DENSITY-IN-PHASE. 1 



In the first place, let us consider the number of systems 

 which in the time dt pass into or out of the specified element 

 by p l passing the limit p^. It will be convenient, and it is 

 evidently allowable, to suppose dt so small that the quantities 

 ^ dt, q l dt, etc., which represent the increments of p l , q l , etc., 

 in the time dt shall be infinitely small in comparison with 

 the infinitesimal differences p p^, q r <?/, etc., which de- 

 termine the magnitude of the element of extension-in-phase. 

 The systems for which p l passes the limit p^ in the interval 

 dt are those for which at the commencement of this interval 

 the value of p 1 lies between p^ and p^ p dt, as is evident 

 if we consider separately the cases in which p l is positive and 

 negative. Those systems for which p 1 lies between these 

 limits, and the other p's and j's between the limits specified in 

 (9), will therefore pass into or out of the element considered 

 according aH^t? is positive or negative, unless indeed they also 

 pass some other limit specified in (9) during the same inter- 



^val of time. But the number which pass any two of these 

 limits will be represented by an expression containing the 

 square of dt as a factor, and is evidently negligible, when dt 



1 is sufficiently small, compared with the number which we are 

 seeking to evaluate, and which (with neglect of terms contain- 

 ing dt 2 ) may be found by substituting p l dt for p^' p^ in 

 (10) or for dp 1 in (11). 

 The expression 



Dpi dt dp z . . . dp n dqi . . . dq n (13) 



will therefore represent, according as it is positive or negative, 

 the increase or decrease of the number of systems within the 

 given limits which is due to systems passing the limit p^. A 

 similar expression, in which however D and p will have 

 slightly different values (being determined for p^' instead of 

 Pi), will represent the decrease or increase of the number of 

 systems due to the passing of the limit p^'. The difference 

 of the two expressions, or 



dpi . . . dp n dqi . . . dq n dt (14) 



