1 68 PARTICULAR CASES OF EQUILIBRIUM. 



field where the force is < , to another where the force is </>, the 

 increase of energy is 



If the sphere is brought from an infinite distance, we shall have 



The sphere left to itself tends of course to expend energy, and 

 therefore, by equation (33), to move in a direction in which the value 

 of <f> 2 increases most rapidly. It tends then to move to points in 

 which the force is a maximum in absolute value. 



179. Let n be the direction in which < 2 varies most rapidly ; the 

 expression for the force acting on the sphere is 



(34) F- 



U'fi x un 



and its components parallel to the co-ordinates are 



(35) 



OZ 



180. In a variable electrical field, the force cannot be a maximum 

 at any point situate outside the acting masses. 



This theorem follows directly from the preceding demonstration. 



We have seen in fact from Earnshaw's theorem (63), that an 

 electrified body cannot be in stable equilibrium in a variable field. 

 Since an infinitely small sphere can only be in stable equilibrium at 

 points where the value of < 2 is an absolute maximum, that is to say, 

 where the value of < is a maximum in absolute value, it follows that 

 this circumstance cannot present itself for any point outside the 

 acting masses. 



