DISTRIBUTION OF THE FICTIVE LAYER. 39 1 



418. DISTRIBUTION OF THE FICTIVE LAYER. The fictive layer 

 is not a layer of equilibrium, but we know that its density at each 

 point (39) satisfies the ratio 



in which F w and F' n denote for two infinitely near points on each 

 side of the surface, the first outside and the second inside, the per- 

 pendicular components of the actions exerted by the external masses, 

 and by the layer. The preceding methods give the component F n , 

 but the component F' n is in general unknown ; hence they only 

 enable us to determine the density of the fictive layer in certain 

 special cases. 



It may happen, in fact, that the fictive layer may replace magnetic 

 masses which really exist in the magnet not only for external, but 

 also for internal points ; and this is what takes place in the phe- 

 nomena of magnetic induction, when the coefficient k is constant. 

 There is then a constant ratio /* between the external and internal 

 perpendicular components, and the expression for the density is 



47T fJ. 



In this case the distribution is completely known when we know 

 the external perpendicular component at every point. This is not 

 the case if the coefficient k is variable, and still less if there is rigid 

 magnetism. 



The ordinary methods do not give directly the distribution 

 of the fictive layer in a magnetised bar; it is incorrect, in par- 

 ticular, to consider the abscissa of the centre of gravity of the 

 curve of the perpendicular components as giving the position of the 

 pole. This is readily seen, if we examine the case of a cylinder 

 magnetised uniformly in a direction parallel to the axis. We have 

 seen (373) that its action may be represented by that of two layers, 

 one negative and the other positive, distributed uniformly on each 

 of the bases. It is easy to see that the flow of force for the lateral 

 surface is not zero, although the density is zero. The centre of 

 gravity of the curve representing the flow across the lateral surface is 

 in the interior of the magnet, while this pole is exactly situate on the 

 terminal surface. 



