Experiments ivith Alternating Currents. 253 



is hard to say what decides the cessation of vibration ; if the 

 platinum point acted perfectly, L e. if the least rise broke 

 the circuit completely, and the least depression made the 

 circuit, there would, I think, be no limit to the frequency 

 obtainable. 



If we assume that it is necessary for the platinum point to 

 move a certain minimum distance before a break is made in 

 the circuit, then, for the wire to continue vibrating, 



E must varv as 



n 



Theory indicates that the frequency obtainable by the appa- 

 ratus discussed could be raised by increasing E and at the 

 same time adding resistance free from induction. 



The preceding discussion applies to the apparatus under 

 consideration ; but the subject can be treated more generally. 



Let it still be assumed that F varies as i ; so that we may 

 putF = Kz. 



Let ,v=a sin.pt, 

 where x= distance of platinum point below its middle position, 

 a = maximum deflexion of vibrating wire. 



2tt 



and T= time of vibration. 



Let i = =5- then 



i=2 (l — e l yix. 



The work done by the wire against magnetic forces whilst 

 going downwards equals \Fdx or [Kidx, taken between the 

 proper limits ; the work done on the wire whilst ascending 

 also equals \Ki da, taken, of course, between other limits. 



I omit the details, which are tedious, and give the final 

 result, which is that if W equals the resultant work done on 

 the wire in one vibration, then 



-RT 



^r Ki pdRL(l + e l ) 

 R 2 -I-Ly ? 



If T is small, it can be shown that this expression agrees with 

 the previously obtained result. 



In order to obtain the greatest amplitude for a particular 

 frequency, W must be a maximum ; its maximum value 

 cannot easily be found — at any rate, by the ordinary treat- 

 ment ; without finding the particular value, however, it can 



