70 STATIC ELECTRICITY 



But <r = strain just outside dS 



= intensity 



J_ dV 



~~ 4"7r dn 



where dfo is an element of the normal. 



Then W-JL//vdS. 



But by Green's theorem this is equal to 



L /" /" [\ 



where the integrations are taken throughout the air. 



Now going back to the experimental b.. :mM suppose 



that we are entitled to assume that the total strain throughout a 

 tube of strain is constant. It is easy to think of < utal 



illustrations of this, such as the equal and opposite charges 

 concentric spheres, and the equality always existing between tin- 

 two mutually inducing charges whatever the conductors, tin 

 it is not very easy to arrange any general proof. From tli 

 follows that the total strain through any closed surface containing 

 no electrification is zero. The total normal intensity is therefore 

 also zero, and applying this to the case of the element dxdydz, 

 it can be shown that V*V = 0. 



Now by Thomson's theorem there is one, and only one, value 

 of V satisfying V 2 V = in the air, and fulfilling the conditions 

 given on the conductors, and this value makes the integral which 

 is equal to W a minimum. In other words, it is the value of V 

 which corresponds to a distribution of electrification making the 

 energy a minimum, and therefore it is a distribution in equi- 

 librium. Hence we require to find a function V satisfying the 

 conditions. 



