CHAPTER III - CONJUGATE FUNCTIONS

Summary

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).