236 Mr HOPKTNS ON AERIAL VIBRATIONS 



The determination of the nature of these vibrations, or of the 

 function expressing the velocity at any instant of the extreme section 

 of the stop, will necessarily depend on the material of which it is made; 

 and any solution of the problem in question, independently of this 

 consideration, cannot be regarded as complete. Still, whatever may be 

 the nature of the stop, we know that the period of its vibrations must 

 be the same as for those in the tube; and it is also manifest, that each 

 vibration of the stop must begin at a time later by an interval at least 



nearly = -, (/= the length of the tube), than the corresponding vibration 



in the diaphragm at A, whence the original disturbance is supposed to 



proceed. I say that this interval is' nearly equal -, because certain 



phenomena, of which I shall speak hereafter, seem inconsistent with 



its being in particular cases exactly = -. I shall therefore, to give the 



ct 



assumption all the generality possible, consider it as generally = — f- arbi- 



trary quantity, to be determined in each particular case by experiment. 

 Hence then, if ^ denote the form of the function of the time expressing 

 the velocity of the extreme section of the stop, we shall have the 

 velocity = v/'l «/ — (/ + c)}, c being arbitrary. This must also be the 

 velocity of the extreme section of the air at B, consequently we have 

 as a second condition 



•^{at-{l-^c)}=f{at-l) + F{at^-l) (2). 



We have from (1) 



(t>{at + l)=f{at + l) + F{at + l); 



and eliminating F(at + l), 



f{at + l)-/{at-l) = <p{ai + l)-f{at-{l + c)\ ; 



or, writing at + 1 (or at, 



f(flt-ir^l)=f{at)-y\f{at-c) + <t>{at + ^l) (B). 



