252 Mr. Albert Griffiths : Some 



In the following the effect of the spark will be entirely 

 omitted. 



As a matter of fact, the oscillation of the wire cannot be 

 perfectly harmonic ; nevertheless, let us assume that the 

 motion of the wire is represented by the equation 



x = d sin pt, 



where d= maximum distance traversed from the middle point, 



2tt 



p = 



T 



x= distance of platinum point below its middle position. 

 The work done bv the electromagnets on the wire can be 



J -J o 



¥doc, taken between the proper limits where 



F varies as i, i. e. F = kt, where k varies as -^. 



±j 



It is needless to go into details, but the result is that the 



resultant work done on the string in one oscillation 



i UTd 

 varies as -^-= — , 

 ±j 



ETd 



i. e, as 



which shows that the energy given to the wire in one oscil- 

 lation varies inversely as the frequency if the amplitude is 

 unaltered. 



When the vibrating wire has settled down, the energy given 

 to the wire in each oscillation by magnetic causes must equal 

 the energy lost through other causes. 



I. Air resistance, which may be taken to vary as the square 

 of the velocity of the string. 



II. Resistance due to induced currents or damping, which 

 may be taken to vary as the velocity. 



III. Other fractional losses, which will be neglected. 



By equating the energy given to the energy lost, and 

 dealing only with the losses under (I.), I obtain the result that 



E varies as n 3 ; 



so that, as 4 cells were used to obtain a frequency of 500, to 

 obtain a frequency of 1000 of the same amplitude 32 cells 

 would be required. 



If the chief losses come under (II.) , then E varies as n 2 . 



In what follows, the losses under (I.) will alone be considered. 



If n increases, the amplitude diminishes, the relation being 

 that d varies as n~z. 



The wire ceases to vibrate if i is not sufficiently great ; it 



