243 
The solution of this problem, when the conditions are arbi- 
trarily 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 Wiliam Thompson; 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 solve. 
The condition that two functions « and 6 of x and y may be 
conjugate is 
a+ J-1B=F(a@+J—1y). 
This condition may be expressed in the form of the two 
equations 
If « 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, 
Professor CAYLEY pointed out a theory of the translation 
of figures, the small parts of which are the same, which Prof. 
Maxwell in his paper appeared to be leading up to. 
Prof. MAXWELL replied that he had prepared a diagram with 
the purpose of illustrating this case of the transformation of 
Conjugate Functions. 
19—2 
