190-192] Earnshaw's Theorem 165 



By Green's Theorem, the last line 



j j 1 

 dxdydz 



the summation of surface integrals being over the surfaces of all the 

 conductors, 



dxdydz 



S W = - -- If I &SK dxdydz. 



by equation (106). Thus equation (107) becomes 



81^= 1 1 \I&&K dxdydz 28 TF", 

 so that 



SW = 



STT. 



Thus 8W is necessarily negative if $K is positive, proving the theorem. ' 



It is worth noticing that, on the molecular theory of dielectrics, the increase in the 

 inductive capacity of the dielectric at any point will be most readily accomplished by 

 introducing new molecules. If, as in Chap, v, these molecules are regarded as uncharged 

 conductors, the theorem just proved becomes identical with that of 190. 



EARNSHAW'S THEOREM. 



192. THEOREM. A charged body placed in an electric field of force 

 cannot rest in stable equilibrium under the influence of the electric forces 

 alone. 



Let us suppose the charged body A to be in any position, in the field 

 of force produced by other bodies B, B', .... First suppose all the elec- 

 tricity on A, B, B', ... to be fixed in position on these conductors. Let 

 V denote the potential, at any point of the field, of the electricity on 

 B, B, .... Let x, y, z be the coordinates of any definite point in A, say its 

 centre of gravity, and let x + a, y+ b, z + c be the coordinates of any other 

 point. The potential energy of any element of charge e at x + a, y + b, z + c 

 is eV, where V is evaluated at x + a, y + b, z+c. Denoting eV by w, we 

 clearly have 



d~w d 2 w d' 2 w 

 1-' - H 0, 



since F is a solution of Laplace's equation. 



