AND DEFORMATION OP A THIN ELASTIC SHELL. 505 



a = 0, 6 = 0, c = nr 2/c, , 



and the potential energy per unit volume is 

 - )8 2 + 2n(e + / 2 + . 9 2 ) + ( s 



-f- a tenn in 2, 

 where z is written for r/A 8 . 



where z is written for r/A 8 . 



Multiplying this expression by dz, and integrating from & to h, the term in z 

 disappears, and we find for the potential energy per unit area 



= *A> + v -i- 2^ + 



or 



. (12) 



The term containing h 3 is the term depending on the bending, and the term con- 

 taining h is the term depending on the stretching of the middle-surface. We shall 

 hereafter denote by W p W 2 the expressions 



/ x\o. m + W /x 0\ 





4. Geometrical Theory of Small Deformation of Extensible Surfaces. 



5. We have now, by means of equations (4) and (5), to express the potential energy 

 in terms of the displacement of a point on the middle-surface. 



Let , v, w denote the displacements of the point P on the middle-surface, u being 



MOCOCLXXXV1II. A. H T 



