INFLUENCE OF THE DURATION OF THE SWING. 273 



Equation (42) enables us to calculate the velocity v with which the 

 movable body passes through its position of equilibrium 



cos y/ t 



v/ 



sin 2 ; 



we get from this, if the time ^ is very short, 



' 1 



( 6} 



~ 



2 (U + Vtf 2 (K 



The velocity V Q is less than the velocity u + T\ which the movable 

 body would have acquired, if the discharge had taken place at the 

 ,moment /=0. 



890. It is easy to generalise these expressions. If we impart to 

 the system any given series of instantaneous impulses u lt u' ly u( .... 

 at the times t lt / 1? t'( . . . the modified motion may still be repre- 

 sented by equations (44), with the following values for the period 



The sums ^ should be replaced by the integrals when the 

 impulse is continuous. Suppose, for instance, that the impulse, 

 starting at the time / lasts for a period 9, and gives an acceleration 

 w at the time /; we have then 



r 

 \ 



t +e 



w cos y/ dt, 



y.rjj = w sin y/ dt. 



J/o 



891. We shall first of all apply these formulas to the case of an 

 impulse kept uniform for the time 0, which represents the case of a 

 constant transient current. The acceleration w is then constant ; 

 and the total impulse given to a free system, or to a system having 

 a very great period of vibration, will be 



v= 



VOL. II. 



