248 Mb HOPKINS ON AERIAL VIBRATIONS 



By our hypotheses, x (*) must be always greater than \// (%) ; and 

 if we suppose c and c^ less than the least value of z, which gives 

 to ^ (%), or X (^) its maximum value, it is manifest that from this 

 last equation, c, must be considerably smaller than c, and must be 



c 

 affected with a different sign. Suppose c^ = j^, where k is consider- 

 ably greater than unity. It follows then that if the phase of the 

 vibration of the extreme section of a stopped tube be retarded by a 

 certain quantity c, the phase of the actually reflected wave will be 



c 

 accelerated by a quantity t. 



18. Giving then the proper sign to c„ we have 



v=f(at-x)-x{at-(2l-^-a;)} (7), 



and to determine the points of minimum vibration, we may observe 

 that this expression is exactly the same, as if the wave for which 



v, = x{at-{2l-^-x)}, 

 were reflected immediately from a section B' whose distance from A = l — —x. 



Suppose a rigid diaphragm at this section constrained to move 

 exactly as the fluid does there ; we may then suppose the actual 

 stop B removed, and the points of minimum vibration will remain 

 the same. 



Now to determine them in this case, we observe that whenever 

 at — x = at—{2l — T — x) + m\. 



the value of v will be the same as when 



c 



at—x = at—{2l— T - x). 



In the latter case 



