89 91] NEWTONIAN POTENTIAL FUNCTION. 



177 



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

 for a continuous function W 



(2) 



dy 



= / W cos (nx) ds, 



Jo 



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

 contour. 



9F 

 Applying this to W = U , we obtain 



(3) 



dap 



Treating the other term in like manner, we obtain 



Interchanging U and V we obtain the second form 

 (5) 

 where we write 



AF^: 



dx 2 y 



91. Application to Logarithmic Potential. If in the 



second form above we put 7=1, we obtain 





which is the divergence theorem in two dimensions. If the 

 function V is harmonic everywhere within the contour, we have 



f 5T"-- 

 Jo on 



FIG. 44. 



W. E. 



12 



