ON FARADAY'S LINES OF FORCE. 



magnetic matter spread over the outer surface, the density being given by the 

 equation 



p = 37 cos 0. 



Suppose the shell now to be converted into a permanent magnet, so that the 

 distribution of imaginary magnetic matter is invariable, then the external poten- 

 tial due to the shell will be 



/= -/cos0, 



and the internal potential ^,= Ir cos 6. 



Now let us investigate the effect of filling up the shell with some substance 

 of which the resistance is k, the resistance in the external medium being k'. 

 The thickness of the magnetized shell may be neglected. Let the magnetic 

 moment of the permanent magnetism be /a', and that of the imaginary super- 



ficial distribution due to the medium k = Aa*. Then the potentials are 



t 

 external p' = (1+ A) -, cos 0, internal p t = (1+ A] r cos 0. 



The distribution of real magnetism is the same before and after the introduc- 

 tion of the medium k, so that 



< A* ~~" A/ f 



or A = , ,, 1. 



2k + k 



The external effect of the magnetized shell is increased or diminished according 

 as k is greater or less than k'. It is therefore increased by filling up the shell 

 with diamagnetic matter, and diminished by filling it with paramagnetic matter, 

 such as iron. 



VIII. Electro-magnetic spheincal shell, 



Let us take as an example of the magnetic effects of electric currents, 

 an electro-magnet in the form of a thin spherical shell. Let its radius be , 

 and its thickness t, and let its external effect be that of a magnet whose 

 moment is /a*. Both within and without the shell the magnetic effect may be 

 represented by a potential, but within the substance of the shell, where there 



