330 GENERALIZED COORDINATES 



To abbreviate, let us denote > - > by x, 0,, - . Then the 



at at 



equation just obtained may be written 



Q/3 1 ' y}/3 ^ ^^ )/) ""* \ / 



so that x is a linear function of 0^ 2 , , B , the coefficients being 

 functions of 1? 2 , -, n . 

 The kinetic energy, 



is now seen to be a quadratic function of V 2) , n) the coeffi- 

 cients being functions of lt 2 , , ^ n . 



The potential energy W depends only on the configuration of 

 the system, so that W is a function of lt 6 2 , , 6 n only. 



Thus the function L, or T W, is a function of 



L = lift, 6 2 , - . ., n , 6 19 6 V . - -, tfj. (160) 



say 



The corresponding function L' in the displaced motion is the 

 same function of 



9, + W,, 2 + S0 2 , ..-, etc., 

 so that 



By Taylor's theorem, we may expand L' in the form 



or, from equation (160), 



