1891.] State of the Theory of Ihin Elastic Shells. 101 



the other contains a function W\ and A 3 as factors. The term W 2 

 depends entirely on quantities <r l5 ff. 2 , -or, expressing the extension of 

 the middle surface, while the form given for Wj contained only 

 quantities expressing the changes of curvature. Some previous 

 theories proceeded as if "W\ alone occurred, and, in fact, this was 

 the case with a paper by Lord Rayleigh in ' Proceedings of the 

 London Mathematical Society,' vol. 13, 1882, on the "Infinitesimal 

 Bending of Surfaces of Revolution." In the latter paper, a theory 

 of the vibrations of bells was founded on an assumed type, viz., 

 it was assumed that the middle surface remains unstretched. In 

 my paper it was shown that this solution of Lord Rayleigh's 

 fails to satisfy the boundary conditions which hold at the free 

 edges of the bell, and further that it is, in general, impossible to 

 satisfy these conditions, except by taking account of the extension. 

 I, therefore, proposed to substitute for the theory of Lord Rayleigh 

 one in which extension of the middle surface of the bell is re- 

 cognised as taking place, and I did not see how to avoid the con- 

 clusion that the term W x must be rejected, and the term W 2 retained, 

 for the purpose of forming the differential equations and boundary 

 conditions that govern the motion, in other words, that the exten- 

 sion practically determines everything the mode of vibration and 

 the pitch. 



Since that paper was written the subject has been investigated by 

 Lord Rayleigh, Mr. Basset, and Professor Lamb, and the results of 

 their work make it necessary to abandon the theory proposed. I 

 had overlooked a circumstance which shews that my theory of exten- 

 sional vibrations is incapable of giving the gravest modes of vibration 

 of which the shell is capable, viz., the period given by Lord 

 Rayleigh's solution, founded on the assumed type, is, in the limiting 

 case of vanishing thickness, infinitely long in comparison with the 

 gravest extensional period. Now it is a general dynamical theorem 

 that the tone obtained by assuming the type cannot be graver than 

 the gravest tone natural to the system, and it follows that the mode 

 of deformation corresponding to the gravest tone is not included 

 among the extensional modes. This was pointed out by Lord 

 Rayleigh in a paper read before the Society in December, 1888, and 

 published in the ' Proceedings.' It had still to be shown, however, 

 that vibrations mainly dependent on the bending could take place, 

 and the boundary conditions be satisfied. Although this has not yet 

 been done in any particular case, the suggestion thrown out by Mr. 

 Basset* and Professor Lamb,t probably contains the solution of the 



* " On the Extension and Flexure of Cylindrical and Spherical Thin Elastic 

 Shells," ' Phil. Trans.,' A, 1890. 



f " On the Deformation of an Elastic Shell," 'London Math. Soc. Proc.,' vol. 21, 

 1890. 



