230.] The methods given by Maxwell for solving problems in Electrostatics by means of Conjugate Functions are somewhat indirect, since there is no rule given for determining the proper transformation for any particular problem. Success in using these methods depends chiefly upon good fortune in guessing the suitable transformation. The use of a general theorem in Transformations given by Schwarz (Ueber einige Abbildungsaufgaben, Crelle 70, pp. 105—120, 1869), and Christoffel (Sul problema delle temperature stazionarie, Annali di Matematica, I. p. 89, 1867), enables us to find by a direct process the proper transformations for electrostatical problems in two dimensions when the lines over which the potential is given are straight. We shall now proceed to the discussion of this method which has been applied to Electrical problems by Kirchhoff (Zur Theorie des Condensators, Gesammelte Abhandlungen, p. 101). and by Potier (Appendix to the French translation of Maxwell's Electricity and Magnetism); it has also been applied to Hydro dynamical problems by Michell (On the Theory of Free Stream Lines, Phil. Trans. 1890, A. p. 389), and Love (Theory of Discontinuous Fluid Motions in two dimensions, Proc. Camb. Phil. Soc. 7, p. 175, 1891).

In Art. 201 of the text there is a description of Perrot's experiments on the electrolysis of steam. As these experiments throw a great deal of light on the way in which electrical discharges pass through gases I have, while this work has been passing through the press, made a series of experiments on the same subject.

The apparatus I used was the same in principle as Perrot's. I made some changes, however, in order to avoid some inconveniences to which it seemed to me Perrot's form was liable. One source of doubt in Perrot's experiments arose from the proximity of the tubes surrounding the electrodes to the surface of the water, and their liability to get damp in consequence. These tubes were narrow, and if they got damp the sparks instead of passing directly through the steam might conceivably have passed from one platinum electrode to the film of moisture on the adjacent tube, then through the steam to the film of moisture on the other tube and thence to the other electrode. If anything of this kind happened it might be urged that since the discharge passed through water in its passage from one terminal to the other, some of the gases collected in the tubes gg (Fig. 84) might have been due to the decomposition of the water and not to that of the steam.

In the twenty years which have elapsed since the first appearance of Maxwell's Treatise on Electricity and Magnetism great progress has been made in these sciences. This progress has been largely—perhaps it would not be too much to say mainly—due to the influence of the views set forth in that Treatise, to the value of which it offers convincing testimony.

In the following work I have endeavoured to give an account of some recent electrical researches, experimental as well as theoretical, in the hope that it may assist students to gain some acquaintance with the recent progress of Electricity and yet retain Maxwell's Treatise as the source from which they learn the great principles of the science. I have adopted exclusively Maxwell's theory, and have not attempted to discuss the consequences which would follow from any other view of electrical action. I have assumed throughout the equations of the Electromagnetic Field given by Maxwell in the ninth chapter of the second volume of his Treatise.

The first chapter of this work contains an account of a method of regarding the Electric Field, which is geometrical and physical rather than analytical. I have been induced to dwell on this because I have found that students, especially those who commence the subject after a long course of mathematical studies, have a great tendency to regard the whole of Maxwell's theory as a matter of the solution of certain differential equations, and to dispense with any attempt to form for themselves a mental picture of the physical processes which accompany the phenomena they are investigating.

1.] The influence which the notation and ideas of the fluid theory of electricity have ever since their introduction exerted over the science of Electricity and Magnetism, is a striking illustration of the benefits conferred upon this science by a concrete representation or ‘construibar vorstellung’ of the symbols, which in the Mathematical Theory of Electricity define the state of the electric field. Indeed the services which the old fluid theory has rendered to Electricity by providing a language in which the facts of the science can be clearly and briefly expressed can hardly be over-rated. A descriptive theory of this kind does more than serve as a vehicle for the clear expression of well-known results, it often renders important services by suggesting the possibility of the existence of new phenomena.

The descriptive hypothesis, that of displacement in a dielectric, used by Maxwell to illustrate his mathematical theory, seems to have been found by many readers neither so simple nor so easy of comprehension as the old fluid theory; indeed this seems to have been one of the chief reasons why his views did not sooner meet with the general acceptance they have since received. As many students find the conception of ‘displacement’ difficult, I venture to give an alternative method of regarding the processes occurring in the electric field, which I have often found useful and which is, from a mathematical point of view, equivalent to Maxwell's Theory.