140 DYNAMICAL THEOEY OF SOUND 



unit area) is assumed to be uniform, and is denoted by p. To 

 form the equations of motion we calculate the forces on the 

 sides of a rectangular element BxSy having its centre at (x, y). 

 In the displaced position, the gradient of a line parallel to x is 

 3?/da?, and that of a line parallel to y is d/dy. Hence the stress 

 across a line through the centre of the element parallel to By, 

 when resolved in the direction of the normal to the plane xy, is 

 Pd^/dx . Sy. The corresponding components of force on the two 

 edges % of the rectangle are 



where the upper signs relate to the edge whose abscissa is 

 x + -J&z?, and the lower to the edge x ^x. The sum of these 

 gives P9 2 f/3# 2 . SxSy. A similar calculation for the two edges &z? 

 gives Pd*/dy 2 . &x$y. The resultant force on the rectangle is 

 therefore 



< 



The above may be compared with the investigation by 

 which, in the theory of Capillarity, the tensions across the 

 boundary of an element 8S of a soap-film are shewn to be 

 equivalent to a normal force 



where B lf R 2 are the principal radii of curvature of the surface. 

 It is shewn in books on solid geometry that, if f denote distance 

 from the plane xy, we have 



R, R, 8# 2 a^ 2 



at points where the inclination of the tangent plane to xy is 

 small. 



Equating the expression (1) to the acceleration of momentum 

 of the element, viz. p&xSy . , we obtain the equation of motion 



This is due to Euler (1766). 



