INDUCED MAGNETISATION 251 



foregoing investigations we have supposed that air is the medium 

 surrounding the body in which magnetisation is induced, and that 

 for air /z = 1 and K = 0. Now we shall consider the case in which 

 the medium has permeability differing from 1. Let us suppose that 

 it has permeability ^ 15 and that the body which it surrounds has 

 permeability /* 2 , and let us find the value of the surface density of 

 magnetism or polarity which we should have to suppose spread 

 over the surface of the body to account for the change of field 

 when the permeability is everywhere reduced to the same value ^ 

 as it has in the medium. This corresponds to the investigations 

 on p. 238, where the imagined surface layer acts as if it were in air 

 on all sides. 



We shall first show that the intensity due to a pole m at 

 distance r in a medium of permeability /^ is no longer r/i/r 2 , but 

 m/fjLjT*. Suppose two poles each m, i.e. each a source of the same 

 total induction, to be placed one in air and the other in 

 medium. At a distance r from the first the induction is 



B = m/r 2 = H. 

 At a distance r from the second the induction is also 



B = wi/r 8 . 

 But if H 1 is the intensity in the second medium 



B . ^H, 

 ' H i = m/W*. 



Then if we apply Gauss's theorem in a system where the permeability 

 is everywhere /u r and regard the field as due to poles acting according 

 to the law jfif* 



where m is the total polarity within the surface S. 



Now suppose that a body of permeability // 2 is placed in a 

 magnetic field and surrounded by a medium of permeability /x r 

 We still assume the principles of continuity of flux of induction 

 and continuity of potential, so that if H x H 2 are the intensities 

 just without and just within the surface, and if they make 1 and $ 2 

 with the normal, 



/^H! cos $!= /u 2 H 2 cos 2 and H 1 sin l = H 2 sin 2 . 



Let Fig. 195 represent the section of a tube of induction passing 

 out of the body and cutting unit area on the surface at AB, and let 

 AC, BD be cross-sections perpendicular to the tube. Apply Gauss's 

 theorem to the surface of which ACBD is a section, and find the 

 value a- of the surface density which would account for the change 

 in intensity at the surface on the supposition that the permeability is 



