446 Lord Rayleigh. Note on the Free [Mar. 14, 



It will be observed that when s is very large, the flexural vibrations 

 tend to become exclusively normal, and the extension al vibrations to 

 become exclusively tangential, as might have been expected from the 

 theory of plane plates. 



Returning now to the general case, the determinant of (9), (10), 

 (11) gives on reduction 



= o ................ (24). 



If p. = 0, we have the three solutions already considered, 



If s = 0, that is, if the deformation be symmetrical about the axis, 

 we have 



& 2 = /* 2 , or &2[& 3 -2(M + l) G*a + l)]+4(2M + lV* = .. (25). 



Corresponding to the first root we have U = 0, W = 0, as is readily 

 proved on reference to the original equations with s = 0. The vibra- 

 tions are the purely torsional ones represented by 



v = sin fiz cos pt .................. (26), 



where _p2 _ ^L _ (27). 



P 



The frequency depends upon the wave-length parallel to the axis, 

 and not upon the radius of the cylinder. 



The remaining roots of (25) correspond to motions for which V= 0, 

 or which take place in planes through the axis. The general character 

 of these vibrations may be illustrated by the case where /u, is small, or 

 the wave-length a large multiple of the radius of the cylinder. We 

 find approximately from the quadratic (on restoration of a) 









The vibrations of (28) are nearly purely radial. If we suppose 

 that fi vanishes, we fall back upon 



= 2(M + 1), 



