from a Vibrating Body to a surrounding Gas. 413 



Secondly, the radius must be sufficiently small, or the pitch suffi- 

 ciently low, to make q small; at the other extremity of the scale, 

 in which c is supposed to be very large, or X very small, Q varies 

 nearly as D* instead of D^, whatever be the order of the Laplace's 

 Function. Hence no simple relation can be expected between 

 the numbers furnished by experiment and the numerical con- 

 stants of the gas in such experiments as those of M. Perolle*, 

 in which the same bell was rung in succession in different gases. 



B. Solution of the Problem in the case of a Vibrating Cylinder. 



I will here suppose the motion to be in two dimensions only. 

 In the case of a vibrating string, which I have mainly in view, it 

 is true that the amplitude of excursion of the string varies sen- 

 sibly on proceeding even a moderate distance along it, and that 

 the propagation of the sound-wave produced by no means takes 

 place in two dimensions only. But the question how far a sound- 

 wave is produced at all, and how far the displacement of the gas 

 by the cylinder merely produces a local motion to and fro, is de- 

 cided by what takes place in the immediate neighbourhood of 

 the string, such as within a distance of a few diameters ; and 

 though the sound-wave, when once produced, in its subsequent 

 progress diverges in three dimensions, the same takes place 

 with the hypothetical sound-wave which would be produced if 

 lateral motion were prevented ; so that the comparison which it is 

 the object of the investigation to institute is not affected thereby. 



Assuming, then, the motion to be in two dimensions, and re- 

 ferring the fluid to polar coordinates, r, 0, r being measured 

 from the axis of the undisturbed cylinder, we shall have for the 

 fundamental equation derived from (1), 



d? \dr* + r dr + r*dd*J> ' ' ' [ ^ } 



and if w f , v ! be the components of the velocity along and perpen- 

 dicular to the radius vector, 



" dr r dd 



If c be the radius of the cylinder, and V the normal compo- 

 nent of the velocity of the surface of the cylinder, we must have 



-¥- =Y when r = c. 

 dr 



As before, I will suppose the motion of the cylinder, and conse- 

 quently of the fluid, to be regularly periodic; but instead of 



* Memoires de VAcademie des Sciences de Turin, vol. iii. (1786-7), Mem. 

 des Correspohdans, p. 1. 



