This thesis is divided into two parts; the first containing theoretical work and discussions; and the second the experimental work. Section 1 contains a discussion of the stationary states in an atom and the kinetic theory expression for the mean free path of an electron in a gas, leading up to the formula for the passage of a beam of electrons through a gas. In section 2, the previous methods of obtaining cross sections, both total and inelastic, are dealt with and the results discussed. An outline of the wave mechanical theory of collision processes is given in section 3, with indications of the agreement obtained with experiment by different methods. The results of some approximate theoretical calculations of phases and an angular scattering curve for krypton are given in section 4. The results of this section have already been published by Dr. F. L. Arnot and the author. Section 5, the first of Part II, contains a description of a new apparatus designed to obtain measurements of the total, elastic and inelastic cross section in gases. The method of using the apparatus and a number of tests of its working are described in section 6. The last section, 7, contains the results obtained with the apparatus for the total, elastic and inelastic cross sections of the mercury atom, along with the cross sections for the ionisation, 2¹P₁, 2³ P₁, 3¹D₂ states. It concludes with an interesting test of the working of the apparatus and a discussion of the possible errors of the method.