272 MEASUREMENT OF CURRENTS. 



We shall assume that the duration of the transient current, as 

 well as the errors made as to the moment at which it starts, are very 

 small compared with the time of oscillation of the needle. It will 

 be sufficient to consider the pendulum movement, which may be 

 represented by the equations 



(40 



yx = U Q sin y/, 



in which x is the distance at the epoch /, u the velocity of dis- 

 placement, and u the velocity at the time /=0, which corresponds 

 to the position of equilibrium, the time T of oscillation being equal 



7T 



to -. 

 7 

 We notice, in the first place, that the ratio 



(42) ^ = * 2 =+yV 



gives the velocity for the position of equilibrium by the velocity at a 

 given instant and the corresponding distance, whatever be the origin 

 of the time. 



Let us assume, as a first case, that at the time / x we give the 

 sys^m an instantaneous velocity v 1 ; we shall have 



! = K O cos 

 (43) 



The motion is pendular, and may be represented by the equations 



u = u' cos yt yx' sin y/, 

 (44) 



yx = UQ sin yt + yx Q cos y/, 



in which u' Q and x' Q are the velocity, and the distance for the moment 

 /=0, which is no longer the epoch of the real or apparent passage 

 through the position of equilibrium. If we make f=f lt and take 

 count of equations (43),. we get 





