167 



§ 3. Proof of formula (8). 



In the expression 2pk(lqk q and p are replaced by their expan- 

 sions (1); in differentiating the Q, P, and ^ are regarded as independ- 

 ent variables, the parameter a being an explicitly given function 

 of t. This gives : 



2 pu . dqT, = ^ fk . dQk + 2 ƒ J . dPk + /; .'-.dt 



^ " i eft 



ƒ 1 f2, f 3 are Fourier series with respect to the Q. 



As for a = constant this substitution is a contact transformation, 

 we must have: 



dW dW 



^7Ï . dQic + :sfk .dP, = :E Pk dQt + 2:---dQk + 2-- dPj, (i2) 



Hence : 



dW , hi cos j 



and : 



W=:E (- Pk + yj) Qjc + ^^ rf,«i....,« . {ni,Q, + . ; 7n„Q„) 

 Furthermore we have: 



oP/t ^ oPi- oPyfc (sm] 



In ƒ2 the (3 occur only under sines and cosines ; from this it 

 follows that the coeflicient of Qk on the second side of the equation 

 must be zero, and hence: 



rj == Pjfc + M^)' 



As the condition (12) determines the P and Q all but the additive 

 constants, it is always possible to include the Jtki^) in the P. If 

 we suppose this to be the case, we get: 



vl = Pk, 



hence : 



I COS i 

 . {m,Q, + ...m,.Q„). . . . (13) 

 sm) 



It follows that : 



oa 



is a Fourier series ivith respect to the Q, and thus the proposition 

 has been proved. 



