254 Methods for the Solution of Special Problems [OH. vm 



The potential at P f of charges e lt e 2 , ... at the inverse points of Q 1} Q. 2 , ... 

 plus a charge 2e at is 



F-f <7+ 



and this by equation (225) is a constant. This result gives the method of 

 inversion in two dimensions : 



If a surface S is an equipotential under the influence of line-charges 

 e 1} e 2 , ... at Qi, Q 2 ,..., then the surface which is the inverse of S about 

 a line will be an equipotential under the influence of line-charges e 1} e 2 , ... 

 on the lines inverse to Q l} Q 2 , ... together with a charge %e at the line 0. 



Two-dimensional Harmonics. 



303. A solution of Laplace's equation can be obtained which is the 

 analogue in two dimensions of the three-dimensional solution in spherical 

 harmonics. 



In two dimensions we have two coordinates, r, 6, these becoming 

 identical with ordinary two-dimensional polar coordinates. Laplace's equa- 

 tion becomes 



ld_f dV\ 8 2 F _ 

 + ~ ' 



and on assuming the form 



V 



in which R is a function of r only, and a function of only, we obtain the 

 solution in the form 



F= "2 (Ar n + =1) (G cos n<f> + D sin n<t>). 



n =0 \ r / 



Thus the " harmonic-functions " in two dimensions are the familiar sine 

 and cosine functions. The functions which correspond to rational integral 

 harmonics are the functions 



r n sin nO, r n cos nO. 



In x, y coordinates these are obviously rational integral functions of x 

 and y of degree n. 



Corresponding to the theorem of 240, that any function of position 

 on the surface of a sphere can (subject to certain restrictions) be expanded 

 in a series of rational integral harmonics, we have the famous theorem of 

 Fourier, that any function of position on the circumference of a circle can 

 (subject to certain restrictions) be expanded in a series of sines and cosines. 

 In the proof which follows (as also in the proof of 240), no attempt is made 

 at absolute mathematical rigour: the forms of proof given are those which 

 seem best suited to the needs of the student of electrical theory. 



