524 ME. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS 



and the equation of motion is 



Hence a general theorem : For any surface of revolution there exists a system of 

 symmetrical vibrations, in which every element moves perpendicular to the meridian 

 plane through a distance which is the same for all points on a parallel of latitude, and 

 the frequency of such vibrations depends only on the rigidity of the material, while 

 the ratios of the intervals are independent of the material. These are the only 

 purely tangential vibrations of which the shell is capable. 



14. Let us examine more minutely the question whether a spherical shell can 

 vibrate in such a manner that no line on the middle-surface is altered in length. 



Taking a = 9, /? = <f>, the colatitude and longitude of a point on the middle-surface, 

 and a the radius, ^ = I/a, h a = l/(a sin 0), thus 



= Be + w ' 



i ^\ 



= W + U COt 6 + - ^ 57 ' 



sin0 d<f> 



(41) 



ttCT = 



a 2 ** = - ; ^ + cot^ ^ ^ , ^ wcot 0, 



d 



a 2 *, = - 



d 



cos a ow 



. . (42) 



^ 





Suppose (T!, cr 2 , TT all zero, then 



and 



. - a / 

 U - 





These are the conditions given by Lord RAYLEIGH, and they show that u cosec and 

 v cosec 6 are conjugate solutions of the equation 



