28 Prof. A. Gray on Canonical 



suppositions as to the mode in which p and b i are expressed, 

 by enclosing the derivatives in brackets, we get 



m-(w • « 



Similarly from (8), § -4, we get the relation 



(A) 



(where S is supposed to be expressed in terms of the r/'s, the 

 h's, and t, and p and a { are expressed in like manner) which 

 retains its form when p is expressed in terms of the initial 

 coordinates and momenta and t, and a* is expressed in terms 

 of the final coordinates and momenta and f, so that 



©-(!?) <•> 



Two other reciprocal relations are derived in the same 

 way from the function S' and also retain their form when a 

 similar change of expression is effected for the quantities 

 differentiated. These are 



w~ W/' \W W/ 



(7) 



7. The parallelism of these equations with the canonical 

 equations is made clearer if we replace t by a symbol u 

 which we regard as a coordinate in the generalised sense, 

 and associate with a momentum v. Then if we put H' = H + r, 

 and write the additional canonical equations, 



dt ~ B« ( — lh dt~ -bu> ' ' * ' } 



we obtain, since ^H'/d^^^H/^^- ^H/^^ (by the canonical 

 equations), 



H + »=const (2) 



since 



(3) 



Thus the 



partial 



differential equation 



^S i XT ; 



^ r-J± = consi.. . 



becomes. 



which takes 



its place 



with others, 



such as 

 = 0, . . 











as the foundation of the various canonical relations. 



