2 i 3 .] LAGRANGE'S EQUATIONS. U5 



as appears by carrying out the indicated differentiations with 

 respect to /, and if we now make use of the relations (4) and (5), 

 our equation assumes the form 



d . te . d . ds\ . dx Q . dz 



The quantities in the two parentheses, each multiplied by m, 

 are easily recognized as the partial derivatives with respect to 

 q and q of the kinetic energy 



hence the equation reduces to the form 



*L-*L=Q (7 ) 



dt dq dq V> 



known as the (second) Lagrangian form of the equation of 

 motion of a particle constrained to a curve. 



213. Particle Subject to One Condition. By Art. 200, the equa- 

 tions of motion are 



(8) 



with the condition 



$(p t y,z,t)=o. (9) 



Let the two generalized co-ordinates q^ q^ be connected with 

 the Cartesian co-ordinates by the equations 



The first of these equations gives 



r 



dt dq 



hence, regarding x as a function of /, q^ q^, 

 ) we find : 



