> m:i ORMATION OP A THIX ELASTIC SHELL. 523 



The complete spherical shell may execute purely radial vibrations, and the frequency 



IS 



1 



(l-o-)aV 

 where a is the radius.* 



The indefinitely long circular cylinder may also execute purely radial vibrations 

 with a frequency 



1 



a being the radius. 



Observing that the more accurate equations of motion and boundary-conditions, 

 which contain the terms in h 3 , will in all such terms have only differential coefficients 

 of to with respect to a or ft, the above theory is seen to hold also if these more 

 accurate equations be considered. 



(2.) Again, consider the possibility t of purely tangential vibrations, the edge being 

 a line of curvature. 



Since w = 0, the third of equations (36) gives 



(o-j + o-trj/p! + (o- 2 + ovj/pz = 



at all points of the surface. 



Now, the boundary-conditions at a = const, are 



and with two functions u, v it will not generally be possible to satisfy these con- 

 ditions. 



If, however, the surface be of revolution, and ft be the longitude, then 

 dh^/dft = 0, and all the conditions can be satisfied by taking 



(1) u = 0, -I 



9a L at all points of the surface, 



< 2 >a =:0 J 



r\ 



(3) ;r (A g v) = at the edge ; 



* [In the paper as read, this result was verified by reference to a question set in the Mathematical 

 Tripos, part III., 1885. It has since been pointed out to me that it coincides with the formula given by 

 LAMB in 'London Math. Soc. Proc.,' vol. 14, p. 50. July, 1888.] 



t MATUIEU deduced the possibility of some purely tangential vibrations from his differential equations. 



3 X '2 



