OF ENERGY IN A SYSTEM OF MATERIAL POINTS. 



719 



Substituting for the elements of this determinant their values as given by 

 equations (7), (9), and (10) it becomes 



dp, dp n dt 



3q?' ~ dq~," ~d 



dfr _dp^ _dt_ 



dqn" ~dq~n' ~dq? 



dp, dp n dt 



~ ~dE 



Now the rows in this determinant are the same as the columns in the former 

 one ; the accented and unaccented letters being exchanged and the signs of 

 all the elements changed. We may therefore express the relation between the 

 two determinants in the abbreviated form 



- /dp/ dp^_dtf\_ ( _ /dp, dp^ dt \ 



- (dq, ' dq n d/~ ( > 2 ' \dq,'" - dq n ' dE)~ 



Hence ds' da-' dt' = ds f ds dEZ + 



- \dq l dq n 



- ( - ) n+l ds ds dE^ + (^ 



- (dq, dq n ' dE 



= ( - ) n+l da- ds dt 



= ds da 1 dt (15). 



If we suppose the time, t t', to be given, dt = dt' and 



ds da-' = ds da- (1 G), 



or dq,' dq n 'dp,' dp n ' = dq, dq n dp, dp n (17). 



The initial state of the system is a function of 2n variables. We have 

 hitherto supposed these to be the n co-ordinates and the n momenta, but 

 since the total energy E is a function of these variables we may substitute 

 for one of the momenta, say _p/, its value in terms of the n co-ordinates, 

 the n-l remaining momenta, and E, and thus express every quantity we 



