Page:Elementary Text-book of Physics (Anthony, 1897).djvu/132

118 The modulus of rigidity is therefore equal to half the tangential stress $$S$$ divided by the elongation of unit length along the axis of the shear.

105. Modulus of Voluminal Elasticity of Gases.—Within certain limits of temperature and pressure, the volume of any gas, at constant temperature, is inversely as the pressure upon it. This law was discovered by Boyle in 1662, and was afterwards fully proved by Mariotte. It is known, from its discoverer, as Boyle's law.

Thus, if $$p$$ and $$p'$$ represent different pressures, $$v$$ and $$v'$$ the corresponding volumes of any gas at constant temperature, then Now, $$p'v'$$ is a constant which may be determined by choosing any pressure $$p'$$ and the corresponding volume $$v'$$ as standards: hence we may say, that, at any given temperature, the product $$pv$$ is a constant. The limitations to this law will be noticed later.

Let $$p$$ and $$v$$ represent the pressure and volume of a unit mass of gas at a constant temperature. A small increase $$\Delta p$$ of the pressure will cause a diminution of volume $$\Delta v$$; by Boyle's law we have the relation $$pv = (p + \Delta p) (v - \Delta v) = pv + v \Delta p - p \Delta v $$. We may assume that the increment $$\Delta p$$ is very small, in which case $$\Delta v$$ will also be small; we may therefore, in the limit, neglect the product of these increments and obtain $$\frac{\Delta p}{\Delta v} = \frac{p}{v}$$. Now $$\frac{\Delta v}{v}$$ is the change of unit volume, and therefore $$\frac{\Delta p}{\Delta v}v = p$$ is the modulus of voluminal elasticity. The elasticity of a gas at constant temperature is therefore equal to its pressure.

106. Modulus of Voluminal Elasticity of Liquids.—When liquids are subjected to voluminal compression, it is found that their modulus of elasticity is much greater than that of gases. For at least a limited range of pressures the modulus of elasticity of any one liquid is constant, the change in volume being proportional to