HAMILTON'S PRINCIPLE 325 



An equation of this kind is true for each particle of the system 

 and at every instant of the motion. It is moreover true whatever 

 the displaced motion may be. On summing this equation for all 

 particles we obtain 



1 di ^ Xl + v ^ y ^ + w z ^ ~ (ufa + vfa + wfiwd 



= (Xfa+Y^y, + ZA)- (153) 



Now let T denote the kinetic energy of the motion, so that 



Then ST= 1 



Now u[ z u\ = fa + u^f u\ = lufiu^, 



if we neglect the small quantity of the second order (fo^) 2 , so that 



we have 



oT = m,i (ufiui + v 1 ov l + w l ow l ). 



265. Assuming for the moment that the system of forces is 

 conservative, let W denote the potential energy of the system at the 

 instant under consideration, and W 1 that of the imaginary system 

 in the slightly displaced configuration. Then, by 118, we have 



= (work done in moving system from actual 



to displaced configuration) 

 = - V (xfa+Yfa +Z 1 Sz l ). (154) 



Substituting into equation (153) for the expressions which have 

 been found to be equal to ST and &W, we find that this equation 

 reduces to the simpler form 



or again, 



