TRANSACTIONS OF SECTION A. 631 



The second term may be written 



dx _ d d.v dx / d ^ _ ^ <^i' \ 



dt dqm tit dt \ dqm dt dt dg^ / 



The two first terms give then the Lagrangian equations 



d dT _ dT 

 dt dq„i dq^ 



It remains to consider the integral corresponding to 

 dx / d d.c _ d d.r \ 

 di V rfy,„ dt di dqm ' 

 with analogvies in y, ». 



It will be convenient at this stage to suppose the variables two in number 

 q , q.^, the spirit of the proof being general. AVe have then 



d dx d ( ' dx ' dx \ ' d dx ^ ' d dx 

 . — = I q ^. q ] = q — + q 



dq^ dt dq^ \ \ dqy 2 dq.2 / 1 dij^ dq^ 2 dq^ dq^ 



d dx _ ' d dx , J ^ ^ 

 dt dq^ I dq^ dq^ 2 dq., dq^ 



the corresponding terms thus becoming 



dx ' / f^ ^ _ ^_ _^ \ 



di 2^ ^9i ^Ui ^2 ^9i ^ 



Similarly 



dx / d dx ^ d^dx\ _ dx ' f <^ dx d dx \ 



dtXdq^^t dt dqj ~ dt i Vrfg'j dq^ dq^ dq,J 



Let now 



d dx d dx 



^9i dl\ ~ ^Qx ^li ~ 



d dt/ _ d di/ _ Y 

 dq^ dq^ dq^ dq„ 



d dz ^' dz _ rj 



dq^ dq^ ~ dq^^ dq^ 



Then we have evidently only to show that 



dt ) 



Now consider the total increment A .t, A «/, A (~) arising from integrating 



dx J , dx , 

 —~dq,+ .~~-dq 

 dqi dq.^ 



along a closed course of ^j, q.y The result by a well known theorem of Stokes 

 gives 



A.I = X<7^, dq.^ 





If we suppose the circuit of q^, q^ indefinitely small these become 



Kdq^ dq.^, Ydq^ dq^ Zdq^ dq.^ 



