122 Lord Rayleigh. On the [Dec. 13, 



"We will now proceed with the calculation for the frequencies of 

 vibration of the complete cylindrical shell of length 21. If the 

 volume-density be />,* we have as the expression of the kinetic energy 

 by means of (35), (36), (37). 



T = J.2fy>. [j 



(50). 



From these expressions for V and T in (39), (50) the types and fre- 

 quencies of vibration can be at once deduced. The fact that the 

 squares, and not the products, of A s , B 5 , are involved, shows that 

 these quantities are really the principal coordinates of the vibrating 

 system. If A,, or A/, vary as cos p s t, we have 



mn 



This is the equation for the frequencies of vibration in two dimen- 

 sions.f For a given material, the frequency is proportional to the 

 thickness and inversely as the square on the diameter of the 

 cylinder.} 



In like manner if B 5 , or B/, vary as cos p s 't, we find 



1+ 



If the cylinder be at all long in proportion to its diameter, the 



According to the equations (in columnar co-ordinates) of my former paper, the 

 conditions that fir, z shall be independent of ^ lead to 



ddz /dy\2 

 r> dz \dz) ~ 



where C is an absolute constant. 



The case where the section is a rhombus (drfdz = tan a) may be mentioned. 



The difficulty referred to above arises when dr/dz = oo. 



* This can scarcely be confused with the notation for the curvature in the pre- 

 ceding parts of the investigation. 



f See ' Theory of Sound,' 233. 



J There is nothing in these laws special to the cylinder. In the case of similar 

 shells of any form, vibrating by pure bending, the frequency will be as the thick- 

 nesses and inversely as corresponding areas. If the similarity extend also to the 

 thickness, then the frequency is inversely as the linear dimension, in accordance 

 with the general law of Cauchy. 



