46 Mr. A. B. Basset. On the Extension and Flexure [Dec. 19, 



normal traction perpendicular to the middle surface, and S and T are 

 the two shearing stresses which tend to produce rotation about two 

 lines of curvature of the middle surface. This hypothesis requires 

 that these stresses should be at least of the order of the square of the 

 thickness of the shell, for when this is the case they give rise to terms 

 in the expression for the potential energy due to strain, which are 

 proportional to the fifth power of the thickness, and which may be 

 neglected, since it is usually unnecessary to retain powers of the 

 thickness higher than the cube. It can be proved directly from the 

 general equations of motion of an elastic solid, that this proposition is 

 true in the case of a plane plate, provided the surfaces of the plate 

 are not subjected to auy pressures or tangential stresses, but there 

 does not appear to be any simple method of establishing a similar 

 proposition in the case of curved shells. I have therefore adopted 

 this proposition as a fundamental hypothesis, and have endeavoured 

 to establish its truth and to obtain a satisfactory theory of cylindrical 

 and spherical shells in the following manner : — 



Taking the case of a cylindrical shell, let OADB be a curvilinear 

 rectangle described on the middle surface, of which the sides OA, 

 BD are generators, and the sides AD, OB are circular sections. The 

 resultant stresses per unit of length across the section AD consist of 

 (1) a tension, T 1 ; (2) a tangential shearing stress, M 2 ; (3) a normal 

 shearing stress, K 2 ; (4) a flexural couple, Gr 2 , about AD ; (5) a 

 torsional couple H 1} perpendicular to AD ; and the stresses across BD 

 may be derived by interchanging the suffixes 1 and 2. Resolving 

 along OA, OB, and the normal, and taking moments about these 

 lines, we obtain the following equations,* viz. : — 



X 



dz a d<p a 



> 



1 



= L 



dz a ad) * 



(Mg-MO a-H 2 = J 



* Compare Besant, " On the Equilibrium of a Bent Lamina," ' Quart. Journ. 

 Math.,' 1860. 



