The year (1846) in which Weber’s law of electric force was promulgated marks also very approximately the beginning of the modern tendency to ignore action at a distance, and to bring the medium across which electric and magnetic actions take place into the reckoning. The elastic analogies of Thomson have been mentioned already. Another analogy, between Electrostatics and Heat-Conduction, had been noted by him a little earlier, and used to illustrate various propositions in Attractions. The mathematical theory of Magnetism, next taken up by Thomson, was set forth in a form free from all hypothesis, the magnetic fluids of Poisson and others being now replaced by the notion of magnetic polarization. He further added to the grammar of continua by developing the conceptions and the properties of solenoidal and lamellar distributions of magnetism, which were suggested by Ampère’s investigations. The two definitions of magnetic force in the interior of a magnet, afterwards distinguished as magnetic force and magnetic induction, are also introduced here for the first time. The whole memoir is a model of scientific exposition, and recalls the ‘grand style’ of the classical mathematicians, and especially of Gauss.
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 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 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.
The paper by Maxwell “On Faraday’s Lines of Force,” written shortly after he had taken his degree, is now perhaps little read, but deserves attention if only for the introduction, written in his own incomparable style, where we find already laid down the lines on which his subsequent speculations were to proceed. From the mathematical standpoint the paper is a comprehensive statement, without a suggestion of theory, describing the known facts of Electro-magnetism in terms of a system of vectors supposed to exist at all points of the field. Precision is here given to Faraday’s idea of lines of force, whether electric or magnetic, by means of the analogy of the motion of an incompressible fluid. The new vector here introduced into Electro-magnetism is that of momentum, and its rate of change is shewn by a dynamical, argument to be responsible for electro-magnetic induction. The proof of this depends on the expression for the energy of the field in terms of an integral extending over space, and is a deduction from the conservation of energy. The dynamical relation between pondero-motive and inductive forces had been indicated in a general way by Helmholtz in his celebrated tract, and this may possibly have been the first suggestion to Maxwell’s subsequent dynamical theory.
The way was in fact now clear, so far as the mathematical scheme is concerned, for Maxwell’s definite theory. He ventured as we all know to go a step further and to look behind the mathematical relations for a deeper insight into the matter, and if possible for a physical or mechanical meaning of the analytical symbols. Regarding the question as a dynamical one he sketched out a mechanical model of the ether which should reproduce the known electrical relations, rather with a view of convincing himself that such a model was possible than as a definite explanation in detail. This was followed by the classical paper in which the laws of electro-magnetism were shewn to be deducible from dynamical considerations, without the assumption of any particular mechanism. The final presentment in his treatise, in which use is made of Lagrange’s generalized equations, is too familiar to need further reference. Whether we prefer to regard it as an analogy or an explanation, it is a striking exemplification of the originality of Maxwell’s genius.
At this point we may appropriately close our survey, for I do not undertake to be a guide in the subsequent history, which is still in the making. It is, however, to be remarked that Maxwell, who placed as it were the crown on one period of Mathematical Physics, was also in a sense the initiator of another, by his work on Gas Theory, which involved the creation of a molecular calculus.
Looking back on this long history we can trace through all the details an increasing tendency. The period we have been reviewing began under the influence of the great achievements of Laplace and Lagrange in the development of the Newtonian Astronomy. The notion of action at a distance, though not regarded by Newton himself as the last word on the matter, had had a great success, and when the field of Physical Astronomy was beginning to be fully occupied, the mathematicians who turned their attention to physical questions very naturally assumed that the same conception would be fruitful in other directions. Fortunately there was one physical process where these ideas obviously did not apply. Heat was indeed imagined to be a material, and moreover a fluid substance, but hardly molecular, and its transmission in conductors was naturally regarded as a continuous process. To this we owe the work of Fourier, which stands by itself, outside the historical order of development, except in so far as the solution of particular problems involved analytical processes, and led to analytical theorems which had a much wider scope. When the molecular structure of bodies was taken into account, as in the early days of Elasticity, the steps were somewhat vague and uncertain, and I think that the writers themselves must have experienced some relief when they had finally arrived at their differential equations, and felt really at home. It was a great improvement when the consideration of molecular forces could be dispensed with and replaced by Cauchy’s theory of stress. The same tendency to discard unnecessary and unverifiable hypothesis has been exemplified in Electricity, in the transition from Poisson and Ampère to Thomson and Maxwell.
One feature which is met with in our period is the frank appeal to intuition. This is noticeable already in the case of Fourier, as has been already indicated, but it runs through the whole school. Even Cauchy, who was or became something of a purist according to the standards of his day, did not shrink on occasion from handling divergent integrals, but managed always to come right in the end. There is this to be said about mathematical work, in any but quite incompetent hands, that a too careless induction sooner or later betrays itself, and leads to a revision of the whole calculation. The great mathematicians, whatever licence they may have allowed themselves, have always had a sure instinct to save them from logical disaster. The rôle which intuition plays in mathematical discovery has sometimes been slighted or even denied. But was it not Gauss who, questioned as to the progress of a research on which he was engaged, replied that he had arrived at the theorems, and that it only remained to find the proofs? For such things as existence-theorems we must of course not look, at all events in the earlier half of our period. The first instance of the consciousness of such a requirement that I can call to mind occurs in Green, but he at once proceeds to appeal to physical conceptions. He wished to satisfy himself as to the existence of a function satisfying Laplace’s equation, which should vanish over a closed surface, and have a definite singularity at a given internal point. He regards it as sufficient to remark that this is the case of an uninsulated conducting surface under the influence of an internal charge. The same use of physical proofs is to be found in Maxwell, and in an especial degree in the writing of the late Lord Rayleigh. The physical mathematician may reasonably claim a certain licence in this respect. He is often in the case of Gauss; the proposition is certain, but having his own business to attend to, he leaves the rigorous proof to the analyst, who ought indeed to be very grateful to him for the exquisite logical exercise which he has provided.
A further feature in the evolution is the gradual recognition of geometrical or physical meanings in various symbols or groups of symbols which are of constant recurrence. This is specially characteristic of the later stages. To Laplace and his school the potential was simply a convenient mathematical entity; the name with its associations came long afterwards from Green. The equation
lost most of its significance when it was transformed, as was necessary for some purposes, to polar co-ordinates, and the recognition of the general properties of the function was delayed. The equation itself first received an explicit interpretation at the hands of Maxwell, and the same holds with regard to the now familiar conceptions of ‘divergence,’ ‘concentration,’ and so on. And it needs hardly to be said that the notion of an operator, as distinguished from the result, belongs to the later period. The terminology of physical entities or qualities such as ‘isotropy,’ ‘permeability,’ and so on is largely due to Kelvin, with his copious onomastic faculty.