1122 



F being the expression for p^ which is obtained bj solving H=Ci 

 p,=F{p,...,p„,q,,...,qu,c,;a) (41) 



When this expression is substitnted in H = c^ the result will be 

 an identity. By differentiating this with respect to c,, f>,, ...,p„, 5',, ... ,(/„, 

 we find 



dF dF , 



d/y_ 1 ^H _ d^idH_ d^.| 

 dF~dF%i~~dF'dp^'~~dF' ... (42) 



dcj dc, dc, 



from which the Hamiltonian equations are easily derived as follows 



dp,__d_F ö^__ö^ d^^dF^ 



dp, dqx dq, dpx dq, Of, 



Let us now form the derivatives of the transformation-function V 

 with respect to all the variables which it contains: 



ör_ bV _rbF _ bV _rbF _ bV_rbF 



ö?i bqx J ^qx ' '' d/>., J dpx ' '' ÖC, J dc^ ' 



Evidently 1^ forms the transition from the variables p^,...,p,„ 

 <{i, ' ■ ■ , qu to the variables />,, . . . ,/;„, q,, . . . , q„, Cj, /,. Of all the 

 rheoparametric differential equations we only need the equation for 

 c' here, viz. 



d rdV\ d f rdF \ 



The integral inside the brackets only depends on t through q^, 

 hence 



«■--ffO '^^'' 



We now form the mean value according to (39): 



r r dF 



— I ... Idp^ . . . dp,, dq^ . . dq,i — 



r r dF 



I ... \dp^. . .dpndq^ . . . dqn 



(45) 



dc. 



where in the denominator Ijq^ has been replaced by ^ accordmg 



to the last equation of the set (42'). It is easily seen, that the 

 numerator and denominator are the partial derivatives with respect 

 to a and c, respectively of a function V of the form 



V z=. j ... I dp^ .. .dp„dq^. .. dqn, .... (46) 



