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 [Pg 28] 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 [Pg 29] 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 [Pg 30] 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.