THE MEASUREMENT OF CURRENT 25 



dealing with both current and ballistic galvanometers. An im- 

 portant special case is that of the critically damped instrument of 

 the D'Arsonval type, as ordinarily used, and also when the 

 period has been so reduced that the instrument has become an 

 oscillograph capable of following a complex wave in its variations. 

 The Equation of Motion. 1. The angular deflection of a 

 reflecting galvanometer is always small so the deflecting moment 

 at any instant may be taken as proportional to the instantaneous 

 value of the current and represented by Ci where C is a constant 

 depending on the construction of the instrument. 



2. For small deflections the restoring moment will be pro- 

 portional to the angle through which the system has been turned, 

 that is, it will be equal to rB where is the angle of deflection and 

 T is the restoring moment for unit angular deflection. 



3. The system as it moves is retarded by air friction, etc., and 

 in some cases by induced currents. It is customary to assume 

 that this retarding moment is proportional to the angular velocity 



of the system, and is therefore represented by k , where k is the 



coefficient of damping.* 



Let P be the moment of inertia of the movable system. The 

 total moment acting to change the angular velocity of a body 

 rotating about a fixed axis is the product of the moment of inertia 

 and the angular acceleration. On equating this product to the 

 sum of the turning moments acting on the system 



Consequently, the motion takes place according to the equation 



*This law of damping was introduced by GAUSS and W. WEBER in their 

 study of the behavior of the vibrating magnets used in their magnetic 

 measurements at Gottingen, 1836-37. With air damping, in order that this 

 law be reasonably well fulfilled, the damping must be slight, the amplitude 

 of the vibration small and the restoring moment due to the suspension 

 large. That this law is not absolutely exact is apparent, for, according to 

 it, the movable system when once set in vibration would continue to swing 

 for an infinite time with a constantly decreasing amplitude. But it is a 

 matter of common experience that the system comes to rest in a compara- 

 tively short time. However, the results obtained by GAUSS'S theory are 

 in close enough agreement with the observed facts to warrant its use, 



