TO THE THEORY OF MAGNETISM. 99 



The value of the total potential function at any point p within 

 the shell, whose polar co-ordinates are r, 0, VT, is 



In a similar way, the value of the same function at a point p' 

 exterior to the shell, all the inducing bodies being within it, is 

 found to be 



r, 6 and OT in this expression representing the polar co-ordinates 

 of/. 



To give a very simple example of the use of the first of 

 these formulas, suppose it were required to determine the total 

 action exerted in the interior of a hollow spherical shell, by the 

 magnetic influence of the earth ; then making the axis of x to 

 coincide with the direction of the dipping needle, and designating 

 by f, the constant force tending to impel a particle of positive 

 fluid in the direction of x positive, the potential function V, due 

 to the exterior bodies, will here become 



V= -/. x = -/cos O.r = U ( *.r. 



The finite integrals expressing the value of V reduce themselves 

 therefore, in this case, to a single term, in which i=l, and the 

 corresponding value of D being 



the total potential function within the shell is 



1 



-(!-,) 



We therefore see that the effect produced by the intervening 

 shell, is to reduce the directive force which would act on a very 

 small magnetic needle, 



from /, to 1+ f- 



72 



