546-548] Lagrange's Equations 479 



Thus each component of velocity of each moving particle will be a linear 

 function of lf 2 , ..., from which it follows that the kinetic energy of motion 

 of the system must be a quadratic function of lf 2 , ..., the coefficients in this 

 function being of course functions of lt 2> .... 



Let us denote T W by L, so that L is a function of lt 2) ...0 n , 

 and of l} 2) ... n , say 



L (j)(0 l) 2 , ... n , 0i, &2, 0n)- 



If L-\-L is the value of L in the displaced configuration l 

 ... O n + S0 n , we have 



_ . 



/= X ...- -^- 



(7C/! 017^ (7 17! 



so that equation (496), which may be put in the form 



f$ = 0, 



Jo 



now assumes the form 



We have 



fr / a/. ra ar, . \ 



Wf-a**?^ 1 *)*- 1 * (498) - 



The last term vanishes since, by hypothesis, #, vanishes at the beginning 

 and end of the motion, and equation (498) now assumes the form 



r|J9 rf^l 

 Jo i (d0 1 dt\d0J) 

 Let us denote the integrand, namely 



90, 



by /, so "that the equation becomes 



The varied motion is entirely at our disposal, except that it must be 

 continuous and must be such that the configurations in the varied motion 



