The next paper which Thomson published, after the two of which a sketch has been given above, was entitled "The Uniform Motion of Heat in Homogeneous Solid Bodies, and its Connection with the Mathematical Theory of Electricity." It is dated "Lamlash, August 1841," so that it followed the first two at an interval of only four months. It appeared in the _Cambridge Mathematical Journal_ in February 1842, and is republished in the "Reprint of Papers on Electrostatics and Magnetism." It will always be a noteworthy paper in the history of physical mathematics. For although, for the most part, only known theorems regarding the conduction of heat were discussed, an analogy was pointed out between the distribution of lines of flow and surfaces of equal temperature in a solid and unequally heated body, with sources of heat in its interior, and the arrangement of lines of forces and equipotential surfaces in an insulating medium surrounding electrified bodies, which correspond to the sources of heat in the thermal case. The distribution of lines of force in a space filled with insulating media of different inductive qualities was shown to be precisely analogous to that of lines of flow of heat in a corresponding arrangement of media of different heat-conducting powers. So the whole analysis and system of solutions in the thermal case could be at once transferred to the electrical one. The idea of the "conduction of lines of force," as Faraday first and Thomson afterwards called it, was further developed in subsequent papers, and threw light on the whole subject of electrostatic force in the "field" surrounding an electric distribution. Moreover, it made the subject definite and quantitative, and not only gave a guide to the interpretation of unexplained facts, but opened a way to new theorems and to further investigation. This paper contains the extremely important theorem of the equivalence, so far as external field is concerned, of any distribution of electricity and a certain definite distribution, over any equipotential surface, of a quantity equal to that contained within the surface. But this general theorem and others contained in the paper had been anticipated in Green's "Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism," in memoirs by Chasles in Liouville's Journal (vols. iii and v), and in the celebrated memoir by Gauss "On General Theorems relating to Attractive and Repulsive Forces varying inversely as the Square of the Distance," published in German in Leipzig in 1840, and in English in Taylor's _Scientific Memoirs_ in 1842. These anticipations are again referred to below. CHAPTER III UNIVERSITY OF CAMBRIDGE. SCIENTIFIC WORK AS UNDERGRADUATE