118 THEORY OF NEWTONIAN FORCES. [PT. I. CH. III. 



vanish at t t and t = t l} so that the integrated part vanishes, 

 and 







Now if all the 8qs are arbitrary, the integral vanishes only if 

 the coefficient of every $q r is equal to zero. 



_ 



dq r dt\ dq r ' 



or if we write L for the Lagrangian function T W 



d /d 



Since the potential energy depends only on the coordinates, 

 , = 0, and we may write the equation (6) 



__ 



~ ~ 



P r is the generalized component of impressed force tending to 

 increase the coordinate q r . 



If the system is not conservative, we must write, instead 

 of - 



and the integral is 



(8) BfTdt + /2 [X r Sx r + Y r Sy 



= S/(T + 2Z r ^ r + F y2 / r 4 Z r z r ) dt. 



-.T cv Sa? r rx 3#v ^ 9^? r P, 



Now t+.gti+gti+ ...... +g^?-. 



so that if we "write 



^ ( v dx s v dy s 7 3^4 



(9) p r = 2g j^_ + Fs _ + ^_| > 



we get the same equations as before. If there is a force-function 



and 



_ 



^ 



dw 



dq r ' 



