THE EQUILIBRIUM OF ELASTIC SOLIDS. 51 



by Tredgold from the extension of rods was that denoted by E, and the value 

 of IJL deduced from E alone by the formulae of Poisson cannot be true, unless 



= $; and as - for lead is probably more than 3, the calculated compressi- 

 bility is much too great. 



A similar experiment was made by Professor Forbes, who used a vessel of 

 caoutchouc. As in this case the apparent compressibility vanishes, it appears 

 that the cubical compressibility of caoutchouc is equal to that of water. 



Some who reject the mathematical theories as unsatisfactory, have conjec- 

 tured that if the sides of the vessel be sufficiently thin, the pressure on both 

 sides being equal, the compressibility of the vessel will not affect the result. 

 The following calculations shew that the apparent compressibility of the liquid 

 depends on the compressibility of the vessel, and is independent of the thickness 

 when the pressures are equal. 



A hollow sphere, whose external and internal radii are a, and 2 , is acted 

 on by external and internal normal pressures h and A 2 , it is required to deter- 

 mine the equilibrium of the elastic solid. 



The pressures at any point in the solid are : 



1. A pressure p in the direction of the radius. 



2. A pressure q in the perpendicular plane. 



These pressures depend on the distance from the centre, which is denoted 

 by r. 



The compressions at any point are r- in the radial direction, and - in 

 the tangent plane, the values of these compressions are : 



_ -- p ...................... (34). 



dr \9/i 3m/ v m * 



L\(p + 2 g) + -tf ............... (35). 



3m/ ^ *' m * 



Multiplying the last equation by r, differentiating with respect to r, and 

 equating the result with that of the first equation, we find 



72 



