496 MR. A. E. II. LOVE ON THE SMALL FREE VIBRATIONS 



2. Theory of Shells. 



In this paper the potential energy of deformation of an isotropic elastic shell is 

 investigated by the same method as that employed by KIKCHHOFF in his discussion of 

 the vibrations of a plane plate.* The shell is supposed to be bounded by two surfaces 

 parallel to its middle-surface, and is deformed in any arbitrary manner. The expres- 

 sion given by KIRCHHOFF for the energy of the plate per unit area of its middle- 

 surface is 



KA + ft + 2 Pl * 4- f ~ (q, - Pa ) 



l + & fa -f 



where 2h is the thickness of the plate, K the rigidity, and 0/(l -f #) = <r/(l cr), o- 

 being the ratio of linear lateral contraction to linear longitudinal extension of the 

 material ; o-j, <r. 2 are the extensions of two line-elements of the middle-surface initially 

 at right angles, and T the complement of the angle between them after strain ; 

 ?i> P& Pi are quantities defining the curvature of the middle-surface after strain, viz. : 



p. 2 q\= sum of principal curvatures, 

 (p 2 q l + pf) = measure of curvature ; 



so that, if p l} p. 2 be the principal radii of curvature after strain, the first term of the 

 above reduces to 



l + e 



A similar expression to that given by KIECHHOFF is obtained below in the case of the 

 shell initially curved ; but here the quantities q lt p 2 , p t are replaced by the difference of 

 their values in the strained and unstrained states, a result which might have been 

 anticipated from the remarks made by KIRCHHOFF (' Vorlesungen,' p. 413) on the strain 

 of a rod initially curved, since the strain of an element is a linear function of these 

 quantities. 



We wish to obtain equations of motion and boundary-conditions in terms of the 

 displacements of a point on the middle-surface of the shell, these being reckoned 

 parallel to the lines of curvature and perpendicular to the tangent plane at the point. 

 For this purpose it is necessary to express all the quantities which occur in the 

 potential-energy-function in terms of these displacements. As the geometrical theory 

 of the deformation of extensible surfaces appears not to have been hitherto made out, 



- * Called above " KIRCHHOKF'S second tlioorv." 

 f- Vorlesungen,' p. 454. 



