﻿164 Mr. C. Chree on the Local Alteration of 



the pressure is applied, the falling off in the radial pressure 

 over a concentric spherical surface varies directly as the 

 volume between the surface considered and that where the 

 pressure is applied, and inversely as the volume within the 

 surface considered. Of two equal volumes of the material, 

 that nearer the centre is the more effective in reducing the 

 radial pressure. 



Linear Relation between three Stresses or three Strains. 



§ 3. It is hardly necessary to point out that if (e.a.a), 

 instead of being a simple shell, were a layer of a compound 

 shell, the results (4) -(10) would still apply to it, lfp and jt/ 

 were taken equal to the pressures exerted on its inner and 

 outer surfaces respectively by the material of the adjacent 

 layers. 



Suppose, now, that in a simple shell, or in a single layer of 

 a compound shell, we take any three concentric; spherical 

 surfaces of radii r l9 r s , r z . Let us denote the volumes they 

 contain, viz. J irr-i &c, by V l5 V 2 , V 3 , and the radial pressures 

 over their surfaces by p x , p. 2 , p 3 respectively. By means of 

 (7) we easily find a linear relation between the three pres- 

 sures. This may be written in various equivalent forms, of 

 winch the most elegant is perhaps 



Kelations of exactly the same form hold between the values 



of 60, or those of any the same strain, over the three surfaces. 



du 



If *j, s 2 , s s , for instance, be the values of s, or — , over the 



three surfaces we have 



s # 2 -^) + *(vr^MvrT>°- ■ (13) 



If, then, we know the values of any strain or stress at any 

 tw y o radial distances in a simple shell, we have at once its 

 value at any other radial distance. The above results (12) 

 and (13) apply equally to a solid sphere or to any single 

 layer of a compound shell. They bear a distant family 

 resemblance to the well-known u theorem of the three mo- 

 ments " * in beams, usually associated with the name of 

 Clapeyron. 



* See Todhunter and Pearson's ' History of Elasticity/ vol. ii. art. 603 ; 

 or Love's "Treatise on Elasticity,' vol. ii. art. 221. 



