180 Dr. J. Hopkinson on the Effect of 



expressed by the equation 



„ A fkrftx nx . , . nx ~\ 



t= — 7—S cos — sin ?« + sm — cos nt > • 



irkn L a a a J 



Let the amplitudes observed at the node and middle of ventral 

 segments of the string be a, /3 ; we have 



Anl 



irkn 



(12) 



therefore 



, 2a a _ a 1 



the result being expressed in seconds. It is worth noticing 

 that the vibrations throughout the ventral segments in this case 

 are nearly a quarter of vibration behind the extremity in phase. 

 If the theory of friction here applied be correct, many impor- 

 tant facts could follow from a determination of the value of k in 

 different substances — for example, the relative duration of the 

 harmonics of a piano-wire. 



Let us now calculate what is the work done by the force 

 maintaining the vibration of the extremity. The force there 

 exerted is 



\dx dx dtP 

 and the work done in time dt is 



*(2 ♦'«£)§* 



x being put equal to /after differentiation. We have then work 

 done from time to time t 



-f{-(f«3)i}* 



i— i 



In estimating the work done in any considerable period, we 

 may exclude the periodic terms as unimportant. Hence work 

 done on extremity of string 



x=l 



{x = l) 



