[From the Proceedings of the Cambridge Philosophical Society, VoL n.] 



XLV. On the Solution of Electrical Problems by the Transformation 



of Conjugate Functions*. 



THE general problem in electricity is to determine a function which shall 

 have given values at the various surfaces which bound a region of space, and 

 which shall satisfy Laplace's partial differential equation at every point within 

 this region. The solution of this problem, when the conditions are arbitrarily 

 given, is beyond the power of any known method, but it is easy to find any 

 number of functions which satisfy Laplace's equation, and from any one of these 

 we may find the form of a system of conductors for which the function is a 

 solution of the problem. 



The only known method for transforming one electrical problem into another 

 is that of Electric Inversion, invented by Sir William Thomson ; but in problems 

 involving only two dimensions, any problem of which we know the solution 

 may be made to furnish an inexhaustible supply of problems which we can solva 



The condition that two functions a and /8 of x and y may be conjugate is 



a + S^lfi = F (x + v^T y). 



This condition may be expressed in the form of the two equations 



da d/3 da. d/3 



~T~ ~~ ~T~ * " J~ T ~7~ = U. 



dx dy ay dx 



If a denotes the " potential function," $ is the " function of induction." 

 As examples of the method, the theory of Thomson's Guard Ring and that 

 of a wire grating, used as an electric screen, were illustrated by drawings of 

 the lines of force and equipotential surfaces. 



* [The author's treatment of this subject and a full explanation of the examples mentioned in 

 the text will be found in the chapter on Conjugate Functions in his treatise on Electricity and 

 Magnetism.] 



