﻿162 Mr. C. Chree on the Local Alteration of 



along any two mutually orthogonal directions perpendicular 

 to r. 



The dilatation A is thus given by 



A=J+2- M (1) 



ar r v ' 



The principal stresses consist of the radial 



£=(*-§») A + 2n J, (2) 



and two equal transverse stresses, 



00=^0= (jfe—fn)A + 2ntt/r, ... (3) 



along any two mutually orthogonal directions perpendicular 

 to r. Here k and n denote respectively the bulk modulus and 

 the rigidity. 



We shall in general find it convenient to consider instead 



of rr the radial pressure (—rr). 



For brevity a shell whose inner and outer surfaces are of 

 radii b and a, and whose elastic constants are k and n, will be 

 represented by (b . a . a) , where a is adopted as representing 

 elastic quality. A solid sphere of radius a will be repre- 

 sented by (0 . a . a) . When there is only one material the shell 

 will be termed simple, as opposed to compound when there are 

 two or more materials. 



§ 2. In a simple shell (e.a.a) exposed to uniform pressures, 

 p over the inner and p' over the outer surface, the following 

 results are easily found : — 



- _ P e% ( * * a3 \ P' aZ /_ * * e *\ 

 S " a 3 -A 3£ + 2^ ^/ + ?-?\ 3* + 2^ ?/" ( ' 



■Vf-»- a 8 -A3* + 4nrV a 3 -* 3 V3yfe + 4n r 8 /' * W 



A -i*S# m 



(--)= a -^ ? p(J-l)+^(l-^)]. • • • (7) 



