﻿178 Mr. C. Chree on the Local Alteration of 



In these formulae 2 denotes summation with respect to q 

 between the limits appearing below and above*. If the 

 material vary slightly as a continuous function of r the sum- 

 mations become replaced by integrations. 



For a solid compound sphere a = 0, and the preceding 

 formulae take the following simpler forms : — 



Subcase (i). 



-{l-2(K A /»)J, (59) 

 Subcase (ii). 



~ 3 k 



(60) 



Subcase (hi). 

 8k 



+i«k{<?(#> + 4(53H*}- <«) 



In (56) and (59) we notice that if the position and material 

 of the sth layer remain unaltered, the displacement throughout 

 it is unaffected by any rearrangement of the material which 



s i 



leaves 2 (Kv/v) and 2 (K q v q /v) unaltered. Thus, if the 



1 8 + 1 



layers of a compound shell or solid sphere differ only in com- 

 pressibility, any redistribution of the material throughout a 

 certain volume which consists in splitting one layer into a 

 number of layers or altering in any way the order in which 

 the different materials occur, while leaving unaltered the 

 volume of each separate material, has no effect on the dis- 

 placement, strains, or stresses throughout any layer no part 

 of which is included within the volume where the redistri- 

 bution has occurred. A particular deduction is that the 

 increment of radius of a solid sphere, or of either surface of a 

 shell, is unaffected by any redistribution in layers which 



i 

 * We must interpret 2 (Kv/v) as but the single terniK^ /v, and similarly 

 l H q 



i 



in other cases. 2 is simply zero. 

 »+i 



