91 93] NEWTONIAN POTENTIAL FUNCTION. 179 



(since V is harmonic in K} and the fourth 



K n K r 



which, when we make e decrease indefinitely, becomes 



Accordingly we obtain the equation 



1 T /, 8F Tr aiogr 

 (12) F P =r- logr^ -- V 



ZTT J c\ dn on 



which is the analogue of equation (6), 83. In a similar way 

 we may find for nearly every theorem on the Newtonian Potential 

 a corresponding theorem for the Logarithmic Potential. A com- 

 parison of the corresponding theorems will be found in C. Neu- 

 mann's work, Untersuchungen uber das logarithmische und das 

 Newton sche Potential*. 



The Kelvin-Dirichlet Problem and Principle may be stated 

 and demonstrated for the logarithmic potential precisely as in 

 86. 



93. Dirichlet's Problem for a Circle. Trigonometric 

 Series. We shall call a homogeneous harmonic function of 

 order n of the coordinates #, y of a point in a plane a Circular 

 Harmonic, since it is equal to p n multiplied by a homogeneous 

 function of cos o> and sin co, and consequently on the circum- 

 ference of a circle about the origin is simply a trigonometric 

 function of the angular coordinate o>. Any homogeneous function 

 V n of degree n satisfies the differential equation 



w n zv n 



(i) x^ + y w = nV n , 



so that a circular harmonic is a solution of this and Laplace's 

 Equation simultaneously. The homogeneous function of degree n 



a n x n + a n ^x n -* y + ...... a^y"- 1 + a y n 



contains n + 1 terms, the sum of its second derivatives is a homo- 

 geneous function of degree n 2 containing n 1 terms, and if this 

 is to vanish identically each of its n 1 coefficients must vanish, 

 consequently there are n 1 relations between the n + 1 co- 

 efficients of V n , or only two are arbitrary. Accordingly all har- 



* See also Harnack, Die Grundlagen der Theorie des logarithmischen Poten- 

 tiates ; Picard, Traite d' Analyse, torn. n. 



122 



