498 MR. A. B. H. LOVE ON THE SMALL FREE VIBRATIONS 



can execute purely tangential vibrations such that every point moves perpendicularly 

 to the meridian through it, and the displacement is symmetrical about the axis of 

 revolution. 



The special problem of the vibrations of a spherical shell has been discussed by 

 Lord RAYLEIGH.* In his paper it is assumed that no line on the middle-surface is 

 altered in length ; the boundary-conditions are not considered. The form of the 

 potential energy taken is a quadratic function of the changes of principal curvature 01 

 the middle-surface, and this I have proved to be in this case the true form in Art. 7. 

 The assumption of inextensibility does in this case lead to expressions for the dis- 

 placements which cannot satisfy the boundary-conditions which hold at a free edge. 



The method developed in this paper is applied to the problem, and the approximate 

 equations integrated. The solution comes out in tesseral harmonics with fractional 

 or imaginary indices, and the frequency is givefa by a transcendental equation ; in 

 case the shell be hemispherical this equation is simplified, and to express the sym- 

 metrical vibrations only the ordinary zonal harmonics with real integral indices are 

 lequired, and the frequency equation can be solved. 



As a further example of the application of the method to small vibrations I have 

 discussed the vibrations of a cylindrical shell. The displacement of a point on the 

 middle-surface is expressed by simple harmonic functions of the cylindrical coordinates 

 of the point. In the case of the symmetrical vibrations the frequency equation is 

 easily solved. 



AKON has applied the method of CLEBSCH to the problem of shells. In his memoir t 

 a point on the middle-surface of the shell is considered as defined by two parameters, 

 as in GAUSS'S theory of the curvature of surfaces ; the displacements are referred to an 

 arbitrary system of fixed axes ; and the expressions found for them are the same as 

 those in Art. 4 of this paper, but the work contains a small error (see note to 

 Art. 4). Free use is made of arbitrary multipliers in order to obtain the equations 

 of equilibrium referred to the fixed axes. As these are in a very unmanageable shape, 

 a method of forming equations referred to moving axes is indicated ; the equations are 

 first obtained with reference to fixed axes, and it is proposed to transform these. The 

 transformation is not effected, but some reductions are made with a view to it 

 (pp. 169 et seq.). In these reductions all effects due to the change of direction of the 

 axes as we go from point to point on the middle-surface are neglected, so that the 

 results are erroneous (see note to Art. 6). 



A theory of the vibrations of a shell whose middle-surface is a surface of revolution 

 has been given by MATHIEU.^ The method is similar to that employed by POISSON 

 for the plate, viz., taking y = for the middle-surface, all the quantities which occur 



* " On the Infinitebimal Bending of Surfaces of Revolution," ' London Math. Soc. Proc.,' vol. 17, 1882. 

 f "Das Gleichgewicht und die Bewegung einer nnendlich diinnen beliebig gekrUmmten elastischen 

 Schale." ' CRKLLE, Journ. Math.,' vol. 78, 1874, p. 138. 



J " Memoire sur le Monvement vibratoire des Cloches," ' Journ. de 1'ficole Poly tech n.,' cahier 51 (1883) . 



