of Energy hi a System of Material Points. 301 



iji . . .p n without index) maybe calculated as functions of these 

 2n + l independent variables. If T be the kinetic energy at 

 the time t, then V + T=E is the whole energy of the system. 

 These quantities may, of course, also be expressed as functions 

 of the 2n + 1 independent variables*. 



If we imagine each of the 2n + l quantities </i . . .p„E actu- 

 ally expressed as a function of the 2n + 1 independent variables, 

 we obtain 2n + 1 equations between 4m + 2 variables. Hamil- 

 ton's method consists in introducing in place of the indepen- 

 dents hitherto chosen, which Are may call the " old indepen- 

 dents," other independents (the Hamiltonian independents). 

 We may, in fact, from the 2^ + 1 equations express any 2/i-f-l 

 variables out of the 4n + 2 variables occurring as func- 

 tions of the remaining 2n + l. Hamilton supposes the vari- 

 ables p x . ..pmp'x . . .p' n , t expressed as functions of g t . . . q n , 

 q\ . . . q r n ^l ; so that the last-named variables play the part of 

 independents. Each of the first-named variables is therefore 

 now to be regarded as a known function of these 2n + 1 inde- 

 pendent variables. Starting from these Hamiltonian indepen- 

 dents, we easily find 



dp' r __ dp s dp' r _ dr dr _ dp r r 

 dq7 s "~~dqTr M ~ dg 7 ^ dj^dW' 



where r and s are any equal or unequal numbers. 



Just as the product of the differentials of three rectangular 

 coordinates dec, dy, dz may be expressed by the product of the 

 differentials of polar coordinates and then becomes equal to 

 r 2 sin 6 dr dd d(f), so if any m variables v l} v 2 . . . v m are func- 

 tions of m other u l} u. 2 . . . u mj the product of the differentials 

 of the first variables may be expressed by the product of the 

 differentials of the 

 tional determinant 



dvidv 2 . . .di' m = diudu 2 . . ,du m ^± -r 1 -^ -—• ' 0-) 



du x au 2 du m 



* E will not contain r, and will therefore simply be a function of 

 g\ . . .p' n , since it remains constant during the whole motion. 



t This follows thus: — If the magnitude A=2 f T Tdt be expressed as a 

 function of the Hamiltonian independents, then Hamilton shows (Thom- 

 son and Tait's ' Natural Philosophy/ new ed. § 330, equation 18) that 



„r__dA 1} _dA dA. 



whence it follows at once that 



dp'r _ dps _ d 2 A 

 !qs~~ dq' r ~ dq r dq' s ' 

 and so on. 



