It may, perhaps, seem as if the value of such an analogy as this consisted merely in the prospect which it offered of comparing, and thereby extending, the mathematical theories of heat and electricity. But to the physicist its chief interest lay rather in the idea that formulae which relate to the electric field, and which had been deduced from laws of action at a distance, were shown to be identical with formulae relating to the theory of heat, which had been deduced from hypotheses of action between contiguous particles.
In 1846—the year after he had taken his degree as second wrangler at Cambridge—Thomson investigated[1] the analogies of electric phenomena with those of elasticity. For this purpose he examined the equations of equilibrium of an incompressible elastic solid which is in a state of strain, and showed that the distribution of the vector which represents the elastic displacement might be assimilated to the distribution of the electric force in an electrostatic system. This, however, as he went on to show, is not the only analogy which may be perceived with the equations of elasticity; for the elastic displacement may equally well be identified with a vector a, defined in terms of the magnetic induction B by the relation
.
The vector a is equivalent to the vector-potential which had been used in the memoirs of Neumann, Weber, and Kirchhoff, on the induction of currents, but Thomson arrived at it independently by a different process, and without being at the time aware of the identification.
The results of Thomson's memoir seemed to suggest a picture of the propagation of electric or magnetic force: might it not take place in somewhat the same way as changes in the elastic displacement are transmitted through an elastic solid? These suggestions were not at the time pursued further by their author; but they helped to inspire another young
- ↑ Camb. and Dub. Math. Journ. ii (1847), p. 61: Thomson's Math, and Phys. Papers, i, p. 76.
