Cylindrically symmetrical electric or magnetic fields are often present in electrical apparatus. It is usually considered to be prudent to study these fields experimentally rather than mathematically, since analytic methods require more advanced mathematics than is normally possessed by practical engineers. The author has derived in the present paper some analytic solutions for a few common field problems of this kind. The paper is essentially in three parts: the first two parts are concerned with electric fields; the last being devoted to a magnetic field. The first part deals with the general theory of zonal toroidal functions and in particular with the application of these functions to the problem of the electric field of an electrode system consisting of a rod and a concentric ring. The experimental investigations undertaken by the author to justify the usefulness of these functions were carried out with the aid of an electrolytic tank. The second part is devoted to the allied problem of the sphere gap. Legendre's functions and dispherical coordinates are used. The third part deals with the exact calculation of the self-inductance of a circular turn of wire by means of toroidal functions. Finally a number of appendices have been included, dealing with a variety of interesting details which it has been considered advisible to separate from the main text of the thesis.