s, 



Relations in General Dynamics. 25 



does not cause any change in the values of the _p's (initial 

 and final) provided the value of the integral 



(' {l(pq)-R\dt, (2) 



is taken as the complete value of the principal function. 

 [Of course it will be understood that the substitution of 



the values of g 1? q 2) , q k , in terms of t and the 2k constants, 



the a's and the 6's ; leads to equations of the form 



«i = X 1 («i J .•->«*> h, ■■..jW, a 2=X 2 ( a n •■'-•> a k, hi ""> Wv •] 



The use of this form of the principal function amounts to 

 the introduction of an additive constant as in the equation 



S 1 = S(*,g'i, ...., g k , ol 1} ...., * k ) — S , ... (3) 



where S is what S becomes when r is substituted for t, and 

 the initial coordinates, the a's, for the ^'s. Now since BS/^a 

 does not change with £, (3) gives the relations 



B(S — S ) _ ^ d(S-Sp) _ ^ d(S — S ) _ ,^ 



Let the a's in S T be replaced by their values as given by 

 (1), thus introducing the c's. Denoting the value of "d&ifdq 

 taken after this substitution by (dSi/cty), we obtain 



/3Si\ _ d§ , SS_i^ai , dSiBf^, .^1^ = ^ (fy\ 



\'dq/ "dq d«i d# 'd^'dq "" cW ~dq 'dq 



since by (4) 



dSi/Botj = BSj/^ao — BSj/^^/r = 0. 



Thus the values of the p's are not altered by the change to 

 new constants. 



The coordinate q appears explicitly only in S, and the 

 corresponding initial coordinate a only in S. As 



BSi/3«i = 'd&ij'dciz = = 0, 



the appearance of the a's in the a's may be neglected ; and 

 so we have 



dS dS 7 , M 



_ 5i = * Ta=''' (,>) 



for the corresponding momenta at times t and r. 



If we suppose the time to flow back from t to t and the 

 momenta to be reversed for every value of t without change 



