AND DEFORMATION OF A THIN ELASTIC SHELL. 543 



9. Summary.* 



This paper is really an attempt to construct a theory of the vibrations of bells. In 

 any actual bell complications will arise, which have been omitted in this discussion, 

 partly from variations of the thickness in different parts, and partly from the 

 want of isotropy in the material. We can hardly expect a metal which has been 

 subjected to the process of bell-manufacture to be other than very seolotropic, while it 

 is notorious that bells are usually thickest at the rim. The difficulty of the problem 

 in its general form seems to make it advisable to begin with the limiting case of an 

 indefinitely thin perfectly isotropic shell, whose thickness is everywhere constant, and 

 so small compared with its linear dimensions, that powers of it above the first may be 

 neglected in mathematical expressions, which contain the first and higher powers 

 multiplied by quantities of the same order of magnitude. 



Of previous theoretical work we have examples in Lord RAYLEIOH'S ' Theory of 

 Sound,' and in his paper on the " Bending of Surfaces of Revolution," in ARON'S and 

 MATHIEU'S memoirs, and in IBBETSON'S treatise on the Mathematical Theory of 

 Elasticity. In the ' Theory of Sound ' Lord RAYLEIGH treats the vibrations of a 

 thin ring or infinite cylinder of matter, supposed to be deformed in such a way that 

 the motion is in one plane and the elements remain unextended, and remarks that at 

 the time of publication this was the nearest approximation to a theoretical treatment 

 of bells. He afterwards applies his theory of the bending of surfaces to obtain a more 

 exact analytical method of treating the problem, but his disregard of the boundary- 

 conditions which hold at a free edge appears to vitiate this theory. AROX can hardly 

 be said to have attained a theory of bells, and the interest of his memoir is mainly 

 mathematical ; his inaccuracies have been already referred to. I have also previously 

 referred to the objection which lies against MATUIEU'S method of treatment ; this and 

 the complexity and difficulty of some of his analysis seem to render a new method 

 desirable. I shall have to refer to IBBETSON later. 



The theory here put forward rests on the form of the function expressing the 

 potential-energy of deformation per unit area of the middle- surface of the shell. 

 Supposing that the surface is stretched and has its curvature changed, we find that 

 the energy consists of two terms. One of these contains only the functions defining 

 the stretching, while the other contains also those defining the bending of the middle- 

 surface. The modulus of stretching is proportional to the thickness, while the 

 modulus of bending is proportional to its cube. Unless, therefore, the functions 

 expressing the stretching, viz., the extensions and shear of rectangular line-elements 

 of the middle-surface, are of a higher order of small quantities than those defining 

 the bending, viz., the changes of the principal curvatures and of the directions of the 

 principal planes, the vibrations depend on the term which involves the stretching, and 

 not on that which involves the bending. Now, it seems to have been universally 



Partly rewritten, July, 1888. 



