LAGRANGE'S EQUATIONS 341 



Since this is true for all possible ranges of time, we must have 



at every instant. 



If the 0's can vary independently, each coefficient must vanish, 

 and the system of equations will be 





etc., while if B6 1) 50 2 , are connected by the constraints implied 

 in equations (166), (167), , we find, as in 272, that the sys- 

 tem of equations must be replaced by 



(177) 



274. These systems of equations reduce to those previously 

 obtained in the special case in which the forces are conservative. 

 For in this case consider the work done in a slight displacement in 

 which l alone varies, l being increased by 80 r It is O^^, and 



is also S0 lt so that] 



L oeingjncreasi 



FES 



rm, 3T _ dT dW dL 



Thus _ + 0l = ____ = __, 



cu_iu. 



since TTdoes not contain r Thus equation (175) reduces to 



d_ fd_L\ _ dL_ _ 

 dt \30J 00! ~ 



as before, and in the same way, equations (177), can be trans- 

 formed into the equations obtained in 272. 



