﻿Isotropic Spheres under Uniform Surface- Pressure. 181 



shell. Supposing these to be of radii e and a, the applied 

 pressures being p and p' y we thus find 



$(k.Sv)=^7r( 2 ye*-2>'a*) (68) 



Since the dilatation of a layer is uniform throughout, it is 

 given by 



A = Sv/v. 



We may thus write (68) in the alternative form 



%{k.&.v)=^7T(pe*-p ! a*). . . . (69) 

 J For a solid sphere we have 



t(k.Sv)=X(k.A.v) = -^7ra?p'. . (70) 



The relation we have established is of course insufficient 

 by itself to determine the dilatation in any one layer of a 

 compound shell or sphere, but it at least supplies a very 

 simple check on the accuracy of results otherwise determined. 

 For the three-layer shell (e . a . c . ol x . b . a . a) it gives, de- 

 noting the dilatations in (e . a. . c) , (c . a x . b), and (b .ex.. a) by 

 A, A 1; A 2 respectively, 



A:A(c 3 -^) + ^ 1 A 1 (6 3 -c 3 ) + M 2 (a 3 -6 3 )=pe 3 -p'a 3 , 



and this will be found consistent with the values of the 

 dilatation supplied by (38), (39), and (40) for the three 

 layers. 



Summary. 



The principal results arrived at for the application of 

 uniform surface-pressure are as follows : — 



A. 



(1) In any simple sphere or spherical shell, or in any one 

 layer of a compound sphere or shell, there is a simple linear 

 relation, independent of the elastic constants of the material, 

 between the values of any the same stress or strain over any 

 three concentric spherical surfaces. 



B. 



For the effects of altering a spherical layer so as to 

 increase one or both of its elastic constants (the effects of 

 diminishing one or both constants, generally speaking, being 

 the exact opposite) : — 



