256 Methods for the Solution of Special Problems [CH. vm 



and on passing to the limit and putting a =/, this becomes 



I /-0-27T 1 oo r8=2n 



F=^ Fd9 + -^ Fcosn(6-a)d0 ......... (227), 



Z7Tj0 = o 7T ! J e = 



expressing F as a series of sines and cosines of multiples of a. 

 We can put this result in the form 



00 



F = F 4 2 (a n cos nx + b n sin no), 

 i 



1 f 27r 

 where a = I F cos 



1 f 27 

 a n = I 



7T Jo 



and 



O 



so that ^ is the mean value of F. 



If F has a discontinuity at any point = fi of the circle, and if F lf F 2 are 

 the values of F at the discontinuity, then obviously at the point 6 {3 on the 

 circle, equation (226) becomes 



so that the value of the series (227) at a discontinuity is the arithmetic mean 

 of the two values of F at the discontinuity (cf. 256). 



305. We could go on to develop the theory of ellipsoidal harmonics etc. 

 in two dimensions, but all such theories are simply particular cases of a very 

 general theory which will now be explained. 



CONJUGATE FUNCTIONS. 

 General Theory. 



306. In two-dimensional problems, the equation to be satisfied by the 

 potential is 



(228); 



and this has a general solution in finite terms, namely 



-iy) ..................... (229), 



where / and F are arbitrary functions, in which the coefficients may of course 

 involve the imaginary i. 



For V to be wholly real, F must be the function obtained from / on 

 changing i into i. Let/*(# + iy) be equal to u + iv where u and v are real, 



