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 [Pg 41] 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. I have referred mainly to the development of general principles and methods, but that is, of course, not the whole of the story. A complete history would have to treat in some detail the special problems which suggested themselves from time to time. The impulse to general theory [Pg 42] indeed often came about in this way. For instance, the problem of the two electrified spheres gave the impulse to Electrostatics, whilst Chladni’s figures of nodal lines led up by degrees to the theory of Elasticity. It is, moreover, in the special applications that the skill of the analyst is particularly evoked, with results often of great interest and value even from the purely mathematical point of view. We need not go back to the theory of Attractions, or of the Figure of the Earth, which evoked Spherical Harmonics. The Conduction of Heat led incidentally to Bessel Functions, and above all to the theorems specially associated with the name of Fourier, whilst Poisson’s problem of the two electrified spheres is a signal instance of the treatment of a functional equation. To Kelvin we owe the method of electric inversion, including the astonishing solution of the problem of the electrified spherical bowl, which had engaged the attention of Green, and the symmetrical treatment of Spherical Harmonics. To Maxwell are due the singularly beautiful solution of the problem of [Pg 43] current sheets, a new interpretation of Spherical Harmonics, and other interesting results and points of view scattered through his treatise. As an example of a more systematic application of mathematical technique we may refer again to Cauchy’s wave-problem, where the integrals afterwards attributed to Fresnel first make their appearance.