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



the plane with a force where r is their distance apart. The 



potential due to m is V= m logr and it satisfies the differential 

 equation 



~ 



x df ~ 



Similarly, in the case of any mass distributed in the plane, 

 with surface-density /*, an element dm = pdS produces the po- 

 tential dm log r, and the whole the potential 



V = - II dm log r = - M//, log rdS, 



where r is the distance from the repelling dm at a, b to the 

 repelled point a, y, so that 



r 2 = (x - a) 2 -h (y - b) 2 . 



We may verify by direct differentiation that, at external 

 points, this V satisfies 



W dy* 



8 a? -a) , ,, 



dadb = ~ ~ ~ ' 



This potential is called the . logarithmic potential and is of 

 great importance in the theory of functions of a complex variable. 



90. Green's Theorem for a Plane. In exactly the same 

 manner that we proved Green's Theorem for three dimensions, 

 we may prove it when the integral is the double integral in a 

 plane 



ff (dUdV 



<i) I=\\ V^--^r- 



JJ A \dx fa 



