GREEN'S THEOREM. 



sections of the cone, and r and r' their distances from the point O. 

 We have then 



/ r i _/>* = , 



and multiplying these equations by each other, we get 



but in virtue of the theorem of the flow of force we have also 



and, therefore, 



f n dS=f n dS' 



If we agree to consider as positive, the perpendicular components 



Fig. 5- 



directed towards the exterior of the surface, and as negative those 

 directed towards the interior, f n and f' n are of opposite signs, which 

 gives 



If the surface, while still continuous, had concave portions, and 

 if the cone in question da) cut it in more than two points, it would 

 meet it an even number of times ; the product f n dS would have the 

 same numerical value for each of the intercepted elements, but these 

 products would have to be taken alternately of opposite signs, and 

 the algebraical sum would still be zero. We have, then, for any 

 closed surface external to the acting mass m, the equation 



