between molecules is a function of their distances, that temperature depends solely upon the kinetic energy of molecular motions, and that the number of molecules which at any moment are so near to each other that they perceptibly influence each other is comparatively so small that it may be neglected. He calculated the average velocities of molecules, and explained evaporation. Objections to his theory, raised by Buy's-Ballot and by Jochmann, were satisfactorily answered by Clausius and Maxwell, except in one case where an additional hypothesis had to be made. Maxwell proposed to himself the problem to determine the average number of molecules, the velocities of which lie between given limits. His expression therefor constitutes the important law of distribution of velocities named after him. By this law the distribution of molecules according to their velocities is determined by the same formula (given in the theory of probability) as the distribution of empirical observations according to the magnitude of their errors. The average molecular velocity as deduced by Maxwell differs from that of Clausius by a constant factor. Maxwell's first deduction of this average from his law of distribution was not rigorous. A sound derivation was given by O. E. Meyer in 1866. Maxwell predicted that so long as Boyle's law is true, the coefficient of viscosity and the coefficient of thermal conductivity remain independent of the pressure. His deduction that the coefficient of viscosity should be proportional to the square root of the absolute temperature appeared to be at variance with results obtained from pendulum experiments. This induced him to alter the very foundation of his kinetic theory of gases by assuming between the molecules a repelling force varying inversely as the fifth power of their distances. The founders of the kinetic theory had assumed the molecules of a gas to be hard elastic spheres; but Maxwell, in his second presentation of the theory in 1866, went on the assumption that the molecules behave like centres of forces. He demonstrated anew the law of distribution of velocities; but the proof had a flaw in argument, pointed out by Boltzmann, and recognised by Maxwell, who adopted a somewhat different form of the distributive function in a paper of 1879, intended to explain mathematically the effects observed in Crookes' radiometer. Boltzmann gave a rigorous general proof of Maxwell's law of the distribution of velocities. None of the fundamental assumptions in the kinetic theory of gases leads by the laws of probability to results in very close agreement with observation. Boltzmann tried to establish kinetic theories of gases by assuming the forces between molecules to act according to different laws from those previously assumed. Clausius, Maxwell, and their predecessors took the mutual action of molecules in collision as repulsive, but Boltzmann assumed that they may be attractive. Experiment of Joule and Lord Kelvin seem to support the latter assumption. Among the latest researches on the kinetic theory is Lord Kelvin's disproof of a general theorem of Maxwell and Boltzmann, asserting that the average kinetic energy of two given portions of a system must be in the ratio of the number of degrees of freedom of those portions. ADDENDA. PAGE 14. The new Akhmim papyrus, written in Greek, is probably the copy of an older papyrus, antedating Heron's works, and is the oldest extant text-book on practical Greek arithmetic. It contains, besides arithmetical examples, a table for finding "unit-fractions," identical in scope with that of Ahmes, and, like Ahmes's, without a clue as to its mode of construction. See BIBLIOTH. MATH., 1893, p. 79-89. The papyrus is edited by J. Baillet (Mémoires publiés par les membres de la mission archéologique française au Caire, T. IX., 1r fascicule, Paris, 1892, p. 1-88). PAGE 39. Chasles's or Simson's definition of a Porism is preferable to Proclus's, given in the text. See Gow, p. 217-221. PAGE 114. Nasir Eddin for the first time elaborated trigonometry independently of astronomy and to such great perfection that, had his work been known, Europeans of the 15th century might have spared their labours. See BIBLIOTH. MATH., 1893, p. 6. PAGE 116. This law of sines was probably known before Gabir ben Aflah to Tabit ben Korra and others. See Biblioth. Math., 1893, p. 7. PAGE 125. Athelard was probably not the first to translate Euclid's Elements from the Arabic. See M. Cantor's VORLESUNGEN, Vol. II., p. 91, 92. PAGE 240. G. Eneström argues that Taylor and not Nicole is the real inventor of finite differences. See BIBLIOTH. MATH., 1893, p. 91. PAGE 250. An earlier publication in which 3.14159 · is designated by π, is W. Jones's Synopsis palmariorum matheseos, London, 1706, p. 243, 263 et seq. See BIBLIOTH. MATн., 1894, p. 106. ... PAGE 335. Before Gauss a theorem on convergence, usually attributed to Cauchy, was given by Maclaurin (Fluxions, § 350). A rule of convergence was deduced also by Stirling. See Bull. N. Y. Math. Soc., Vol. III., p. 186. PAGE 358. The surface of a solid with p holes was considered before Clifford by Tonelli, and was probably used by Riemann himself. See MATH. ANNALEN, Vol. 45, p. 142. PAGE 361. As early as 1835, Lobachevsky showed in a memoir the necessity of distinguishing between continuity and differentiability. See G. B. Halsted's transl. of A. Vasiliev's Address on Lobachevsky, p. 23. Recent deaths. Johann Rudolf Wolf, Dec. 6, 1893; Heinrich Hertz, Jan. 1, 1894; Eugène Catalan, Feb. 14, 1894; Hermann von Helmholtz, Sept. 8, 1894; Arthur Cayley, Jan. 26, 1895. Absolutely convergent series, 335, 337, Algorithm, origin of term, 106; Mid- Absolute geometry, 301. 338. Abul Gud, 111; ref. to, 113. Abul Hasan, 115. Abul Wefa, 110; ref. to, 112, 113. dle Ages, 126, 129. Al Haitam, 115; ref. to, 112. Al Hayyami, 112; ref. to, 113. Achilles and tortoise, paradox of, 27. Al Hogendi, 111. Acoustics, 262, 270, 278, 386. Action, least, 253, 366, 401; varying, 292, 318, 379. Adams, 375; ref. to, 214. Al Karhi, 111, 113. Al Kaschi, 114. Al Kuhi, 111; ref. to, 112. Addition theorem of elliptic integrals, Allman, IX., 36. Analytical Society (in Cambridge), | Astronomy: Babylonian, 8; Egyptian, Arabic numerals and notation, 3, 73, Bachet de Méziriac. See Méziriac. Aristotle, 34; ref. to, 9, 17, 27, 43, 61, Bauer, XII. 68, 125. Arneth, X. Aronhold, 327. Aryabhatta, 86; ref. to, 88, 91, 98. Assumption, tentative, 75, 92. Astrology, 155. Baumgart, XI. Beaune, De. See De Beaune. Beha Eddin, 114. Bellavitis, 322; ref. to, 300, 304, 317. Bernelinus, 122. Bernoulli, Daniel, 238; ref. to, 255, See Bernoulli, Nicolaus (born 1695), 238. |