﻿180 Mr. C. Chree on the Local Alteration oj 



provided rr—Sku/r be not zero*. This applies at any point 

 of a simple shell or of a layer of material k, n in a compound 

 shell. 

 We may write the above in the form 



U 



— ( k + b n ) sec 2 = rr-i- (u/r). 



Now rr and u/r are continuous at a common surface of two 

 contiguous layers of a compound shell. Thus, distinguishing 

 corresponding quantities in two such layers by dashed and 

 undashed letters respectively, we find at the common surface 



{M + M$QGW-(M' + ±ri)sQcW = §(k-k f ). . {66) 



This relation connecting 6 and 6' depends solely on the 

 material of the layers, and not at all on the radius of their 

 common surface. It should, however, be noticed that if a 

 layer of material k', n' were intercalated between two layers 

 of material k, n, the values of cos & at its two surfaces would 

 differ, so that 6 has different values at the inner surface of 

 the outer layer of material k, n and at the outer surface of the 

 inner layer of the same material. 



Linear Relation between Changes of Volume. 



§ 14. In conclusion, I would call attention to a simple and 

 concise relation between the changes of volume or the dilata- 

 tions of the layers composing a compound shell or solid sphere 

 exposed to uniform pressure. 



If k be the bulk modulus of a volume v of an isotropic 

 material exposed to surface-forces whose components at the 

 point x, y, z on the element d$ of surface are F, G, H per 

 unit of surface, then the increment of the total volume, 8v, . 

 due to the action of these forces, is given by 



Sk8v=$(Fa: + Gy + 'Rz)d&, . . . (67) t 



the integral being taken over the entire surface. 



Suppose now we apply this to the several layers of which 

 a compound shell is composed. The surface-integrals taken 

 for any two contiguous layers over their common surface 

 clearly vanish. Thus, summing the equations like (67) for 

 the several layers, nothing remains on the right-hand side but 

 the integrals referring to the inner and outer surfaces of the 



* It is zero in the core of a compound sphere, and in every layer of a 

 compound shell of uniform compressibility. 



t See Trans. Camb. Phil. Soc. vol. xv., equation (23) p. 318. 



