AND DEFORMATION OP A THIN ELASTIC SHKI.L. 497 



it was necessary to give the elements of such a theory for small deformations. The 

 expressions obtained for the principal radii of curvature show that the potential energy 

 due to bending is never the same quadratic function of the changes of principal 

 curvature as for a plane plate, except in the single case where the middle-surface is a 

 sphere and unstretchc'l. 



The general variational equation of motion is developed in the foi-m of surface and 

 line integrals, and the equations reduce to those of CLEBSCH * in the case of a plane 

 plate. The terms herein which depend on externally applied forces are obtained 

 directly, without the use of the arbitrary multipliers which render the calculations of 

 CLEBSCH so tedious, and without the necessity which he finds for correcting an 

 " error " t as regards the distribution of force at the edge, thus avoiding some of the 

 criticisms of DE ST. VENANT.J 



We know that when a plane plate vibrates the transverse displacement is indepen- 

 dent of the displacements parallel to the plane of the plate ; and when the transverse 

 vibrations alone are taking place no line on the ujiddle-snrface is altered in length. 

 I discuss the question whether vibrations of the shell are possible in which this last 

 condition holds good, and show that it leads to three partial differential equations 

 giving the displacements as functions of the position of a point on the middle-surface, 

 and that these equations are not in general of a sufficiently high order to admit of 

 solutions which shall also satisfy the conditions which hold at a free edge. This 

 result is quite independent of the theory adopted, as the equations of inextensibility 

 are in the most general case a system of the third order, while the boundary-conditions 

 are four in number. It would, of course, be possible to find a system of forces applied 

 to the boundary which could artificially maintain this kind of vibration. It appears, 

 then, that the term of the potential eneigy which depends on the bending, which is 

 multiplied by A s , is small compared with the term depending on the stretching, which 

 is multiplied by h ; and, in order to obtain the limiting form of the theory when h = 0, 

 we may form approximate equations of equilibrium and motion and boundary-con- 

 ditions by omitting the term in h a . Having formed these equations, I proceed to 

 discuss the question whether the shell can execute vibrations in which there shall be 

 no tangential displacement, and it is shown that this requires both the principal 

 radii of curvature of the middle-surface to be constant at every point. The 

 frequencies of the purely radial vibrations of a sphere and an infinitely long circular 

 cylinder are given ; the displacement is a simple harmonic function of the time, and 

 is the same at all points of the sphere or cylinder. The formula for the frequency 

 admits of independent verification. Another general result deduced from the 

 approximate equations is that any shell whose middle-surface is a surface of revolution 



Elasticitat,' pp. 306, 307; Equations (105), (106). 

 t Ibid., p. 284. 



J Translation of CLEBSCH, p. 691. The method of CLEBSCH is styled " ohucure, indirecte, fort 

 cmnpliqueW 



MDCCOLXXXVITT. A. 3 S 



