24 Prof. A. Gray on Canonical 



It is in general to be understood that the coordinates, the 

 <?'s, at time t, are independent functions of the initial co- 

 ordinates and momenta. That is, varying these coordinates, 

 subject, say, to the condition that only the b's are changed, 

 we get k equations of the form 



*=IW»> (3) 



The condition for the independence of 8q u Sq 2 , , $q k is 



the non-evanescence of the determinant 



w - d&i "dh 'db^' 



Since ^1,^2, ..-., q k are independent the speeds q ly q 2 , . .., q k 

 are also independent ; so also are the momenta for time t, 

 namely, 



BT BT ... 



as can easily be proved. Now by the canonical equations 

 . BH . BH . dH 



we have & equations of the form 



=1 'dpjdpi 



>k 



&Q; = 2 ^^r % ( 6 ) 



and the determinant 



d 2 H B 2 H d 2 H 



2 + 



dpi 3^i c^ 2 Bp 2 " " " cfe dp * 



does not vanish. 



It is to be understood in what follows, except where the 

 contrary is specified, that these conditions of independence 

 are fulfilled. 



4. Reserving then a 5 , a 2 , , a& to denote the initial co- 

 ordinates, we shall denote the constants in S by a 1} a 2 , ...., a*, 

 and prove that the substitution of any other set of con- 

 stants c 1? c l3 , Cfa say, connected with the us by /j inde- 

 pendent equations of the form 



a l=</>l(^ £l5 ••••5 ^, C l3 , Cfc), 



a 2 = ^ s (*,^i, , £/„., c l3 , c A ), etc. . (J) 



