356 GENERALIZED COORDINATES 



Since the differentials dd l} d& 2 , - . do not occur, it appears that T' 

 can be expressed as a function of u v u 2 , - , 6 lt 2 , . . . only. We 

 can easily find its value; we have 



= 2 T, since T is a homogeneous quadratic func- 



tion of 15 2 , + n . 

 Thus T' = 2 T T = I 7 , 



showing that T' is equal to T, but is expressed as a function of 

 **!, tt a , -, 0J, 2 , .... Thus 



To illustrate, let 



so that M! = 2 (a^i + h6 2 ) , w 2 = 2 (7^i + 



Then, by definition, 



T = tii*! + tiai + ---- ^ 



+ 2 &, (A^ + 6 



2 

 J,/ ^From equation (201), we have 



dT 



In Lagrange's equations 



_^ /^A _ ^ _ 



^WJ ^~ 



ai d(T-W) dT 

 we have = i - '- = = 



