512 MR. A. E. H. LOVE ON THE SMALL FREE VIBRATIONS 



The equation for the radii of curvature being 



(E> + EV) (G> + GV) - (F> + FV) 2 = 0, 

 where 



V = v/(EG - F*) = 1 ~f 1 ,~ g> nearly, 



A|A| 



we find that to the first order 



I=-<' !(1+ , 1 + ^) = i + ^- s . 



We have already found expressions for K. Z , X 1( KJ in terms of the displacements; 

 hence, we have found expressions for the new principal curvatures and the position of 

 the new principal planes, in terms of the displacements, for the position of these 

 planes depends only on F' or on KJ ; we have also found the interpretation of the 

 *.,, X 1? jq in terms of the quantities defining the curvature and the extension. 



In the case of an inextensible sphere, the potential energy due to bending is 





For any other surface, whether extensible or not, this will not be the case. If the 

 middle-surface were unextended, the above would be right to small quantities of the 

 first order, but we always require the potential energy correct to small quantities of 

 the second order. 



5. Equations of Motion and Boundary-Conditions. 



8. Following KIRCHHOFF'S method, we are going to apply the principle of virtual 

 work to obtain the differential equations of motion and equilibrium, and the boundary- 

 conditions. 



Let Xj, Yj, Zj be the components of the bodily force per unit mass parallel to the 

 lines of curvature ft = const., a = const., and perpendicular to the tangent plane to 

 the middle-surface, acting at any point Q of the shell. Let QP be perpendicular to 

 the middle-surface before strain, and let 1 3 , m s , 3 be the direction cosines of QP after 

 strain referred to axes at P, as in Artt. 1, 6 ; if u, v, w be the displacements of P, 

 and 2 the distance PQ, then, when a small variation in the configuration is made, the 

 displacements of Q will be found from equations (1), dropping the p, q, to be 



