386 VIII. NEWTONIAN POTENTIAL FUNCTION. 



2(i/-&) 2 



This potential is called the logarithmic potential and is of great 

 importance in the theory of functions of a complex variable. 



137. 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 



over an area A bounded by any closed contour C. Since we have 

 for a continuous function TF 



53) dxdy =[W, -W, + . + TF 2w - W 2n ^] dy 



= I TFcos (nx) dSj 

 c j 



where n is the inward normal, ds the element of arc of the contour. 



dV 

 Applying this to W=U-^--y we obtain 



54) 



Treating the other term in like manner, we obtain 



C A 



Interchanging U and V we obtain the second form 



ds= I I (V4U U4V)dxdy, 

 c A 



where we write ^y 



'-^ 



