In a previous paper it was shown that the quantum theory of conduction leads naturally to a division of crystals into conductors and insulators, and various properties of insulators were worked out. Since that paper was written, experimental material has come to my notice which necessitates an extension of the theory to include the effect of impurities, as it appears that impurities dominate the electrical properties of the semi-conductors. As the substances which show a negative temperature coefficient of the electrical resistance fall into two main classes, it will be as well to define what we mean by an electronic semi-conductor. In the first place, there are substances such as silicon which show a negative temperature coefficient in the impure state, but which are good metallic conductors in the pure state and are therefore to be classed as metals. The negative temperature coefficient is probably due to surface effects caused by the presence of oxide, and a tentative theory in this direction has been recently proposed by Frenkel. Secondly, there are substances such as cuprous oxide which always show a negative temperature coefficient and which become much worse conductors when the amount of impurity present is reduced. Only these latter substances are to be regarded as semi-conductors and it is with them that we shall deal in this paper. Lastly, there are some substances such as germanium which probably belong to both classes. That is, in some modifications they are metallic and in others insulating. The treatment of semi-conductors given in the previous paper depends on the fact that the energy spectrum of an electron moving in a perfect lattice splits up into bands of allowed and disallowed energies, and if there are just sufficient electrons present to fill up one of the allowed bands there can be no conductivity at absolute zero temperature.