This is the final result of the newest measurements made by W. Westphal, according to information kindly furnished by him and Mr. H. Rubens. (Nov., 1912). [Compare p. 490, footnote. Tr.]

⁵⁸F. Richarz and O. Krigar-Menzel, Wied. Ann. 66, p. 190, 1898.

⁵⁹Nevertheless regular refraction and reflection are not irreversible processes; for the refracted and the reflected rays are coherent and the entropy of two coherent rays is not equal to the sum of the entropies of the separate rays. (Compare above, Sec. 104.) On the other hand, diffraction is an irreversible process. M. Laue, Ann. d. Phys. 31, p. 547, 1910.

⁶⁰E. Hagen und H. Rubens, Ann. d. Phys. 11, p. 873, 1903.

⁶¹E. Aschkinass, Ann. d. Phys. 17, p. 960, 1905.

⁶²E. Riecke, Wied. Ann. 66, p. 353, 1898.

⁶³P. Drude, Ann. d. Phys. 1, p. 566, 1900.

⁶⁴H. A. Lorentz, Proc. Kon. Akad. v. Wet. Amsterdam, 1903, p. 666.

⁶⁵J. H. Jeans, Phil. Mag. 10, p. 91, 1905.

⁶⁶Lord Rayleigh, Nature 72, p. 54 and p. 243, 1905.

⁶⁷Not to be confused with the “field intensity” (field-strength) $E_z$ of the exciting vibration.

⁶⁸Compare P. Ehrenfest, Wien. Ber. 114 [2a], p. 1301, 1905. Ann. d. Phys. 36, p. 91, 1911. H. A. Lorentz, Phys. Zeitschr. 11, p. 1244, 1910. H. Poincaré, Journ. de Phys. (5) 2, p. 5, p. 347, 1912.

⁶⁹Not yet published (Jan. 26, 1914. Tr.)

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