1888.] Bending and Vibration of thin Elastic Shells. 



113 



any lamina within the thickness of the shell are In, hr, and the 

 corresponding energy (as in the case of the sphere just considered) 

 takes the form 



(18). 



This energy is to be added* to that already found in (14) ; and we 

 g3t finally 



3 



Pl 



. (19), 



is the complete expression of the energy, when the deformation is 

 ruch that the middle surface is unextended. We may interpret T by 

 leans of the angle x> through which the principal planes are shifted ; 

 '-,hus 



P\ 



It will now be in our power to treat more completely a problem of 

 great interest, viz., the deformation and vibration of a cylindrical 

 shell. In my former paper I investigated the types of bending, but 

 without a calculation of the corresponding energy. The results were 

 as follows. f If the cylinder be referred to columnar coordinates 

 z, r, 0, so that the displacements of a point whose equilibrium co- 

 ordinates arei, a, are denoted by &z, 8r, a0, the equations express- 

 ing inextensihility take the form 



, 



dz 



from which we may deduce 



= 0, +> = <>.. .(21), 



<*. 



By (22), if 0oc coss0, we may take 



and then, by (21) 



cos 50 ............ ..... (23), 



s (A*a+B,) sin 

 = s~ l B s a sin 50 



(24), 

 (25). 



* There are clearly no terms involving the products of T with the changes of 

 'principal curvature ^'(p^ 1 ), $(p 2 ~ 2 ) ; for a change in the sign of T can have no 

 influence upon the energy of the deformation defined by (7). 



f The method of investigation is similar to that employed by Jellet in his 

 memoir (" On the Properties of Inextensible Surfaces," ' Irish Acad. Trans.,' vol. 22, 

 1855, p. 179), to which reference should have been made. 



