the symbols are such as present themselves in very different fields, [Pg 23] and it is to be remembered that it was from this very example that Thomson, in his early speculations, constructed analogies between elastic displacements and rotations on the one hand, and distributions of electric and magnetic force in free space on the other. To observe the growth of mathematical Electricity we must go back to the year 1811, when Poisson laid the foundations of Electrostatics as a branch of the theory of Attractions. Adopting the hypothesis of two electric fluids, he remarks that the resultant electric force at any point in the interior of a conductor must be zero. Combined with Coulomb’s law of electric force, and Laplace’s relation between normal force and surface density, this led at once to the distribution of electricity on a charged conductor in the form of an ellipsoid. Poisson further introduces the conception (but not the name) of the electric potential, and lays down the conditions which it has to satisfy at any point of the field due to a system of electrified conductors. In particular he investigates the induced distribution on a sphere due [Pg 24] to any system of external charges. Finally, by a triumph of analytical skill, he solved the classical problem of two electrified spheres. From the present point of view there is little further to record till Oersted’s discovery of the action of an electric current on a magnetic needle (1820). This was followed almost immediately by Savart’s analysis of the magnetic force into forces due to the infinitesimal elements of the electric circuit, and the simple rule which he formulated. This led Ampère to study the mechanical action between electric circuits. He analysed this into forces between the elements of the circuits, acting in the lines joining them, and subject to the law of action and re-action. His theory was based on a few plausible assumptions, and on a series of experiments devised in a strictly mathematical spirit to narrow down the various issues to be decided. His work is now seldom referred to, but it exhibits the [Pg 25] mathematical skill which he had exercised before in the Calculus of Variations, as well as in other directions. It is true that we are still in the atmosphere of action at a distance, and Ampère appeals in fact to the example of Newton and Gravitation, but only with Newton’s qualification. He does not claim to have arrived at an ultimate explanation of phenomena, but only to have established a formula from which these can be calculated. The consequences which he deduced are more significant than the formula of elementary attraction itself. In the first place he finds that the resultant effect of a closed circuit on an element of another circuit depends on a vector which is afterwards identified with magnetic force. He then finds the force exerted on a small closed circuit, and proves it to be identical with the force on an elementary magnet.