EXTENSION-IN-PHASE. 15 



d d(p lt ... 



dtd( Pl ,...y n ") 



which substituted in (34) will give 

 d 



_ 

 - 



... n _ 



dtd( Pl ',...q n ') 



The determinant in this equation is therefore a constant, the 

 value of which may be determined at the instant when t = ', 

 when it is evidently unity. Equation (33) is therefore 

 demonstrated. 



Again, if we write a, ... h for a system of 2 n arbitrary con- 

 stants of the integral equations of motion, p v q v etc. will be 

 functions of. a, ... h, and t, and we may express an extension- 

 in-phase in the form 



/rd(p 

 "V *(< 



,, ^|T da - - dh - ( 35 > 



d(a, ...h) 



If we suppose the limits specified by values of a, . . . ^, a 

 system initially at the limits will remain at the limits. 

 The principle of conservation of extension-in-phase requires 

 that an extension thus bounded shall have a constant value. 

 This requires that the determinant under the integral sign 

 shall be constant, which may be written 



... n 

 dt d(a,...h) = * (36) 



This equation, which may be regarded as expressing the prin- 

 ciple of conservation of extension-in-phase, may be derived 

 directly from the identity 



gj <*(pi, ...g n ) d(pi', . . . q n r ) 



d(a, ...h) ' d(p l f , . . . q n ') d(a, ... h) 

 in connection with equation (33). 



Since the coordinates and momenta are functions of a, ... . h, 

 and t, the determinant in (36) must be a function of the same 

 variables, and since it does not vary with the time, it must 

 be a function of a, ... h alone. We have therefore 



...*). ' (37) 



