﻿174 Mr. C. Ohree on the Local Alteration of 



where 



n=(3* + 4»)[^»(3i 1+ 4 Bl )+ 6 -^{^=^ 





+±=a£=S}]. • . (42) 



Substituting from (41) for 8p b in equations (19) to (24), 

 we get the increments which, added to the corresponding re- 

 sults (4) to (9) , give the complete values of the displacements 

 &c. throughout the unaltered material (b . a . a) . The incre- 

 ments themselves measure, of course, the effects of an altera- J 

 tion of the material at the inner surface from k, n to & 1? ?i v 

 The value of hp c and the corresponding increments (25) to 

 (30) obtained for (e .a . c . a x . a), as explained above, would 

 give the effects of a similar alteration at the outer surface. 



A comparison of these results will be found to emphasize 

 the fact that a change of rigidity is much more effective at 

 the inner than at the outer surface of a thick shell. 



Solid Sphere. 



§ 11. The results for a solid sphere under uniform surface- 

 pressure ma}' easily be derived from those for a shell. 



In a simple sphere (0 . a . a) , under surface-pressure p', the 

 displacement is deduced from (5) by taking e zero, and the 

 strains and stresses are thence easily written down. We 



thus get : — 



r */ ? '= -**=*> < 43 ) 



*=-///*, (44) 



^=00=--;/, (45) 



S = 0. . r (46) 



These results may also be derived from the corresponding 

 strains and stresses in the shell (e . a, . a), by omitting all 

 terms multiplied by e, even though a power of r occurs in the 

 denominator. Perhaps an even simpler method of reduction 

 consists in supposing p—y l . 



For the compound sphere (0 . a . c . a x . b . a . a) we may, as 

 in the case of the compound shell, proceed in either of two 

 ways. We may start with the results for the simple sphere 

 (0 . a . a) and find the increments in the displacements, strains, 

 &c, depending on the increments 8pb and 8p c of pressure at 



