AMPERE'S THEORY OF MAGNETISM. 481 



The internal field would also be uniform in the case of an 

 ellipsoid (356). 



From this we arrive at a new way of regarding terrestrial 

 magnetism ; the magnetic action of the earth is equivalent to that 

 of a series of circular currents situate in equidistant planes perpen- 

 dicular to the magnetic axis, these currents circulating from east 

 to west. 



498. AMPERE'S THEORY OF MAGNETISM. We see that it is 

 possible, by means of currents situate in parallel planes, to realise 

 a system equivalent to a uniform magnet, which has the same 

 external surface ; the two systems are equivalent for all external 

 points, and produce the same induction in the interior. 



Any given magnet may, in like manner, be replaced by a system 

 of superficial currents, in so far, at least, as the external action is 

 concerned. 



This action, in fact (315), is equivalent to that of a layer of 

 total mass null, distributed on the surface. If o- be the density 

 of the layer at a point, F H and ' n the perpendicular components, 

 measured from the surface, of the actions which it exerts outwards 

 as well as inwards, we have (38) 



Let us consider the internal potential V of the layer, and the 

 equipotential surfaces to which the force F' is perpendicular, and 

 suppose that on each of these surfaces we place equal and opposite 

 magnetic layers, the density of which, at each point, is determined 

 by the condition 



The external action of this system of surfaces is null. We 

 observe now that the product cr'dri = -- dV is constant between 



4 7T 



two equipotential surfaces. If then we connect the negative layer 

 of the surface Jjf, where the potential is V, with the positive layer of 

 the following surface ^ 5 at the potential V + dV, we form a shell, the 

 magnetic power of which, o-'dn', is constant. A current, of the same 

 strength, which followed on the surface, the curve formed by the in- 

 tersection of the shell, would have the same action on the outside ; 

 we could proceed in the same way with all other shells. But in 



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