112 EEPOKT — 1891. 



energy acquired by the system from A, and Q2 the energy given out to B, 

 it is easy enough to show that 



^=^2 ("71) 



0)1 0)2 



corresponding to the well-known thermodynamic equation. At the 

 same time the external work performed by the string is Qj — Q2. 



If ,Si and ^2 be the angular momenta of the shaft and spoke during 

 the operations (i.) and (iii.) respectively, either member of (71) is equal 

 to §2 — ^1- 



If two discs were brought into contact when their angular velocities 

 were unequal, there would be a loss of energy by friction, so that the 

 analogy with an irreversible cycle would not be complete. 



Section III. Statistical Hypotheses. 



39. The investigations now to be considered depend on the existence 

 of a certain law of average distribution of speed, which holds whenever 

 an enormously large number of molecules is in a state of steady or 

 stationary motion. This remark applies to the Kinetic Theory of Gases, 

 and the methods are only applicable when the nature of the molecules is 

 such that the law of distribution in question is capable of investigation. 



Among the more recent researches bearing on the subject may be 

 particularly mentioned Professor Tait's papers ' On the Foundations of 

 the Kinetic Theory of Gases,' ' Dr. Boltzmann's papers on the ' Analogies 

 of the Second Law ' '^ and on the ' Properties of Monocyclic and other 

 Belated Systems,' ^ and Sir William Thomson's recent communication to 

 the Boyal Society ' On some Test Cases for the Maxwell-Boltzmann Doc- 

 trine regarding Distribution of Energy.' ■* 



40. The BoUzmann,-3Iaxivell Bodr-ine. — The law of distribution of speed 

 is variously known as Boltzmann's Theorem and Clerk Maxwell's Theorem, 

 being due in part to one writer and in part to the other. It seems to 

 have been first discovered by Clerk Maxwell for the case of a number of 

 perfectly elastic smooth colliding spheres of two or more different 

 magnitudes, or, if preferred, a number of simple particles which repel 

 one another when at a certain distance apart, after the manner of 

 perfectly elastic spheres.^ The theorem was subsequently generalised 

 by Boltzmann ^ for the case of a system of pai'ticles repelling one another 

 according to any law, and was finally generalised still further by Max- 

 well ^ for a number of molecules, each consisting of a dynamical system 



' Trans. M.S. Edinburgh, 188G-91. 



- ' Analogien des zweiten Hauptsatzes der Thermodynamik,' Crelle, Journal, c. 

 p. 213. 



' 'Ueber die Eigenschaften monocyclischer und anderer damit verwandter 

 Systeme,' Crelle, Journal, xcviii. p. 68. 



* Nature, August 13, 1891. 



'' 'On the Collisions of Elastic Spheres,' Pldl. Mag. 18G0; 'On the Dynamical 

 Theory of Gases,' PMl. Trans. R.S. May 1866. 



« 'Ueber die mech. Bedeut. des 2"^° Haupts d. mech. Wilrmelehre,' Wiener 

 Sitzb. Bd. 53, pp. 195-220. ' Studien iiber das Gleichgew. d. leb. Kraft zwischen 

 beweg. mater. Punkten,' ibidem, Bd. .58 (1868). ' Ueber das Gleichgew. zwischen 

 mehratom. Gasmolekiilen ' ; ' Analyt. Beweis des 2"" Haupts d. mech. Warmetheorie 

 aus d. Siitzen fiir den Gleichgew. d. leb. Kraft'; ' Einige allgem. Satze iiber 

 Warmegleichgewicbt,' ]Viener Sitzb. Mathem. Naturw. Klasse, Band 63. 



' ' On Boltzmann's Theorem,' &c., Trans. Camb. Pldl. Soc. 1878. 



