It marked a significant advance in astrophysics when M. N. Saha first showed how the degree of ionization in stellar material could be calculated in terms of its temperature and pressure (or density). In the usual theory of thermal ionization the free electrons are treated as a classical perfect gas and, therefore, the theory is applicable only so long as the temperature and density of the material are such that the free electrons are non-degenerate in the sense of the Fermi-Dirac statistics. In the outer atmosphere of a star the condition of non-degeneracy is always satisfied, but in the interior of the white dwarf stars and the planets (and possibly in the interior of other stars as well) the conditions of temperature and density are such that the free electrons form a degenerate gas and their behaviour can no longer be described in terms of classical perfect gas. In the case of cold matter Saha’s theory loses its validity and the degree of ionization in degenerate matter must be investigated on other lines. This investigation of the ionization in degenerate matter is of importance in astrophysics, particularly in researches dealing with the internal constitution of the white dwarf stars and also, as has been recently shown (Kothari and Majumdar 1936 a , 1936 b ; Kothari 1936), in predicting the maximum radius for a cold body.