SummaryAn attempt is made to reduce the generality of the usual theories of finite strain and to concentrate on the case of hydrostatic pressure. The method used has been based on the works of Murnaghan (1951) and Birch (1952), and yields a general relation giving the elastic constants in terms of the strain and the derivatives of pressure with respect to strain.It becomes evident that a law in lieu of Hooke's law is needed to proceed with a mathematical theory; and so various elasticity equations in the literature are investigated to determine the special assumptions used to derive them from the present general equations. A linear relation between the second and third order elastic constants is then proposed and the resulting relation between pressure and density compared with Bridgman's experimental results for the alkali metals. It is found that the proposed law of finite hydrostatic strain agrees favourably with experiment and also with deductions from the atomic theory of solids.The paper concludes with a few relevant remarks on the implications of the present theory to certain seismological problems.