THEORY OF INTEGRATION. 27 



remaining constant (a) will then be introduced in the final 

 integration, (viz., that of an equation containing dt,} and will 

 be added to or subtracted from t in the integral equation. 

 Let us have it subtracted from t. It is evident then that 



Moreover, since 5, ... h and t a are independent functions 

 of r l , . . . r 2n , the latter variables are functions of the former. 

 The Jacobian in (71) is therefore function of 6, . . . ^, and 

 t a, and since it does not vary with t it cannot vary with #. 

 We have therefore in the case considered, viz., where the 

 forces are functions of the coordinates alone, 



Now let us suppose that of the first 2 n 1 integrations we 

 have accomplished all but one, determining 2 n 2 arbitrary 

 constants (say c?, ... h) as functions of r^ , . . . r 2n , leaving b as 

 well as a to be determined. Our 2 w 2 finite equations en- 

 able us to regard all the variables r^ , . . . r 2n , and all functions 

 of these variables as functions of two of them, (say r l and r 2 ,) 

 with the arbitrary constants <?,... h. To determine 5, we 

 have the following equations for constant values of <?, ... h. 



u-/ 2 ~~; ** T ~77~ t * v * 



da db 



df^i , r 2 ) c?7* 2 7 dTi . . 



whence -^7 TT- db ^- dr^-\- -= r 2 . (74) 



d(a, 6) c?a c?a 



Now, by the ordinary formula for the change of variables, 



= r 



J 



zn ) 



a ^ 



