102 Theory of Thin /;/,**> S/ull*. [Jan. 15, 



difficulty. Each of these writers has shown that, in particular statical 

 problems relating to cylinders, the quantities expressing the exten- 

 sion can be very small everywhere except in the neighbourhood of an 

 edge, and there they may increase with such rapidity as to secure the 

 satisfaction of the boundary conditions, the total potential energy due 

 to extension, which varies as the surface integral of ^W 2 over the 

 middle surface, being, nevertheless negligible in comparison with 

 that due to bending, which varies as the surface integral of /r"NY,. 

 Mi Basset and Professor Lamb both suggest that this may be the 

 solution of the difficulty in the case of vibrations also, and their 

 n-Milts point to a method of approximation which might be applied 

 to the general case, and such that it could be verified by mathematical 

 analysis that Lord Rayleigh's solution, founded on an assumed t \ p>< -. 

 is actually a very close approximation to the state of things in any 

 part of a vibrating bell not very close to a free edge. 



It may be as well to point out what parts of the theory put forwaid 

 in my paper specially require revision. (1.) On p. 500 the alteration 

 suggested in Kirchhoff' s theory is erroneous ; the quantities ', v, w' 

 are functions of , /3, and their differential coefficients must be intro- 

 duced as by Kirchhoff, and afterwards neglected ; this correction 

 makes no difference to any of the n suits. (2.) On p. 503. Art. 4, the 

 " products " neglected are such as occur in the equations when 

 account is taken of the fact that the axes of reference are really not 

 in fixed directions. If they had been retained, the part of the 

 potential energy which is multiplied by A 3 would have contained 

 terms depending on the extension as well as terms depending on the 

 bending. Mr. Basset has obtained, by a different method, the form 

 of this function for cylindrical and spherical shells, with these terms 

 expressed. It follows that the form given for the potential energy 

 in equation (12), p. 505, is only correct in case either (a) the shell 

 is unextended, when its second line vanishes, or (6) the extension is 

 the important thing, when its first line may be neglected ; but it 

 would most probably be sufficiently exact for the application of a 

 method of approximation. (3.) The first paragraph of Art. 13, 

 ]>. ''-.' 1, is wrong, and so are all other paragraphs to the same effect ; 

 viz., it is incorrect to conclude that, because a^ a.-,, w do not every- 

 where vanish, therefore W-Ji s is infinitely small in comparison with 

 \\ .,h. It appears, on the contrary, that the values of <TJ, <r 4 , v can be 

 very small indeed everywhere except close to the edges, in such a 

 way that the integral of W 2 ft, taken over the middle surface, is very 

 small in comparison with that of W t A 3 . 



The remainder of the paper must be understood as giving a theory 

 of the extensional vibrations of the shell. Such vibrations undoubtedly 

 can exist, but they would be difficult to excite, and the theory of 

 t'tem has no application to vibi-atirig bells under ordinary conditions. 



