﻿17(3 Mr. C. Chree on the Local Alteration of 



The value of S in the compound sphere is thus of special 

 interest. For it we easily find : — 



in (0 . a . c), 



S = 0; 



in (c . «! . b) , 



S,6/g)'(*i~*> n I g* + 4w) ; . . . (53) 



in (b . a . a), 



S = 6 /yfo~* )w g* + 4ni l . . . (54) 



Since S varies as b d — c 3 in (b . a . a) , it follows that the 

 stress-difference will there remain small so long as the altered 

 layer is of small volume until p' becomes very large. 



In (c . «! . b) , however, S depends on the volume of the 

 altered layer only in so far as that enters into II', and is thus 

 of importance, however thin the layer may be, provided it 

 differ considerably from the rest in compressibility. 



It is hardly necessary to point out the loophole this affords 

 to supporters of the stress-difference theory * of rupture, if 

 experiment should seem to decide against them. 



Any number of Layers. 



§ 12. The compound shell formed of any number of con- 

 centric layers of different materials may be treated in the 

 same way as the three -layer shell ; but probably it is quite as 

 easy to express the displacement in each layer in terms of 

 arbitrary constants and determine these from the continuity 

 of the radial displacement and stress. 



Let there be i layers, the radii of the (^+1) surfaces 

 reckoned from within outwards being a , a 1? . ... a*. The 5th j 

 layer is thus that the radii of whose bounding surfaces are 

 a s -i and a s . The volume of the 5th layer, i. e. 4z7r(al—-a^_ l )/3, 

 is denoted by v g , and its elastic constants by k 8 and n g . The 

 notation 



4» S /(3* S + 4« S ) = K„ -j 



3V(3* s + 4»,)=N s =(^)m 8 | • • (55) 



is also used. 



I shall merely record the value of the displacement in the 

 sth layer in some special forms of the general case, the pres- 

 sures p and p' being applied as before over the inner and 

 outer surfaces respectively of the compound shell : — 

 * See Phil. Mag. September 1891, pp. 240-2. 



