12 EXTENSION-IN-PHASE 



second configuration will coincide with the third system of 

 coordinates. This will give D% = D s ". We have therefore 

 2V = 2>J. 



It follows, or it may be proved in the same way, that the 

 value of an extension-in-phase is independent of the system 

 of coordinates which is used in its evaluation. This may 

 easily be verified directly. If g 1 ^ . . ,q n ^ Q lt . . . Q n are two 

 systems of coordinates, and Pi, p n > P\i - P n the cor- 

 responding momenta, we have to prove that 



J'...Jdp 1 ...dp n d qi ...d qn =j*...fdP l ...dP n dQ 1 ...dQ n ,(2) 



when the multiple integrals are taken within limits consisting 

 of the same phases. And this will be evident from the prin- 

 ciple on which we change the variables in a multiple integral, 

 if we prove that 



. . P., ft, . . . ft) = 1 



>P n >2i, - 2V) 



where the first member of the equation represents a Jacobian 

 or functional determinant. Since all its elements of the form 

 dQ/dp are equal to zero, the determinant reduces to a product 

 of two, and we have to prove that 



d(Q l9 





We may transform any element of the first of these deter- 

 minants as follows. By equations (2) and (3), and in 

 view of the fact that the (j's are linear functions of the <?'s 

 and therefore of the _p's, with coefficients involving the <?'s, 

 so that a differential coefficient of the form dQ r /dp y is function 

 of the <?'s alone, we get * 



* The form of the equation 



d de p _ d df p 

 dp y dQ x dQx dp v 



in (27) reminds us of the fundamental identity in the differential calculus 

 relating to the order of differentiation with respect to independent variables. 

 But it will be observed that here the variables Qx and p y are not independent 

 and that the proof depends on the linear relation between the Q's and the p's. 



