36 Prof. A. Gray on Canonical 



Since S is a function of the g-'s, the as, and t, we have 

 when t and t are unvaried, 



8S 1 =2(^g ? )+2(^8a)=2(^ ? )-2(&8a), (7) 



which is again (1) of § 13. It follows that S 2 is here also 

 stationary with respect to paths which nearly coincide with 

 the actual trajectory and lie between the same terminal 

 points. 



The analysis for this more general case is thus in all 

 respects the same as before, provided we use H' = H— 2(Q^) 

 instead of H, and, if required, L' = L + 2(Q#) instead of L 

 (see § 13) as the value of the so-called kinetic potential. 



17. So far no use has been made in these variations of the 

 second Hamiltonian function S\, defined by the equation 



& 1= P{2(i>?) + H}eft (I) 



This gives for a varied trajectory 



^=2(1^) +S(^S&) =2( ? a i >)-2(«S6), (2) 



when S' x is expressed in terms of the p's and the b f s. 



18. Let now A be defined by the equation 



A=S 1 +pH<ft 5 (1) 



•s r 



then we have 



A=( t -Z(pq)dt=Z($pdq) (2) 



A is what is usually called the action for the motion from 

 the epoch t to the epoch t. It may be described either as 

 the time integral of %(%>q) [which, when T is a homogeneous 

 quadratic function of the q's, is 2T] or as the sum of the 

 displacement integrals of the p's. 

 Integrating in (1) by parts we get 



A+p t^dt^ + m-ILoT, . . . (3) 



where on the right H refers to the epoch t. From this we 



