MEMBRANES AND PLATES 141 



The kinetic energy is given by 



taken over the area of the membrane. 



The potential energy is found most easily as the work 

 required to stretch the membrane. As in the theory of 

 capillarity this is equal to the tension P multiplied by the 

 increase of area. Now if a prism be constructed on a rectangular 

 element Sx&y of the plane xy as base, this will cut out from the 

 displaced membrane a nearly rectangular portion whose sides 

 are 



~ VK-S'h 



and whose area is therefore, to the second order, 





The same expression is obtained by calculating, from the 

 expression (1), the work done by normal pressures applied 

 (as in 22) to deform the membrane into its actual shape, the 

 ratio of f to its final value being, at any stage of the process, 

 the same all over the membrane. The result is 



The reader who is familiar with the theory of attractions will 

 recognize that this is equal to 



where in the first term the integration extends over all the 

 elements Ss of the contour, and 8n is an element of the normal 

 to &s drawn inwards, in the plane of the membrane. Since 

 at a fixed edge f = 0, the formula agrees with (5). 



