14 CONSERVATION OF 



the limiting phases being those which belong to the same 

 systems at the times t and If respectively. But we have 

 identically 



/.../*,..., ,-/.. 



for such limits. The principle of conservation of extension-in- 

 phase may therefore be expressed in the form 



g) -, xooN 



..g.9 = 1 ' 



This equation is easily proved directly. For we have 

 identically 



d( Pl ,...q n ) _ d( Pl ,...q n ) 



g.'O <*(M g.O ' 



where the double accents distinguish the values of the momenta 

 and coordinates for a time if'. If we vary t, while if and t" 

 remain constant, we have 



d_ d( Pl , ...q n ) _ d( Pl " 9 . . . q n ") d_ d( Pl , ...q n ) 



Now since the time if' is entirely arbitrary, nothing prevents 

 us from making if 1 identical with t at the moment considered. 

 Then the determinant 



- ?") 



will have unity for each of the elements on the principal 

 diagonal, and zero for all the other elements. Since every 

 term of the determinant except the product of the elements 

 on the principal diagonal will have two zero factors, the differen- 

 tial of the determinant will reduce to that of the product of 

 these elements, i. e., to the sum of the differentials of these 

 elements. This gives the equation 



d 



_. 

 dt d(pj>, . . . q n ) dp," ' dp n " dqj* ' dq n 



Now since t = t" , the double accents in the second member 

 of this equation may evidently be neglected. This will give, 

 in virtue of such relations as (16), 



