46-A Messrs. Watson and Burbury on the Law 



20. It would appear at first sight that we have omitted the 

 energy of the action of each element on itself] which is prima 

 facie infinite. Exactly in the same way, in finding 



Iff 



mm 



: dx ay dz 



\/x 2 + zf + z 2 



as the expression for the whole potential energy of a mass of 

 matter, we appear to obtain the energy only of the mutual 

 actions of each pair of particles, and not that of each particle 

 on itself. But if the matter have at every point finite density, 

 it is easily seen that the above expression gives us the whole 

 potential. 



In like manner, if the current u or idx be conceived as 

 flowing, not through an infinitely thin line, but through a 

 tube of finite section, so that the density, so to speak, of the 

 current is finite at every point, the expression 



$$(Fu + Gv + H.w)ds 



gives us the whole electrokinetic energy of all the currents in 

 the field. 



F, G, H will now be called the components of vector poten- 

 tial. 



21. We have hitherto called the above expression 



p JJJ (Fu + Gv + Kiv) dx dy dz 



the potential energy of the system ; and we have seen that it 

 is deduced from our proposed law by methods applicable to 

 potential of mass. We now assume, with Maxwell, that this 

 energy, consisting, as he says it does, of " something in motion 

 and not a mere arrangement," may be treated as kinetic energy 

 for the purpose of applying to it the equation of Lagrange. 

 By this means, as Maxwell has shown, the phenomena of induc- 

 tion are capable of explanation. 



22. It is now to be observed that, as above mentioned, and 

 as Maxwell points out, the vector potential of an elementary 

 current in any direction, as x, stands in the same relation to 

 the current as the potential of a mass of matter situated at the 

 middle point of the element stands to that matter. It follows 

 that the vector potential has all the properties of the potential 

 of mass. Hence can be most easily deduced many most im- 

 portant theorems in electrodynamics. 



For instance F, the vector potential of currents parallel to x, 

 must satisfy the equation 



4tt^ + V 2 F = 



(where u is the current in x), corresponding to Poisson's equa- 



tion 47rp + V 2 F = 0, 



