560 Mr. M. Solomon on the 



Examining the question theoretically, and making the usual 

 assumption that the retarding forces are proportional to the 

 first power of the velocity, one arrives at the conclusion that 

 the time taken for the amplitude to become IJm of its initial 

 value is independent of the strength of the controlling field, 

 and so the time taken by the needle in coming to rest from 

 a given initial deflexion should be the same whether the 

 period of vibration is long or short. 



For, making the assumption stated above, we have as the 



. equation of motion for a needle of magnetic moment M, and 



of moment of inertia I, swinging in a uniform magnetic field 



of strength H, 



T d~a , AT da TT _ r . _ 



I—, +^tt +HMsina=0, 

 at- at 



where a is the deflexion at time t, and N is the coefficient 

 allowing for the dampiiiy, by which is meant the resistance 

 tending to destroy the motion of the needle. This resistance is 

 due to the viscosity of the air and of the suspension fibre, and 

 also to the eddy-currents set up by the swinging needle in 

 neighbouring metallic circuits. Of these retarding forces 

 that due to the eddy-currents must be proportional to the 

 first power of the velocity : but it is possible that those due 

 to the viscosity of the air and the suspension fibre vary as 

 some function of the velocity other than its first power. 



When the angle of swing is small we may take a instead of 

 sin a., and if we put 



A N/I = 2w, 



and i ' 



HM/1=/, 

 the above equation becomes 



d' 2 a „ da <, 



_ +2n _ + y a=0 . 



This equation can easily be solved and gives for the periodic 

 time reckoned either from one maximum deflexion to the 

 next maximum deflexion on the same side, or, as the time 

 between two successive crossings of the zero in the same 

 direction, 



277 





Vr 



and if 8 be the decrement, or ratio of one complete swing to 

 the next, we obtain 



))TT 



B = e V ^ 2 > 



