﻿Isotropic Spheres under Uniform Surface-Pressure. 169 



whether expansion following internal pressure p, or contrac- 

 tion following external pressure p'— throughout the unaltered 

 material both outside and inside of it. 



The exact opposite holds if ?i x — n and k l — k be one negative 

 the other zero, or if both be negative, a possible exception 

 arising in the latter case when e/c and b/a are both very 

 small. 



Whilst the changes of the dilatation in (b . a . a) and (e.a.c) 

 due to the existence of the altered layer agree in general 

 qualitatively, they usuallv differ quantitatively. 



For if A + SA refer to (e.a.c) and ( A + SA 2 ) to (b . a . a) 

 — the dilatation in the simple shell (e.a.a) under the same 

 pressures being A — we easily find 



-p'^XM + ^+Mk-k)^]]. . (31) 



Thus in general there is a difference between the dilata- 

 tions in (b . a . a) and (e.a.c) which increases with the 

 volume of the altered layer, and is more conspicuous the 

 closer this layer to the inner surface of the compound shell. 



If n x - n, 



then SA 2 = $A. 



Thus if the altered layer retain the same rigidity as the rest 

 the dilatations in (b . a . a) and (e.a.c) remain equal, however 

 much the altered layer may differ from the rest in compres- 

 sibility. 



The coefficient of p in (31) is easily shown to be essentially 

 positive, and that of p' is obviously so. Thus when n 1 — n 

 is positive, and k x — k is positive or zero, 5A 2 — 8 A has the 

 same sign as p when p is zero, and the opposite sign to p 

 when p is zero. 



Now when p only exists A is positive, and 8A 2 and 8 A 

 are both negative when n x — n is positive and'^ — & is positive 

 or zero. It follows that under these circumstances the 

 presence of the altered layer is more effective in reducing the 

 dilatation in (e.a.c) than in reducing that of (b . a . a). 



exceptional case when p only exists, viz. when the intercalated layer is of 

 exceptionally large rigidity and lies close to the inner surface of the 

 shell, while a/b is large. The expansion naturally accompanying pres- 

 sure on the inner surface may then be converted into an absolute con- 

 traction throughout the thin inner layer (c . a . c). 



