180 THEORY OF NEWTONIAN FORCES. [PT. I. CH. IV. 



monies of degree n can be expressed in terms of two independent 

 ones. We have found in 44 that the real and imaginary parts 

 of the function (x-\-iy) n are harmonic functions of x, y, being 

 respectively equal to 



p n cos nco and p n sin no). 



Accordingly the general harmonic of degree n is 



(2) V n = p n [A n cos nco + B n sin nco} = p n T n . 



We may call the trigonometric factor T n , which is the value of 

 the harmonic on the circumference of a circle of radius unity, the 

 peripheral harmonic of degree n. 



If a function which is harmonic in a circular area can be 

 developed in an infinite trigonometric series 



W= oo 



(3) V ( x > y) = % {A n cos nco + B n sin nco} = ZT n 



=o o 



on the circumference of the circle of radius R, the solution of 

 Dirichlet's Problem for the interior of the circle is given by the 

 series 



(4) r-T.+%T l +T t +.... 



For every term is harmonic, and therefore the series, if con- 

 vergent, is harmonic. At the circumference p = E, and the series 

 takes the given values of V. The absolute value of every term 

 is less than the absolute value of the corresponding term in the 

 series (3), in virtue of the factor p n /fi n , therefore if the series (3) 

 converges, the series (4) does as well. Since the series fulfils all 

 conditions, by Dirichlet's principle it is the only function satis- 

 fying them. 



We may fulfil the outer problem by means of harmonics of 

 negative degree. Taking n negative, the series 



(5) F=r +|r 1 + ^r 2 +... ' .-' 



is convergent, takes the required values on the circumference, and 

 vanishes at infinity except the constant term. For a ring-shaped 

 area between two concentric circles, we may satisfy the conditions 

 by a series in both positive and negative harmonics, 



00 



(6) F= 2p n {A n cos nco + B n sin nco} 

 o 



00 



H- 2p~ n {A n f cos nco + B n ' sin nco}. 

 i 



