20 Mr. S. B. McLaren on Hamilton s Equations and the 



should satisfy the same general conditions as before. For a 

 finite value of H all the q's must he finite. (12) is 

 evidently connected with the virial of Olausius, which is, in 

 fact, equal to 



^~ qr dq r 



Thus the virial is equally distributed and it is equal to the 

 kinetic energy. In particular, if His a quadratic function 

 the virial is identical with the potential energy and the 

 potential and kinetic energies are equal. 

 One more point may be noticed, 



ffx^dxdtf = (13) 



In (13) the integration is over the same area as in (8). 

 For 



x^dx dy =^ x{H q - H v )dx = 0, 



dy. 



1 



t 



(14) 



since H q = H v = H 2 . 



Similarly the second term on the right of (14) vanishes 

 and (13) is proved. 



For x and y in (14) we may substitute any two of the p's, 

 or if the necessary conditions hold also for the q's, any two 

 of the 2n coordinates. So 



>'f^ = °- < 15 > 



V^N = 0, (16) 



J*f£«r~o, ..... (17) 



j 3r ^N = 0, (18) 



(15) and (16) hold so long as r and s differ ; (17) and (18) 

 hold for all values. 



