of Energy in a System of Material Points. 303 



■which equation expresses Boltzniann's theorem in its fullest 

 generality w . 



In this equation the old independents q\ . . . </„, p\ . . .p n r 

 appear again. Since we divided by dr, and consequently dr 

 appears no more in the equation, this is equivalent to saying 

 that the time of the whole motion is to be regarded as a con- 

 stant. On the other hand, all the coordinates and momenta 

 holding good for the instant of commencement of motion (i. e. 

 all the quantities cf\ . . . p' n ) are to be increased by infinitely 

 small amounts. The values q 1 . . ,p n of the coordinates and 

 momenta at the time t will therefore also undergo infinitely 

 small increase: and, according to equation (5), the product of 

 the first must be put equal to the product of the latter incre- 

 ments; consequently, if we choose the old independents, we 

 must have 



% dp n _ 

 ^dffdjA- 



For the sake of a clear view of the meaning of equation (5), 

 let us imagine, instead of one system S, an infinitely large 

 number of exactly similar systems S. Let the law of action 

 of the forces be precisely the same for all the systems (of 

 course without any two systems having any action upon each 

 other). Let the duration of motion t be exactly the same for 

 all the systems — but the conditions of the systems at the in- 

 stant of commencement of motion not the same for all the 

 systems, but having at the instant of commencement of motion 

 the values of coordinates and momenta between the limits q\ 

 and q\ + dq\. . ,p' n and p'„ + dj/n for all the systems. Then 

 also at the instant at which the motion ends the conditions of 

 all the systems will not be the same, and coordinates and mo- 

 menta may lie between the limits q x and qi+dq l ...p„ and 



* In Watson's excellent took, { A Treatise on the Kinetic Theory of 

 Gases ' (Clarendon Press, 1876), p. 13, there is an error, or at least an 

 inaccuracy of expression, in the derivation of this equation. In the partial 

 differential quotients of the functional determinant, at the head of that 

 page, besides p and P, the time t of the motion is to be regarded as an 

 independent variable ; but the equation fohowing from this, 



dq T _ (PA. _ _ clQs 



w~ ~dp~dp s - dp~;: 



only holds good when E is variable independently of p and P. Conse- 

 quently, hi forming the partial differential quotients of this equation, E is 

 to be regarded as constant ; in forming those of the functional determi- 

 nant, r is to be regarded as constant : and the applicability of an equation 

 holding good between the first partial differential quotients to the latter 

 requires still to be proved. 



