KINETICS OF A PARTICLE. 



[214. 



The right-hand member of the former of these equations is 



equivalent to -j ~ As similar relations hold for y and z, we 

 dtdq l 



find again the relations (4) and (5), with q l substituted for q. 

 It can be shown in the same way that these relations also hold 

 if <7 2 be written for q. 



214. Let us now multiply the equations (8) by 5/i/d^, 

 !, and add them. This gives 



where 



Similarly, multiplying (8) by dfi/dg 2 , 

 adding, we find 



2' 5 /3/%2> and 



m 



where 



Each of these equations (n) and (12) can be treated by the 

 method used in Art. 212, and we find as the final equations of 

 motion on the surface (9) in the Lagrangian form : 



dT 



d_ 



dt 



(13) 



215. Free Particle. In this case three variables q v q^ <? 3 are 

 required to determine the position of the particle. If the 

 expressions of x, y y z in terms of these new variables do not 

 contain the time explicitly, the introduction of the new varia- 

 bles consists merely in a change of co-ordinates. If they do 

 contain the time, i.e. if we have 



the new system of co-ordinates is a moving system. 



