482 PARTICULAR CASES. 



forming this shell a negative corona has been left corresponding to 

 the difference 2? - J5f' of the two surfaces, and the sum of the external 

 actions of these corona is equal and of opposite sign to that of the 

 superficial currents. If d^> and dS are the two corresponding 

 elements of this corona, and of the surface of the magnet deter- 

 mined by a tube of force, we have 



which gives 



<r'd2= dS. 



4 7T 



The quantity of magnetism contained in the element d^, is 

 therefore the same as that which the element dS would contain 



K 



with a density equal to --- - . 



4?r 



It follows from this that the external action of all the surface 

 currents defined by the preceding shells, is equivalent to that of a 



Fl 



layer of total mass zero, the density of which at every point is H -- . 



4?r 



By continuing the equipotential surfaces, which correspond to the 

 external potential V, by continuous surfaces, we shall see in like 



F n 



manner that a layer whose density at each point is + - , may be 



47T 



replaced by a system of surface currents. 



The combination of the two systems of currents is a new system 

 of surface currents having the same external action as a layer whose 



density is (F n + F' n ), that is to say, the density a- of the magnetic 

 4?r 



layer, which is equivalent to the proposed magnet. 



The external action of any magnet may accordingly be replaced 

 by that of a system of surface currents ; but this equivalence does 

 not hold for points in the interior. 



It is possible, however, to obtain an exact representation of all 

 the magnetic effects, both on the inside and on the outside, if we 

 replace the magnetic filaments into which any given magnet may be 

 decomposed (313), by the corresponding electromagnetic solenoids. 



Ampere assumes that these elementary solenoids are formed of 

 real currents circulating in the interior of molecules ; these currents 

 never pass from one molecule to another, and they exist before 

 magnetisation, the only effect of which is to direct them. 



