An excerpt from the beginning ofCHAPTER I. GENERAL PRINCIPLES. THEORY OF FREE ELECTRONS: THE theory of electrons, on which I shall have the honor to lecture before you, already forms so vast a subject, that it will be impossible for me to treat it quite completely. Even if I confine myself to a general review of this youngest branch of the science of electricity, to its more important applications in the domain of light and radiant heat, and to the discussion of some of the difficulties that still remain, I shall have to express myself as concisely as possible, and to use to the best advantage the time at our disposal. In this, as in every other chapter of mathematical physics, we may distinguish on the one hand the general ideas and hypotheses of a physical nature involved, and on the other the array of mathematical formulae and developments by which these ideas and hypotheses are expressed and worked out. I shall try to throw a clear light on the former part of the subject, leaving the latter part somewhat in the background and omitting all lengthy calculations, which indeed may better be presented in a book than in a lecture. 1. As to its physical basis, the theory of electrons is an offspring of the great theory of electricity to which the names of Faraday and Maxwell will be for ever attached. You all know this theory of Maxwell, which we may call the general theory of the electromagnetic field, and in which we constantly have in view the state of the matter or the medium by which the field is occupied. While speaking of this state, I must immediately call your attention to the curious fact that, although we never lose sight of it, we need by no means go far in attempting to form an image of it and, in fact, we cannot say much about it. It is true that we may represent to ourselves internal stresses existing in the medium surrounding an electrified body or a magnet, that we may think of electricity as of some substance or fluid, free to move in a conductor and bound to positions of equilibrium in a dielectric, and that we may also conceive a magnetic field as the seat of certain invisible motions, rotations for example around the lines of force. All this has been done by many physicists and Maxwell himself has set the example. Yet, it must not be considered as really necessary; we can develop the theory to a large extent and elucidate a great number of phenomena, without entering upon speculations of this kind. Indeed, on account of the difficulties into which they lead us, there has of late years been a tendency to avoid them altogether and to establish the theory on a few assumptions of a more general nature. The first of these is, that in an electric field there is a certain state of things which gives rise to a force acting on an electrified body and which may therefore be symbolically represented by the force acting on such a body per unit of charge. This is what we call the electric force, the symbol for a state in the medium about whose nature we shall not venture any further statement. The second assumption relates to a magnetic field. Without thinking of those hidden rotations of which I have just spoken, we can define this by the so called magnetic force, i. e. the force acting on a pole of unit strength. After having introduced these two fundamental quantities, we try to express their mutual connexions by a set of equations which are then to be applied to a large variety of phenomena. The mathematical relations have thus come to take a very prominent place, so that Hertz even went so far as to say that, after all, the theory of Maxwell is best defined as the system of Maxwell's equations.