26 Prof. A. Gray on Canonical 



in amount, the initial and final configurations become inter- 

 changed, and the sequence of configurations is the former 

 sequence taken in the reverse order. The momentum cor- 

 responding to the coordinate a is then 'dS/'da ; and so in 

 the forward motion we have 



P--J (7) 



Oa K J 



If the constants are the initial coordinates (or as we shall 

 see the initial momenta) S may be dispensed with. But 

 then the values of "dS/'da obtained by putting t=r, and 



(qi, q 2 , )=(«!, a 2 , ) do not fix the values of the b's ;. 



on the contrary the values of the 6's, otherwise assigned, 

 fix the values of the derivatives. The values of the a's, or c 9 s > 

 must be so chosen as to make the derivatives, dS /3a, have 

 the proper assigned values. 



Considerations similar to these apply to the function S' 

 and the constants used in its expression. 



It is important to notice that by wdiat precedes the /' con- 

 stants used may be the initial momenta. S is then expressed 

 as a function of the initial momenta and the final coordinates. 

 But S' [§ 5, (1)] may be expressed in terms of the p's and the 

 mutual coordinates, that is of the initial coordinates and the 

 final momenta. Thus in a reversed motion when the time 

 flows back from t to t the function S for the forward motion 

 becomes the function S' for the backward motion, and the 

 reciprocal nature of the functions is further emphasized. 



Now in the forward motion ~d$'/"dp = q. Hence in the 

 backward motion we have dSyd&;= —at ; or by the remark 

 just made 



f h = ai ....... (8) 



when S is expressed in terms of the ^'s and the 6's. 

 5. The functions defined by the equations 



8=C{%(pq)-K}dt, 8' = C{Z(pq) + H.}dt , (1) 



Jr Jt 



when S is determined as a function of the q's and k con- 

 stants, and S' as a function of the p's and k constants, with 



s*-* * P = q > (2) 



obviously satisfy the partial differential equations 



1 +H=0 > if- H = (3) 



