178 THEOKY OF NEWTONIAN FORCES. [PT. I. CH. IV. 



Applying this to the harmonic function logr, where P, the 

 fixed pole from which r is measured, is outside the contour, 



f 81ogr 7 f 19r 7 f cos(rn) 7 



(7) 5 (fe = -^-ds= ^ } ds = 0. 

 Jc J<?rara Jo r 



If the pole P is within the contour, we draw a circle K of any 

 radius about the pole, and apply the theorem to the area outside 

 of this circle and within the contour, obtaining 



f aiogr f coa(m) , r 2 " 



(8) _ & ds = -I - - / ds = - dO = 2-7T. 

 J c on ] K r J 



These two results are Gauss's theorem for two dimensions. 

 They may of course be deduced geometrically. We may now 

 deduce Poisson's equation for the logarithmic potential as in 77 

 for the Newtonian Potential. The logarithmic potential due to 

 a mass dm being dmlogr gives .rise to the flux of force 2-TrcZm 

 outward through any closed contour surrounding it, and a total 

 mass m causes the flux 



2-Trm = 2?? 1 1 pdxdy. 

 Put in terms of the potential this is 



(9) I d ^ds = -j( &Vdxdy = 27rfj pdxdy, 



and since this is true for any area of the plane, we must have 



(10) AF=-27r/i. 



This is Poisson's equation for the logarithmic potential. 



92. Green's formula for Logarithmic Potential. Apply- 

 ing Green's Theorem (5) to the functions log r and any harmonic 

 function F, 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 e about the pole, 



The third term is 



