262 MEASUREMENT OF CURRENTS. 



lation T, the needle has acquired sufficient velocity to complete 

 the arc 8 that is to say, the position of equilibrium with velocity 



2T 



zero at the period r ; but if the current is re-established the 



5 

 needle will be stationary. As there is always a little damping, 



and the current is not opened and closed at exactly the proper 

 times, the needle still makes small oscillations of which the mean 

 may be taken. 



In like manner to reduce the needle to zero, with velocity zero, 

 the current must be opened for a third of the oscillation, closed for 

 a second third, and then definitely opened. 



This rule is not sufficient if the damping is considerable. By 

 the formulas of (681), and for different values of A, Gauss calculated 



the fraction of the time of an oscillation during which the 



t t 

 current must be closed, as well as the fraction 2 l during which 



it must be broken, in order that at the time / 2 the needle reaches 

 its position of equilibrium with no velocity. 



The following is an extract from this table : 



The fraction l increases with the damping, and the frac- 



t t 

 tion - - varies in the contrary direction. Their sum is at first 



2 



equal to - , and increases very slowly with the value of A. 



o 



