112 Lord Rayleigh. On the [Dec. 13, 



and on integration over the whole thickness of the shell (2/i) * 



f / 1 \2 / , 1 \8 m M- /I , 1 \ 2 1 



i (* I +{* I +~ - (* +^~ ) f ...(14). 

 IV />!/ V Pfcf m + nV Pl p 9 J J 



This conclusion may be applied at once, so as to give the fesult 

 applicable to a spherical shell ; for, since the original principal planes 

 are arbitrary, they can be taken so as to coincide with the principal 

 planes after bending. Thus T = ; and by Gauss's theorem 



+s = o, 



ft P-2 



so that 



_ 



where fy" 1 denotes the change of principal curvature. Since e = /, 

 g = 0, the various Iamina3 are simply sheared, p,nd that in proportion 

 to their distance from the middle surface. The energy is thus a 

 function of the constant of rigidity only. 



The result (14) is applicable directly to the plane plate ; but this 

 case is peculiar in that, on account of the infinitude of P}t /? 2 (8) is 

 satisfied without any relation between fy^ and fy 3 . Thus for a plane 

 plate 



where Pl ~ l , pz~ l , are the two independent principal curvatures after 

 bending. 



We have thus far considered T to vanish; and it remains to inves- 

 tigate the effect of the deformations expressed by 



& = TXIJ = IT^-^S) ............ .... (] 7) f 



where f, rj relate to new axes inclined at 45 to those of x, y. The 

 curvatures defined by (17) are in the planes of f , */, equal in numerical 

 value and opposite in sign. The elongations in these directions for 



* It is here assumed that m and n are independent of h, that is, that the material 

 is homogeneous. If we discard this restriction, we may form the conception of a 

 shell of given thickness, whose middle surface is physically inextensible, while yet 

 the resistance to bending is moderate. In this way we may realise the types of 

 deformation discussed in the present paper, tvithout supposing the thickness to be 

 infinitely small ; and the independence of such types upon conditions to be satisfied 

 at a free edge is perhaps rendered more apparent. 



