from a Vibrating Body to a surrounding Gas. 403 



supply the deficiency behind. Now conceive the periodic time 

 of the motion to be continually diminished. Gradually the alter- 

 nation of movement becomes too rapid to permit of the full 

 establishment of the merely local reciprocating flow; the air is 

 sensibly compressed and rarefied, and a sensible sound-wave (or 

 wave of the same nature, in case the periodic time be beyond the 

 limits suitable to hearing) is propagated to a distance. The 

 same takes place in any gas ; and the more rapid the propaga- 

 tion of condensations and rarefactions in the gas, the more nearly 

 will it approach, in relation to the motions we have under consi- 

 deration, to the condition of an incompressible fluid ; the more 

 nearly will the conditions of the displacement of the gas at the 

 surface of the solid be satisfied by a merely local reciprocating flow. 



This explanation, when once it suggested itself, seemed so 

 simple and obvious that I could not doubt that it afforded the 

 true mode of accounting for the phenomenon. It remained only 

 to test the accuracy of the assigned cause by actual numerical 

 calculation in some case or cases sufficiently simple to permit of 

 precise analytical determination. The result of calculations of 

 the kind applied to a sphere proved that the assigned cause was 

 abundantly sufficient to account for the observed result. I have 

 not hitherto published these results ; but, as the phenomenon 

 has not to my knowledge been satisfactorily explained by others, 

 I venture to hope that the explanation I have to offer, simple as 

 it is in principle, may not be unworthy of the notice of the Royal 

 Society. 



For the purpose of exact analytical investigation I have taken 

 the two cases of a vibrating sphere and a long vibrating cylinder, 

 the motion of the fluid in the latter case being supposed to be 

 in two dimensions. The sphere is chosen as the best represen- 

 tative of a bell, among the few geometrical forms of body for 

 which the problem can be solved. The cylinder is chosen as the 

 representative of a vibrating string. In the case of the sphere 

 the problem is identical with that solved by Poisson in his me- 

 moir " Sur les mouvements simultanes d'un pendule et de Fair 

 environnant-" * ; but the solution is discussed with a totally dif- 

 ferent object in view, and is obtained from the beginning, to 

 avoid the needless complexity introduced by taking account of 

 the initial circumstances, instead of supposing the motion already 

 going on. 



A. Solution of the Problem in the case of a Vibrating Sphere. 



Suppose an elastic solid, spherical externally in its undisturbed 

 position, to vibrate in the manner of a bell, the amplitude of vi- 

 bration being very small. Suppose it surrounded by a homo- 

 * Memoires de VAcademie des Sciences, vol. xi. p. 521. 

 2 D 2 



