699 



Better results are obtained if the effect is magnified by reso- 

 nance. For this purpose an alternating current having the same or 

 nearly the same frequency as the osciUations of C about the wire 

 D is made to flow through the coil. 



For the oscillations of C about the vei'tical axis under the influence 

 of the couple 6 we have the equation 



6 =QÏt ^ &a + Pa (6) 



in which the angle «, the deviation from the position of equilibrium 

 is reckoned positive in the same dij-ection as the current in the wind- 

 ings. Q is tlie moment of inertia. & the torsion constant of the 

 wire and P a small coefficient of friction. Instead of and P we 

 shall introduce two new constants 



'0 P 



Q' "-=2^ ^'^ 



the first of wliicii is 2.t times the free frequency, as it would be 



in the absence of friction, wiiereas y. is the constant of damping. 



Indeed the free oscillations (the equation for which is deduced from 



(6) by putting 8 = 0) are given by 



a = Ce~'' cos {V u)„^ — tt t -\- p). 



The differential equation (6) is easily solved if we develop & as 



a function of t in a Foukier series. Now according to (5) has 



dl 

 the same phase as . Hence, if the magnetisation were proportional 



dt 

 to the current we could directly represent Ö as a harmonic function 

 whose phase would be i .t in advance of that of the current i in 

 the coil. The proportionality will, however, hold for small intensities 

 only. If the amplitude of i is made to increase so that the magneti- 

 sation approaches saturation, the magnetisation cnrve takes a diffei'- 

 ent form. Finally, for very large amplitudes of /, the magnelisation 

 will suddenly pass from one saturation value into the opposite one, 

 simultaneously (except for a small difference of phase) with the 

 change of direction of the current. F'or this limiting case the cal- 

 culation will now be made. 



The couple acting on the cylinder may be represented by fig. 1, 

 in which the sinusoid refers to the current / '). 



dl 

 1) The curve with the sharp peaks represents the value of -r-, to which the couple 



dt 



is proportional. It was obtained in the following way. The iron cylinder, which 



had its right position along the axis of the coil K, was surrounded by a narrow 



glass tube covered with windings and immediately beside this tube a similar one, 



equal to it and covered in the same way, was placed. The windings of the two 



