Law of Partition of Energy.- 589 



of the imaginary system /is constant. I think that f(qp) in 

 case of a single system represents the time during which, on 

 an average of the cycle, it is in the phase {qp>)- If then E 

 he the only other constant, we should have /= 0(E), and that 

 would, as it seems to me at present, lead to the law of equal 

 partition. 



With great reluctance I am compelled here to differ from 

 those hioh authorities. I think the method as a whole fails. 

 Firstly because Maxwell's conditions are not fulfilled by any 

 existing system. Secondly because, given Maxwell's con- 

 dition alone, we have no right to assume E to be the only 

 constant. Every system must, to go no further, have 

 constant parameters ; for instance (1) the masses m^ in 2j &c, 

 of its molecules, (2) their force constants fa /a 25 <& c -> if they 

 are centres of finite force, (3) their radii c x c 2 , &c, in the 

 limiting case of elastic spheres. These parameters should 

 prima fade appear in /. Now m and /j, do appear in E, and 

 therefore in f as a function of E. But why may we assume 

 that they appear in that form only? And why may we 

 assume that in the limiting case of elastic spheres the c's do 

 not appear at all ? These restrictions on the form of f are 

 indeed justified mathematically by Boltzmann's method, 

 if his fundamental assumption be true, and our habit of 

 accepting them as proved by Boltzmann predisposes us to 

 accept them when assumed in a totally different case. 

 I think, however, they cannot be justified in any other way 

 than by Boltzmann's method. At all events Maxwell's 

 principle, taken, as we are ordered to take it, alone, seems 

 to me not to justify them. 



Boltzmann' s Assumption. 

 11. Boltzmann formally announces that he shall assume 

 that the motion of his molecules is, and for all time continues 

 to be " molecular ungeordnet." This expression is intended 

 to define some property which the system possesses. It 

 cannot be, and in fact is not, used as a substantive assumption. 

 For any special case a separate assumption has to be made, 

 which may be regarded as the interpretation of " molecular 

 nngeordnet" as applied to the special case in question. 



In the case of elastic spheres or binary encounters generally, 

 the thing assumed is as follows : — 



The number per unit of volume of spheres of mass M, 

 whose velocities lie between the limits 

 U . . U + </U -) 

 V . . V-MV ^A, 

 W . . W + c/WJ 



