28 CONSERVATION OF EXTENSION-IN-PHASE 



where the limits of the multiple integrals are formed by the 

 same phases. Hence 



d(ri,r z ) d(r^ ...r Zn ) d(c, ... h) 

 d(a,b) " d(a,...h) d(r 99 ...rj 



With the aid of this equation, which is an identity, and (72), 

 we may write equation (74) hi the form 



The separation of the variables is now easy. The differen- 

 tial equations of motion give r l and r z in terms of 'r^ , . . . r 2n . 

 The integral equations already obtained give <?,... h and 

 therefore the Jacobian d(c, . . . A)/c?(r 3 , . . . r 2n ), in terms of 

 the same variables. But in virtue of these same integral 

 equations, we may regard functions of r 19 . . . r 2n as functions 

 of r l and r% with the constants c, . . . h. If therefore we write 

 the equation in the form 



d(ri, . . .r 2n ) _ r 2 r i , 



' ~ **- ..h) dr *> (77) 



d(r s , ..r 2n ) d(r 8 , . . . r 2n ) 



the coefficients of dr l and dr% may be regarded as known func- 

 tions of r x and r 2 with the constants <?,... h. The coefficient 

 of db is by (73) a function of 6, . . . h. It is not indeed a 

 known function of these quantities, but since <?,... h are 

 regarded as constant in the equation, we know that the first 

 member must represent the differential of some function of 

 5, ... A, for which we may write b'. We have thus 



db ' = r * dr ~ ..h) dr *> (78) 



d(r 8 , . ..r an ) d(r 8 , ...r 2n ) 



which may be integrated by quadratures and gives V as func- 

 tions of r 1? r 2 , ...<?,... A, and thus as function of r 1? . . . r 2n . 

 This integration gives us the last of the arbitrary constants 

 which are functions of the coordinates and momenta without 

 the time. The final integration, which introduces the remain- 



