388 VIII. NEWTONIAN POTENTIAL FUNCTION. 



139. Green's Formula for Logarithmic Potential. Apply- 

 ing Green's Theorem 56) to the functions logr and any harmonic 

 function V, supposing the pole of P to be within the contour, and 

 extending the integral to the area within the contour and without a 

 circle K of radius about the pole, 



c 

 The third term is 



(since V is harmonic in K) and the fourth, 



_ ( Y ^ds = - f^rd^ = - Cvd, 



J dn J r J 



K K K 



which, when we make e decrease indefinitely, becomes 



Accordingly we obtain the equation 

 63) . F, 



which is the analogue of equation 6), 128. In a similar way we 

 may find for nearly every theorem on the Newtonian Potential a 

 corresponding theorem for the Logarithmic Potential. A comparison 

 of the corresponding theorems will be found in C. Neumann's work, 

 Untersuchungen uber das logarithmische und das Newtonsche Potential. 1 ) 

 The Kelyin-Dirichlet Problem and Principle may be stated and 

 demonstrated for the logarithmic potential precisely as in 132. 



14O. Dirichlet's Problem for a Circle. Trigonometric 

 Series. We shall call a homogeneous harmonic function of order n 

 of the coordinates x, y of a point in a plane a Circular Harmonic, 

 since it is equal to $ n multiplied by a homogeneous function of 

 cos G) and sin 09, and consequently on the circumference of a circle 

 about the origin is simply a trigonometric function of the angular 

 coordinate ra. Any homogeneous function V n of degree n satisfies 

 the differential equation 



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

 tiates; Picard, Traite d } Analyse, torn. II; Poincare, Theorie du Potentiel Newtomen. 



