A final step towards a complete formulation on modern lines of the mathematical relations of Electricity consisted in the expression of magnetic force, or rather magnetic induction, in terms of the vector now known by the name of electric momentum. This vector, or its analogues, presented itself in various ways. We have [Pg 33] first an investigation by Kirchhoff on the laws of induction in three-dimensional conductors, based on Weber’s law of electric force. Almost simultaneously we have Stokes’s paper on the Dynamical Theory of Diffraction, which is not so important nowadays as a contribution to Optics, but as containing a calculation of the waves in an elastic medium due to any initial disturbance. This was made to depend on Poisson’s integration of the general equation of sound, and it is here that we meet for the first time with a full interpretation of this solution, which led up to that of the elastic wave-problem. The relation to the present matter consists, however, in the kinematical process by which displacements in any medium are expressed in terms of expansions and rotations, so that in Clifford’s language everything is reduced to “squirts and whirls.” The same process occurs again some years later in Helmholtz’s great memoir on Vortex Motion, where we meet explicitly with the analogy of the relations between electric currents [Pg 34] and magnetic force to those between vortices and fluid velocities. This analogy is developed towards the close of the investigation, but we can now see that it was implicit from the beginning in the very definition of a vortex. In both investigations the connection is established by means of a subsidiary vector, which in the electric analogy corresponds to the electric momentum of Maxwell.