18 CONSERVATION OF 



J. . . JV dtf . . . dqj =f. . . Jp" dp{' . . . dq n ", (45) 



where the limits in the two cases are formed by corresponding 

 phases. When the integrations cover infinitely small vari- 

 ations of the momenta and coordinates, we may regard P* and 

 P" as constant in the integrations and write 



P'f. . .fd Pl > <%" = 



Now the principle of the conservation of extension-in-phase, 

 which has been proved (viz., in the second demonstration given 

 above) independently of any reference to an ensemble of 

 systems, requires that the values of the multiple integrals in 

 this equation shall be equal. This gives 



P 1 ' = P f . 



With reference to an important class of cases this principle 

 may be enunciated as follows. 



When the differential equations of motion are exactly known, 

 but the constants of the integral equations imperfectly deter- 

 mined, the coefficient of probability of any phase at any time is 

 equal to the coefficient of probability of the corresponding phase 

 at any other time. By corresponding phases are meant those 

 which are calculated for different times from the same values 

 of the arbitrary constants of the integral equations. 



Since the sum of the probabilities of all possible cases is 

 necessarily unity, it is evident that we must have 



all 



f...fpd Pl ...dq n = l, (46) 



phases 



where the integration extends over all phases. This is indeed 

 only a different form of the equation 



811 



phases 



which we may regard as defining 



