AND DEFORMATION OF A THIN KLASTIC SHELL. 4S)'J 



ore expanded in powers of y, and approximate equations taken. These equations are 

 included in those given in the present paper for shells whose middle-surface is any 

 whatever. MATHIEU gives for the special case some of the theorems on purely 

 normal and purely tangential vibrations here proved (see notes to Art. 13). The 

 solution for spherical shells is given in his paper. The introduction of the generalised 

 tesseral harmonic into this solution enables us to recognise that a certain type ot 

 vibration given by MATHIEU cannot exist (see note to Art. 18). The objections raised 

 by DE ST. VENANT to POISSON'S method for plates seem to lie equally against its 

 extension to sheila 



3. Internal Strain in an Element of the Shell. 



1. Suppose the lines of curvature on the middle-surface of the shell to be drawn ; 

 let these be the curves a = const., ft = const. ; then any point on the middle-surface 

 is given by its a, ft. At each intersection of a curve a with a curve ft suppose the 

 normal to the middle-surface drawn and lengths h marked off upon it inwards and 

 outwards from the surface, the loci of the extremities of these lines are two surfaces 

 parallel to the middle-surface. If we suppose the space between these surfaces filled 

 with isotropic elastic material we obtain the elastic solid shell which we wish to treat. 



Let the middle-surface be covered with a network of the lines a = const., ft = const, 

 at distances from each other comparable with the thickness of the shell, and suppose 

 the normals drawn as above described at all the points of these curves. The shell 

 will thus be divided into a great number of elementary prisms ; and, according to 

 KIRCHHOFF'S general method, we must first discuss the equilibrium of one of these 

 elementary prisms. 



Let a, ft bo the parameters of the centre P of one of these elementary prisms before 

 strain. Imagine three line-elements of the shell (1, 2, 3) to proceed from P, the 

 elements (1) and (2) being along the lines ft, a. through P, and (3) along the normal 

 at P to the middle-surface. Then after strain these lines are not in general co- 

 orthogonal, but by means of them we can construct a system of rectangular axes to 

 which we can refer points in the prism whose centre is P. Thus, P is to be the 

 origin, the axis of x is to lie along the line-element (1), and the plane of x, y is to contain 

 the line-elements (1) and (2); then the line-element (2) will make an indefinitely 

 small angle with the axis y, and the line-element (3) will make an indefinitely small 

 angle with the axis z. 



By means of the lines of curvature and the middle-surface we can construct a 

 system of orthogonal surfaces (a, ft, y), so that we may use the formulae of orthogonal 

 coordinates with reference to a, ft, y. 



We write for the distance between two near points 



382 



