785 



indiictioneqiiation. For in consequence of the motion of the string 

 in the magnetic field the number of lines of force passing through 

 the circuit changes to an amount proportional to 







Expressed in the units used by Dr. Okkhoke, the induction-equation 



now takes the form : 



/ 

 dJ rdu 



^ ==''■' +"-1^ + "J i''^ («) 







vv^here E is an external electromotive force acting on the circuit. 



§ 2. The problem of finding the vibrations governed by the 

 equations (1), (3) and the condition y = for .i' = and x = I, 

 can be easily solved. First, let E be 0, and so the question of free 

 (damped) vibrations may be put. Suppose that 



where (^ is a function of ./; and 1 is a constant. Then the equations 

 change into 



dV/) HI 

 — oi //) -[- wixtp — a" - — 1= 



I 

 =zRl -\- LimI + iBto I (f dx . 







Hence 



/ 



(a>^ — iox.) (f -\- cr 



d'cp HH(o 







^x' (>{E+L 



o r 



I (f dx . 



Putting at'' — iuix in the first member ?r and =: n we 



Q{R^Li(Jt)) 



have 



cf + a' — =pj <fdx. 



This equation may be satisfied by 



. n ^ n 



(f =z A cos ~ X -\- B sin — x -{- C 

 a a 



provided that 



52* 



