﻿Isotropic Spheres under Uniform Surface- Pressure. 171 



The coefficients of k x — k and of n x — n inside the square 

 brackets in the coefficients of both p and p' may be shown to 

 be essentially positive. 



Having regard to the signs of Aj in the several cases we 

 at once obtain the result that when n x — n and k x — k are both 

 positive, or one positive and the other zero, the layer (c .a x . b) 

 suffers more alteration of density in consequence of the 

 application of p or p' than if the whole shell were of material 

 hi, n x . In other words, when the layer has larger elastic 

 constants than the remainder it suffers more alteration in its 

 density than if the entire shell consisted of the same material 

 as the layer. 



Alteration of material small. 



§ 8. The preceding results apply irrespective of the magni- 

 tudes of (ki — fy/k and (n x — n)/n, provided these be neither 

 infinitely great nor infinitely small, and so hold however 

 great be the differences of the two materials. 



The case when the material is nominally the same through- 

 out, but the elastic constants vary slightly with r, seems not 

 unlikely to be of frequent occurrence in practice. The law 

 of variation of the elastic constants in such a case is probably 

 in general a continuous function of r, still the variation may 

 not infrequently be practically restricted to a thin layer. It 

 thus seems worth while glancing briefly at the special case 

 when (fti — n)/n and [k-^ — k)^ are so small that terms con- 

 taining their squares or product may be neglected. In this 

 case, putting for shortness 



3£/(3& + 4n)=N,l 



4n/(3* + 4rc) = K,J ' ' ' ' K * 



we get from (16) and (17) 



(rf-b*)(»-*) [n,-n (e*ay k,-k p'a*- P e* \ 



3 _^2 



(a 3 -* 3 ) 



to S )(c 3_, } { nj __ n n ^ p _ pi) + ^ K ^V } (35) 



If ni = n, 



or the altered layer differ from the rest only in compressibility, 

 we have 



b* s -c 3 . W — c^^-k Tjr p'a*—pe* ,_.. 



