1889.] of Cylindrical and Spherical Thin Elastic Shells. 4:7 



where X,Y . . . denote certain expressions involving the bodily forces, 

 such as gravity and the like, and the time variations of the displace- 

 ments. 



The values of the four couples may be calculated by a direct 

 method, and are 



.... (ii). 



To explain the symbols involved in these equations, let u, v, w be 

 the displacements along OA, OB, and the normal; a-^, <r%, -str 3 the 

 extensional and shearing strains along and about these directions 

 respectively; then putting E = (m—n)l(m-\-n), the symbols in (ii) 

 are denned by the following equations : — 



% = ffl + E Oi + tfg), § = * 2 + E (^H-^) 



dhjo 1 / d?w 



\(d?w \ E , x t /..-x 

 = -^(w +W )-a ( ^ ) > (m) - 



_ 2 d 2 w ^1 dv 1 du 



a dzd<p a dz a 2 d(f) 



It is important to notice that the couples involve the extension of 

 the middle surface as well as the change of curvature. 



The expression for the potential energy is next found, and its value 

 per unit of area of the middle surface is 



+ -3j(&* + 8^+i*aP) ... (iv). 



The quantities V, fi', p', depend partly upon quantities which define 

 the bending, and partly upon the extension of the middle surface. 



This expression is different from that obtained by Mr. Love, 'which 

 arises from the fact that he has omitted to take into account several 

 terms depending upon the product of the extensions and h s . It will 

 be noticed that (iv) reduces to the second line when the middle surface 

 is inextensible, and in this case agrees with the expression obtained 

 by Lord Rayleigh.* 



* f Kay. Soc. Proc.,' Dec, 1888. 



