464-4(37] Mathematical Theory 401 



MATHEMATICAL THEORY. 



466. If ft is the magnetic potential, supposed to be denned at points 

 inside magnetic matter by equation (348), we have, as in equations (341) 



(cf. 430), a= - g- etc., so that 



an 

 a = -Tx> 



an 



The quantities a, ft, c, as we have seen ( 434), satisfy 



GCL GO v/C s\ /c\f\c\\ 



.f. 1 = (393) 



at every point, and 



.(394), 



where the integration is taken over any closed surface. In terms of the 

 potential, equation (393) becomes 



a / an\ a / an\ a / an\ Q , 



^~l/ A "5~M"^~(A t 3~)+5~' (t JL ~5~ = ^ V* y& )> 



ox \ oxj oy\ oy J oz \ oz / 

 while equation (394) becomes 



f(a^dS = 0.. ...(396). 



J j on 



If fji is constant throughout any volume, equation (395) becomes 



Thus inside a mass of homogeneous non -magnetised matter, the magnetic 

 potential satisfies Laplace's Equation. 



467. At a surface at which the value of /it changes abruptly we may 

 take a closed surface formed of two areas fitting closely about an element dS 

 of the boundary, these two areas being on opposite sides of the boundary. 

 On applying equation (396), we obtain 



where fr, ^ are the permeabilities on the two sides, and , ^ denote 



(jV\ QV<i 



differentiations with respect to normals to the surface drawn into the two 

 media respectively. 



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