Partition of Kinetic Energy. 101 



already by Prof. Boltzmann *. But his method, though 

 (I believe) quite satisfactory, is somewhat special. My object 

 is rather to follow closely the steps of the general theory. If 

 objections are taken to the argument of the particular case, 

 they should be easy to specify. If, on the other hand, the 

 argument of the particular case is admitted, the issue is much 

 narrowed. I shall have occasion myself to make some 

 comments relating to one point in the general theory not 

 raised by the particular case. 



In the general theory the coordinates f of the system at 

 time t are denoted by q v q 2 , . . . q n , and the momenta by p u 

 P2y ■ • - Pn- At an earlier time t r the coordinates and momenta 

 of the same motion are represented by corresponding letters 

 accented, and the first step is the establishment of the theorem 

 usually, if somewhat enigmatically, expressed 



ch/\ d(/ 2 . . . dq'n dp\ dp' 2 . . . dp' n =dq 1 dq 2 . . . dq n dp l dp 2 . . dp n 



. . . (1) 



In the present case q lt q 2 are the ordinary Cartesian co- 

 ordinates («£, y) of the particle ; and if we identify the mass 

 with unity, p l9 p 2 are simply the corresponding velocity- 

 components {u, v); so that (1) becomes 



dx 1 dy r du' dv' '=■ dx dy du dv (2) 



For the sake of completeness I will now establish (2) 

 de novo. 



In a possible motion the particle passes from the phase 

 (x f , y', u r , v 4 ) at time t' to the phase (x, y, u, v) at time t. In 

 the following discussion t' and t are absolutely fixed times, but 

 the other quantities are regarded as susceptible of variation. 

 These variations are of course not independent. The whole 

 motion is determined if either the four accented, or the four 

 unaccented, symbols be given. Either set may therefore be 

 regarded as definite functions of the other set. Or again, 

 the four coordinates x\ y' y x, y may be regarded as inde- 

 pendent variables, of which vf, v\ u, v are then functions. 



The relations which we require are readily obtained by 

 means of Hamilton's principal function S, where 



S=f(T.-V)<ft (3) 



In this Y denotes the potential energy in any position, and T 



* Phil. Mag. vol. xxxv. p. 156 (] 

 t Generalized coordinates appear to' have been first applied to these 

 problems by Boltzmann. 



