30 CONSERVATION OF EXTENSION-IN-PHASE 



taken within limits formed by phases regarded as contempo- 

 raneous represents the extension-in-phase within those limits. 



The case is somewhat different when the forces are not de- 

 termined by the coordinates alone, but are functions of the 

 coordinates with the time. All the arbitrary constants of the 

 integral equations must then be regarded in the general case 

 as functions of r v . . . r 2n , and t. We cannot use the princi- 

 ple of conservation of extension-in-phase until we have made 

 2n ~L integrations. Let us suppose that the constants 6, ... h 

 have been determined by integration in terms of r v . . . r 2w , and 

 t, leaving a single constant (a) to be thus determined. Our 

 2 % 1 finite equations enable us to regard all the variables 

 r v . . . r 2n as functions of a single one, say r r 



For constant values of 5, ... A, we have 



**-* + ft* (82) 



Now 



* * \MI 1 , _ 



-5 da dr* . . . dr zn = 



t 



da . . dh 



d(a, ...h) 



^"^ I f " 



J J d(a } ... A) d(r 2 , . . . r 2n ) 



where the limits of the integrals are formed by the same 

 phases. We have therefore 



^' A >, (83) 



da " d(a,...h) d(r t , . . . r, n ) 

 by which equation (82) may be reduced to the form 



da = 



M M 

 a, . . . h) d(b, ... A) 



d(r 2 , . . . 



Now we know by (71) that the coefficient of da is a func- 

 tion of a, ... h. Therefore, as , ... h are regarded as constant 

 in the equation, the first number represents the differential 



