352 Small Free Vibrations of a Thin Elastic Shell. [Feb. 9, 



February 9, 1888. 



Dr. E. FRANKLAKD, Vice-President, in the Chair. 



The Presents received were laid on the table, and thanks ordered 

 for them. 



The following Papers were read : — 



I. " The Small Free Vibrations and Deformation of a Thin 

 Elastic Shell/' By A. E. H. Love, B.A., Fellow of St. John's 

 College, Cambridge. Received January 19, 1888. 



(Abstract.) 



In this paper the method employed by Kirchhoff and Clebsch for 

 the treatment of a thin plane plate is applied to the case of a thin 

 shell, or plate of finite curvature. The form of the potential-energy- 

 function for the strain in an element of the shell is the same as that 

 obtained by Kirchhoff for a plate, the quantities depending on the 

 curvature of the surface being replaced by the difference of their 

 values in the strained and unstrained states. It is proved that only 

 for an inextensible spherical surface is this function the same function 

 of the changes of principal curvature as for a plane plate. The 

 general equations of equilibrium and small motion under the action 

 of any system of forces are formed. It is shown that in general the 

 shell cannot vibrate in such a manner that no line on the middle- 

 surface is altered in length, because this condition makes it impos- 

 sible to satisfy the boundary conditions which hold at a free edge. 

 It then appears that approximate equations of motion may be taken, 

 in which the terms of the potential energy depending on the bending 

 may be neglected, and only those depending on the stretching need 

 be retained. It is shown that surfaces of uniform curvature with no 

 bounding edges are the only ones which admit of purely normal 

 vibrations, and that vibrations in which the displacement is purely 

 tangential are possible on all shells whose middle-surfaces are surfaces 

 of revolution bounded by small circles. The cases of the spherical 

 and cylindrical shell receive special discussion. The equations 

 of motion can always be solved, but solutions of the frequency equa- 

 tions could only be obtained in case the displacement was symmetrical 

 about the axis. The application of the general equations to problems 

 of equilibrium is illustrated in the case of spherical shells, for which 



