Partition of Kinetic Energy. 109 



it may not be thought at undue — length, I pass on to some 

 remarks on the general investigation. This proceeds upon 

 precisely parallel lines, and the additional difficulties are 

 merely those entailed by the use of generalized coordinates. 

 Thus (1) follows from (7) by substantially the same process 

 (given in my former note) that (22) follows from (6). 

 Again, if E denote the total energy of a system, the dis- 

 tribution 



f(E)dq 1 ...dtidp 1 ...dp n , . . . (33) 



where / is an arbitrary function, satisfies the condition of 

 permanency; and, if Maxwell's assumption be applicable, it 

 is the only form of distribution that can be permanent. 



As I hinted before, some of the difficulties that have been 

 felt upon this subject may be met by a fuller recognition 

 of the invariantic character of the expressions. This point 

 has been ably developed by Prof. Bryan, who has given 

 (loc. cit. § 14) a formal verification that (33) is unaltered by 

 a change of coordinates. If we follow attentively the process 

 by which (I s ) is established, we see that in (3) there is no 

 assumption that the system of coordinates is the same at times 

 t' and tj and that accordingly we are not tied to one system 

 in (33). Indeed, so far as 1 can see, there would be no 

 meaning in the assertion that the system of generalized coor- 

 dinates employed for two different configurations was the 

 same *. 



We come now to the deduction from (33) of Maxwell's law 

 of partition of energy. On this Prof. Bryan (loc. cit. § 20, 

 remarks : — " Objections have been raised to this step in 

 Max we IPs work by myself (' Report on Thermodynamics,' 

 Part I. § 44) on the ground that the kinetic energy cannot in 

 general be expressed as the sum of squares of generalized 

 momenta corresponding to generalized coordinates of the 

 system, and by Lord Kelvin (Nature, Aug. 13, 1891) on the 

 ground that the conclusion to which it leads has no intelligible 

 meaning. Boltzmann (Phil. Mag. March 1893) has put the 

 investigation into a slightly modified form which meets the 

 first objection, and which imposes a certain restriction upon 

 the generality of the result. Under this limitation the result 

 is perfectly intelligible, and the second objection is therefore 

 also met." At this point I find myself in disagreement with 

 all the above quoted authorities, and in the position of 

 maintaining the correctness of Maxwell's original deduction. 



Prof. Boltzmann considers that " Maxwell committed an 



* It would be like saying that two points lie upon the same curve, 

 when the character of the curve is not defined. 



