﻿170 Mr. C. Chree on the Local Alteration of 



Again, when p' only exists A is negative, and SA 2 and 

 8 A are positive when n x — n is positive and k t — k is positive 

 or zero ; thus the presence of the altered layer reduces the 

 contraction in (e .a . c) more than that in (b . a. . a). 



Thus, whether there be pressure on the outer surface alone 

 or the inner surface alone, the presence of the layer (b . ol x . c), 

 in which the rigidity exceeds and the bulk modulus equals or 

 exceeds that of the remainder, has a greater effect on the 

 density of the contained layer (e .oc .c) than on that of the 

 surrounding layer (b . a . a). 



It is equally easy to show that the density of (e . a . c) is in 

 general more affected by the existence of the altered layer 

 than is that of (b.a.a) when n x —n is negative, with a 

 possible exception in the case of pressure on the outer 

 surface when k x — k is also negative and ejc is very small. 



§ 7. The general effect of the existence of the altered layer 

 is, as we have seen, to diminish or to increase the effects of 

 surface-pressure in changing the density of the unaltered 

 material according as the elastic constants of the altered layer 

 are greater or less than those of the unaltered material. 



The converse question naturally presents itself, viz., how 

 the change of density of the altered layer compares with 

 that in the simple shell {e . a 2 . a), all of the same material as 

 the layer, exposed to the surface-pressures p and p. 



If we regard the dilatation as A l in (e.a^.a), and as 

 Aj-fSAx in (c.a^.b) when there is the compound shell 

 (e . a . c . «j . b . a . a) , we easily find from what precedes * that 



nSAA(a 3 -e 3 ) = 

 pUnn,(h-k)(Zk + 4n) ^-^^-e 3 



[eft (£ — b 3 e 3 

 Mk^ni -n)(3£ + 4w) 8 



+ * n ( k y k) i*7i 1 (3yfc + 4n)(a 3 -6 3 + c 3 - e 3 ) 



-^-n ^-^-W-* }].. . (32) 



* Since neither k nor n appears in (7) the radial pressures over r—b 

 and r—c are the same in (e.ec l . a) as in (e . et . a) so that SA, equals the 

 dilatation in a simple shell (c . u x . b) due to surface-pressures 8p h and hp c 

 as given by (10) and (17). 



