DENSITY-IN-PHASE. 9 



would alter the values of tlie jt?'s as determined by equations 

 (3), and thus disturb the relation expressed in the last equation. 

 If we write equation (19) in the form 



it will be seen to express a theorem of remarkable simplicity. 

 Since D is a function of t, p l , . . . p n , q l , . . . q n , its complete 

 differential will consist of parts due to the variations of all 

 these quantities. Now the first term of the equation repre- 

 sents the increment of D due to an increment of t (with con- 

 stant values of them's and ^'s), and the rest of the first member 

 represents the increments of D due to increments of the p's 

 and g's, expressed by p l dt, q l dt, etc. But these are precisely 

 the increments which the jt?'s and #'s receive in the movement 

 of a system in the tune dt. The whole expression represents 

 the total increment of D for the varying phase of a moving 

 system. We have therefore the theorem : 



In an ensemble of mechanical systems identical in nature and 

 subject to forces determined by identical laws, but distributed 

 in phase in any continuous manner, the density-in-phase is 

 constant in time for the varying phases of a moving system ; 

 provided, that the forces of a system are functions of its co- 

 ordinates, either alone or with the time.* 



This may be called the principle of conservation of density- 

 in-phase. It may also be written 



(fL.,=- 



where a, . . . h represent the arbitrary constants of the integral 

 equations of motion, and are suffixed to the differential co- 



* The condition that the forces F lt ...F n are functions of q 1 , . . . q n and 

 a lf a 2 , etc., which last are functions of the time, is analytically equivalent 

 to the condition that F lf . . . F n are functions of qi, ...q n and the time. 

 Explicit mention of the external coordinates, a 1? 2 , etc., has been made in 

 the preceding pages, because our purpose will require us hereafter to con- 

 sider these coordinates and the connected forces, A lt A 2 , etc., which repre- 

 sent the action of the systems on external bodies. 



