ON OUR KNOWLEDGE OF THERMODYNAMICS. 71 



forces without ever colliding with one another. If u, v, tv be the velocity 

 components of any particle, these are constant throughout all time, and 

 therefore the theorem asserts that any law of distribution of the form 



f {u, V, lo) du dv dw dx dy dz 



is permanent ; a conclusion which is obviously correct. 



18. In Maxwell's paper / is to be taken constant for one particular 

 value of E, and zero for all others. 



In the Kinetic Theory of Gases /is proportional to e~'"^. 



The question as to how far this law is unique has been raised by 

 Messrs. Watson and Burbury,' who quote Boltzmann's demonstration of 

 the proposition, but admit tliat there may be exceptions to its truth. 

 That demonstration is, however, based on a consideration of the collisions 

 between the molecules of a gas, and has no application to a problem like 

 the present, where all the systems considere4 are independent conservative 

 systems, and no transference of energy takes place between the bodies 

 of one system and those of the other. It will be considered fully in 

 Section II. § 42. 



19. Since for a conservative system E is always constant, there will 

 always exist possible laws of permanent distribution, for which / is any 

 function of E ; but the possibility of other distributions will depend on the 

 nature of the system and the existence of other integrals of the equations 

 of motion. 



The Reduction of the Kinetic Energy to a Sum of Squares. — Maxwell's 

 Laiv of Partition of Energy. 



20. Objections have been raised to this step in Maxwell's work by 

 myself ^ on the ground that the kinetic energy cannot in general be 

 expressed as the sum of squares of generalised momenta corresponding to 

 generalised co-ordinates of the system, and by Lord Kelvin ^ on the ground 

 that the conclusion to which it leads has no intelligible meaning. Boltz- 

 mann ■* has put the investigation into a slightly modified form which meets 

 the first objection, and which imposes a certain restriction on the generality 

 of the result. Under this limitation the result is perfectly intelligible, 

 and the second objection is therefore also met. 



21. Boltzmann reduces the kinetic energy to the form 



but he does not assume the quantities a^ to be generalised momenta. He 

 calls these quantities ' niomentoids.' They are linear functions of the 

 generalised momenta of the system, and calling these latter ^^i, jSj, . . . , 

 the momentoids are supposed chosen so that the determinant 



Q_S ("1, 02 • • • "«) _i . 



'^{PuP-l ■ ■ ■ Pn) 



' Nature, June 2, 1892, p. 101. 



* Eeport on Thermodynamics, Part I. § 44. 

 » Nature, August 13, 1891. 



* ' On the Equilibrium of Vis Viva,' Part III. Phil. Mag., March 1893. The 

 original is in the SitzungshericMe of the Academy of Munich (not Vienna), and forms 

 the third part to the author's ' Studieu iiber das Gleichgewicht, &c.' (^Wiener Sitzh., 

 Iviii. (ii.), Oct. 1868), and his ' Weitere Studien ' ( Wieww Sitzh., Ixvi. (ii.), Oct. 1872). 



