LAGRANGE'S EQUATIONS 335 



and then find the values of &0 l} S0 2 , -, 80 m from equations (166), 

 (167) . Let us, moreover, choose the m undetermined multi- 

 pliers X, ft, so that they satisfy the m equations 



4 (^-}~ IF +*i + /*&i+ = 0, (170) 



dt\xAj fa 



the suffixes ranging from 1 to m. Then equation (169) reduces to 



Inasmuch as S# TO+1 , S0 m+2 , , W n are all arbitrary, we may take 



and obtain 



dL 





and similarly we may obtain the same equation for all suffixes from 

 m + 1 to n. The equation has, however, already been supposed 

 true for suffixes I to m [cf. equations (170) , (171)]. 

 Thus we have the complete system of equations 



d 



in which the suffixes range from 1 to n. On eliminating the 

 m multipliers X, //., from these n equations, we are left with 

 n m equations, which enable us to determine the changes in 

 the coordinates. 



