The evolution of mathematical physics

by Sir Horace Lamb

the symbols are such as present themselves in very different fields, 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 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 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. The familiar representation of a current by a magnetic shell follows, as well as the theory that the properties of a magnetized body are due to currents circulating in the molecules. Two provinces of physics, hitherto distinct, were here for the first time co-ordinated.

The views of Ampère, owing to their novelty, naturally excited at first some distrust. Preconceptions, especially when they have a definite form, die hard; and it is to be remarked that Poisson’s great memoir on Magnetism, in which the hypothesis of two magnetic fluids is supposed to be verified, coincides almost in time with the latest publication of Ampère.

A good deal of Poisson’s work on Magnetism has become classical, in the sense that subsequent writers have found nothing better than to reproduce it. It is largely independent of the two-fluid theory, and is really a theory of magnetic elements, afterwards treated explicitly as such, without further hypothesis, in the extensions given later by Thomson. The transformation by which the potential of a continuous arrangement of magnetic elements is expressed as due to distribution of imaginary magnetic matter through the volume and over the surface now appears for the first time. In his treatment of magnetic induction Poisson imagines his two fluids to be free to move within molecular spaces which for definiteness he treats as spherical. This latter assumption may be taken as merely illustrative, although it leads to a definite and sometimes impossible value of a coefficient. The particular problems solved, viz. the magnetization of a spherical shell, and of an ellipsoid, by a uniform field retain an interest independent of this special hypothesis.

The years which immediately followed were marked chiefly by the researches of Navier, Cauchy, and Poisson on Elasticity which have already been noticed. We come next to Green’s Essay on Electricity and Magnetism (1828). The mathematical theory of Electrostatics, which had been initiated by Poisson, is here resumed and in a sense completed. The treatment is based on the theorem now generally quoted by the author’s name. The novel point here is not the transformation from volume- to surface-integrals, for this was to be found in Poisson, but that it is the first example of the reciprocal relations which pervade not only Dynamics, but all branches of Physics. In the present case it is a relation between two different distributions of Electricity, but it only needs to give suitable meanings to the symbols to translate it into the language of Hydrodynamics or Acoustics. From the mathematical standpoint we have, further, the treatment of singularities of harmonic functions. The electrostatic theorems due to Green are reproduced in most modern text-books. Among original results we may notice the screening effect of conducting surfaces, the distribution of electricity on a spherical conductor due to internal or external charges, and the theory of condensers.

The phenomena of mutual induction and self-induction of electric currents were discovered by Faraday in 1831-35, but a long period elapsed before these received explicit mathematical investigation, and a longer still before it was recognized that Faraday’s own description in terms of lines of force could be put in an exact mathematical form. The work of F. Neumann (1845-47) was the complement of that of Ampère and involved the same kind of ideas. The additional experimental fact adduced was Lenz’s law. When there is relative motion of two circuits, or of a circuit and a magnet, currents are induced and there are consequent mechanical forces, which can be calculated from the formulae of Ampère. The law referred to is that the sense of the induced currents is such that these mechanical forces act in opposition to the relative motion. Neumann assumes this to be true also as regards the infinitesimal elements into which the circuits may be resolved, and further that the electro-motive force of induction is proportional to the velocity of the relative motion, to the strength of the inducing current or magnet, and to the component (with sign reversed) of the mechanical force in the direction of the relative motion. For the two former of these assumptions there was the experimental evidence of Faraday and others, the latter was adopted as the simplest supposition consistent with the law of Lenz. From this basis he proves that the total current induced in a circuit by the motion of a magnetic pole is proportional to the change in the potential of the pole in relation to a unit current in the circuit, and again to the change in the flux of magnetic force through the circuit. This is really Faraday’s rule, except that it is not expressed in so many words in terms of lines of force. In the second paper he shews that the mechanical action between two currents depends on the mutual potential of the two circuits, viz.

and refers the electro-motive forces of induction to changes in the value of this function.

We are still in the atmosphere of action at a distance, and it was therefore not unnatural that Weber and others should have looked for an explanation both of the mechanical and the inductive effects in a modification of Coulomb’s law of force between electric charges. Since the actions to be explained depend on rates of change, violence had to be done to previous notions, and terms depending on mutual velocities and accelerations were introduced. The resulting law of Weber, which happened to be so framed as not to conflict with the conservation of energy, long exercised a fascination on continental writers, owing to the mathematical neatness of the processes by which the results of Ampère and Neumann could be deduced from it. It was not finally abandoned until Helmholtz shewed that under certain conditions it implied unstable electrical equilibrium, as well as other paradoxical consequences.