246 Mr HOPKINS ON AERIAL VIBRATIONS 



Since b is less than unity, and n soon becomes a very high 



number, after an extremely short time the first line in this expression 



may be neglected, as may also all the terms in the other lines in- 

 volving high powers oi h. 



Whence it follows that the original disturbance (on which the 

 form of the function f will depend), will cease in an extremely short 

 space of time to have any effect on the form of the existing vi- 

 bration, supposing the vibrations maintained by some cause distinct 

 from that producing the original disturbance. 



Also, if the cause maintaining the vibrations cease, the vibrations 

 themselves may cease in an extremely small space of time. 



The inferences we have drawn from the former developement (E) 

 of the expression for v, may be drawn from this and perhaps with 

 still greater facility. 



15. If we suppose >|/' (ss) always = 0, the expression for v will 

 reduce itself to the same as that given by M. Poisson. But in this 

 case it will be observed that all the functions involving the quantity c" 

 disappear, which renders it impossible to account on this theory for the 

 position of the modes or points of minimum vibration as determined 

 by experiment*. For the purpose of determining the positions' of 

 these points theoretically we will recur to the equations (C), the first 

 of which is 



~ v = f{at-x)-f{at-{^l- X)} +^ {at -{2l + c - X)} (6). 



If we neglect ^{at-{2l+c — x)}, (or suppose the substance with 

 which the tube is stopped perfectly rigid) we shall have » = 0, when- 

 ever 



{at — x) - {at — {^l- x)}=Q, or mX, 



{m being any whole number), or when 



{l-x) = m.-. 



* See Art. 36, Sec. II. 



