492 Mr. L. K". G. Filon. Approximate Solution for [June 12, 



(b.) When, on the other hand, the applied stresses still being in the 

 plane of xy, the thickness of the bar, in the direction of the axis of z, 

 is small compared with the other dimensions of the bar, so that we 

 approximate to the case of a thin plate under thrust in its own plane. 



It is shown that, if in case (b) we assume (which will be very nearly 

 true, the thinner the lamina) that the normal traction across a face 

 perpendicular to z is zero throughout the thickness, then the equations 

 connecting the mean displacements U, V with the mean stresses P, Q, S 

 in the plane of the lamina (the mean here being taken with regard to 

 the thickness of the lamina) are of the same form as the equations in 

 case (a) connecting the actual displacements u, v with the three stresses 

 in the plane of xy, provided only that we make an alteration in one of 

 the elastic constants, u, v, w being displacements parallel to x } y, z 

 according to the usual notation. 



Of course, the lamina being thin, the displacements v., v will prob- 

 ably vary little as we go across it, so that the mean values will give 

 us an approximation to the displacements at every point. 



In like manner the stresses in the planes parallel to xy will not 

 differ greatly from their mean values P, Q, S. 



The equations used in the paper correspond to case (b), but all the 

 results are applicable to case (a) by merely changing the elastic constant 

 mentioned. The body stress equations are 



where A' = 2 A/a/ (A. + 2/a) and A, /x are the elastic constants of Lame. 



General solutions of these equations are found in terms of conjugate 

 functions. These solutions are then applied to the case of a rectangular 

 bar bounded by the planes x = ±a, y = ±b. 



The surface stresses applied to the faces y = ±b are supposed given 

 at every point, but over the faces x = + a only the statical stress- 

 resultants (total tension, total shear, total bending moment) are 

 supposed given. 



This last condition is sufficient, provided a is large compared with b. 

 This is assumed in every case ; eventually the boundaries x = ±a are 

 removed to infinity. 



The first part of the paper is occupied in establishing the formal 



dx dy 



= 0, 



where 



