In the commencement of 1845 Thomson told Liouville of the method of _Electric Images_ which he had discovered for the solution of problems of electric distribution. On October 8, 1845, after his return to Cambridge, he wrote to Liouville a short account of the results of the method in a number of different cases, and in two letters written on June 26 and September 16 of the following year, he stated some further results, including the solution of the problem of the distribution upon a spherical bowl (a segment of a spherical conducting shell made by a plane section) insulated and electrified. This last very remarkable result was given without proof, and remained unproved until Thomson published his demonstration twenty-three years later in the _Philosophical Magazine_.[8] This had been preceded by a series of papers in March, May, and November 1848, November 1849, and February 1850, in the _Cambridge and Dublin Mathematical Journal_, on various parts of the mathematical theory of electricity in equilibrium,[9] in which the theory of images is dealt with. The letters to Liouville promptly appeared in the Journal, and the veteran analyst wrote a long Note on their subject, which concludes as follows: "Mon but sera rempli, je le répéte, s'ils [ces développements] peuvent aider à bien faire comprendre la haute importance du travail de ce jeune géomètre, et si M. Thomson lui-même veut bien y voir une preuve nouvelle de l'amitié que je lui porte et de l'estime qui j'ai pour son talent." The method of images may be regarded as a development in a particular direction of the paper "On the Uniform Motion of Heat" already referred to, and, taken along with this latter paper, forms the most striking indication afforded by the whole range of Thomson's earlier work of the strength and originality of his mathematical genius. Accordingly a chapter is here devoted to a more complete explanation of the first paper and the developments which flowed from it. The general reader may pass over the chapter, and return to it from time to time as he finds opportunity, until it is completely understood. CHAPTER IV THE MATHEMATICAL THEORY OF ELECTRICITY IN EQUILIBRIUM. ELECTRIC IMAGES. ELECTRIC INVERSION In describing Thomson's early electrical researches we shall not enter into detailed calculations, but merely explain the methods employed. The meaning of certain technical terms may be recalled in the first place.