﻿372 Mr. C. Chree on the Relation between the 



pressure ; then 



1 8Vi (1) 



1 8V? (S) 



It is assumed that the change of volume does not exceed 

 the elastic limits, and that strain varies directly as stress ; in 

 other words, that a t and a e are constants independent of the 

 absolute magnitude of p r or p". 



The experimental determination of a e appears to be easy ; 

 but that of cti is, according to Dr. Guillaume, so troublesome* 

 that he prefers to deduce it by the theoretical relation 



ct e —Ui=l/k, (3) 



where k is the bulk-modulus of glass, assumed to be an 

 isotropic elastic material. 



On reading Dr. Guillaume's book I was struck by the 

 simplicity of the result (3), but was unable to feel confidence 

 in the proof, of which the following is a brief outline: — 



Let Rj and R e denote the radii of the inner and outer 

 surfaces of an isotropic spherical shell, and let it become 

 exposed to uniform internal and external pressures P t and P e . 

 Then the elastic displacement u is along the radius, and is 

 given at a distance r from the centre by 



1 PjR^ — P e R e J_ (Pj—.P e ) R e Rj 1^ ... 



U 3X+2/* b»_r» r± 4,fi R3_ R 3 r v • U 



where X and /jl are Lame's elastic constants. Noticing 



it is easy to prove that the relation (3) holds exactly. 



Take next a hollow circular cylinder extending from the 

 plane ^ = in the direction of z positive. Let R; and R e be 

 the radii of its inner and outer cylindrical boundaries, and let 

 uniform internal and external pressures Pi and P e be applied. 

 Then, according to Dr. Guillaume, there is at any point not 

 too near the ends of the cylinder, and at distance r from the 

 axis, a displacement w parallel to the axis and a displacement 



* See, however, vol. i. p. 79 of the Wiss. Abhand. Physik. Tech. 

 Reichsanstalt at Charlottentmrg, with description of the direct determi- 

 nation of the internal coefficient and results. 



