312 SURFACE OF EQIILIKKIUM. 



The constant being th :,incl, we have t'..r all values of 

 r greater than r., 



dV C 



and the attraction of the cylinder will be 



C 



\Ve shall now give some propositions extracted from an 

 article by Professor Stokes, in tin- fourth volume of the Cam- 

 bridge and Dublin Mathematical Journal, to which we have- 

 been already indebted in Art. 236. 



242. A surface of </*' W>r turn is one on which a 

 would rest in equilibrium if acted on by the forces of the 

 system, the surface being supposed fixed. 



If V be the potential of an attracting body on a particle, 

 thru V constant, is the equation to a surface of equilibrium 

 with respect to the attraction of the body. For we have 



shewn in Art 235, that -7- is equal to the attraction resolved 



along the tangent to a curve drawn through the atn. 

 particle, but if this curve be on the surface F= constant, 



then -y- =0; that is, there is no force acting on P in tin- 



direction of any tangent to the surface V= constant. 1 1 



it ' /' be, placed on the surface, it will remain in equilibrium. 



(Art. 33.) 



Lines of force are curves traced so that the tangent at 

 any point is the direction of the resultant force at that point. 

 Hence the lines of force are perpendicular to the surfaces of 

 equilibrium. 



. If S be any closed surface to which all tlie attra 

 mass is external, dS an element of S, and dn an element of the 

 normal drawn outwards at dS t then 



the integral being taken throughout the whole surface S. 



