TO THE THEORY OF MAGNETISM. 97 



Equating the coefficients of like powers of the variable r, we 

 have generally, whatever i may be, 



neglecting the constant on the right side of the equation in r as 

 superfluous, since it may always be made to enter into $ (0) . If 

 now, for abridgment, we make 



we shall obtain by elimination 



_ tt 



4?r Z> 4-7T 



A(i) = _M fTfr(2t'+l)a aHa 3 



47T D 47T ' 



These values substituted in the expression 



give the general value of <f> in a series of the powers of r, when 

 the potential function due to the bodies inducing a magnetic 

 state in the shell is known, and thence we may determine the 

 value of the potential function ty arising from the shell itself, 

 for any point whatever, either within or without it. 



When all the bodies are situate in the space exterior to the 

 shell, we may obtain the total actions exerted on a magnetic 

 particle in its exterior, by the following simple method, appli- 

 cable to hollow shells of any shape and thickness. 



.The equation (c) Art. 15 becomes, by neglecting the super- 

 fluous constant, 



If now (<) represent the value of the potential function, cor- 

 responding to < the value of <j> at the inner surface of the shell, 

 each of the functions (<), -^ and F, will satisfy the equations 



7 



