DAMPING. 221 



On the other hand, the value of the moment of the couple which 

 acts on the needle at the time in question is 



(I + 1) MG cos (a + X Q + x) - Cj - HM sin (X Q + x) ; 



the equation of the motion of the needle is then (675) 

 (14) K- 



The elimination of I between equations (12) and (14) leads 

 to a differential equation of the third order, with transcendental 

 coefficients, even if we suppose the value of G to be a constant. 



The equation becomes somewhat simpler when we consider 

 very small oscillations about the position of equilibrium, but the 

 coefficients of the differential equations only become constant when 

 the deflection a and the initial displacement are themselves very 

 small. Taking into account equation (13), we may then write 



(.2)' 

 (14)' 



In this particular case, the motion of the needle on either side 

 of its position of equilibrium is independent of the original cur- 

 rent I . 



If we put 



C HM M 2 G 2 



and eliminate the intensity I of the induced current between 

 equations (12) and (14), we get the linear differential equation 

 of the third order 



T, 



+R 



