1904.] the Volume Elasticity of Elastic Solids. 521 



under similar limitations, to a cylindrical tube closed by caps and 

 exposed to uniform external pressure p e . The tube exposed to pressure 

 would have to be contained in a strong vessel, having a glass roof or 

 side. This might present disadvantages, but the test tube itself would 

 be less exposed to temperature changes than in Case (i) and might be 

 of a very simple type ; so that once the containing vessel was built 

 experiments on a variety of materials would be simple. 



Case (iii) is only a special instance of the obvious result that when 

 an isotropic solid of any shape, bounded by any number of surfaces, is 

 exposed to pressure p, the same at every point of each and all of its 

 bounding surfaces, every line in it whose original length is I suffers a 

 shortening given by 81/1 = -p/3L 



In Case (iii) any body will serve the purpose, but a long rod would 

 be the natural form to adopt. This means a simpler test object than 

 in Cases (i) and (ii), and there is the further recommendation that the 

 mathematical solution is exact right up to the extremities of the 

 object. The compensating disadvantage is that it requires a very long 

 test object, or a very high pressure, to give sufficient extension, unless 

 extremely refined methods of measurement exist. In Cases (i) and (ii) 

 by using a thin walled tube one has much greater sensitiveness. 



By subjecting a cylindrical tube successively to internal and external 

 pressures, and determining values for k by both the methods (i) and (ii), 

 interesting information could be obtained as to whether the bulk 

 modulus is or is not the same for extension and compression. 

 Eemoving the caps, and determining E by simple longitudinal tension, 

 one would thus determine fully the elastic properties of the material, 

 assuming it isotropic. 



In any case 81 should be measured from the equilibrium position of 

 the test object after it is supported, and if it is hollow and exposed to 

 internal fluid pressure after it has been filled with liquid. 



Eesults in some respects more general than the preceding are 

 obtained very simply from the formulae which I have given for the 

 mean change of length in elastic solids under any given system of 

 loading. 



Not confining ourselves to isotropy, suppose first that the material 

 is merely symmetrical with respect to three rectangular planes, which 

 we may suppose perpendicular to the axes of x, y and z. 



Let E 3 be Young's modulus for traction parallel to the axis of z, 

 and 7731, that one of the (six) Poisson's ratios which answers to traction 

 parallel to z and contraction (strain) parallel to x. 



Let g = dyjdz represent the extensional strain parallel to z. Then 

 'in a body of any shape exposed to surface forces whose components at 

 any point parallel to x, y, and z are denoted by F, G, H, the mean 

 value g of g throughout the whole volume v is given by 



