344 GENERALIZED COORDINATES 



If we give the system a small displacement, in which t is 

 increased to 1 + &0 l> 2 to 2 + 80 2 , etc., we have, on equating 

 two different expressions for the work done : 



On substituting the value of &x obtained on page 340, this 



and the equation reduces to 



This is the same equation as equation (175), and the different 

 forms of Lagrange's equations can be deduced as before. 



Lagrange's Equations for Impulsive Forces 



276. Let a system of impulses act during the short interval 

 from t = t t to t = t 2 . Let lt 2 , , n now be supposed to be 

 independent coordinates, so that Lagrange's equations are 



d 



If we multiply by dt, and integrate from t = t l tot = t t) we have 



r 



1, 



, e ,, 



The value of the first term is 



