IS AN INVARIANT. 13 



r dQ x dp y 



^^ n /^dQ L \ = _d_de, == d^ 

 dQ x r^i W& %J d& cZft, d& ' 



But since f' 



r i \ a (j/ r / 



d -k = ^. (28) 



*& ^0. 



Therefore, 



...g n ) 



... Q n ) 

 The equation to be proved is thus reduced to 



which is easily proved by the ordinary rule for the multiplica- 

 tion of determinants. 



The numerical value of an extension-in-phase will however 

 depend on the units in which we measure energy and time. 

 For a product of the form dp dq has the dimensions of energy 

 multiplied by time, as appears from equation (2), by which 

 the momenta are defined. Hence an extension-in-phase has 

 the dimensions of the nth power of the product of energy 

 and time. In other words, it has the dimensions of the nth 

 power of action, as the term is used in the ' principle of Least 

 Action.' 



If we distinguish by accents the values of the momenta 

 and coordinates which belong to a time ?, the unaccented 

 letters relating to the time , the principle of the conserva- 

 tion of extension-in-phase may be written v * < 



//" /* /% 



... I dpi . . . dp n dqi . . . dq n = I ... I dpj . . . dp n f dqi r . . , dq n ' } (31) 

 *J *J *J 



or more briefly 



r r r 



>! 7 . . . dq^ (32) 



