98 APPLICATION OF THE PRELIMINARY RESULTS 



= (<), Q = &/r and = SF, and moreover, have no singular 

 values in the space within the shell; the same may therefore 

 be said of the function 



and as this function is equal to zero at the inner surface, it 

 follows (Art. 5) that it is so for any point p of the interior 

 space. Hence 



But ijr + V is the value of the total potential function at the 

 point p, arising from the exterior bodies and shell itself: this 

 function will therefore be expressed by 



In precisely the same way, the value of the total potential func- 

 tion at any point p' 9 exterior to the shell, when the inducing 

 bodies are all within it, is shown to be 



being the potential function corresponding to the value of </> 

 at the exterior surface of the shell. Having thus the total po- 

 tential functions, the total action exerted on a magnetic particle 

 in any direction, is immediately given by differentiation. 



To apply this general solution to our spherical shell, the 

 inducing bodies being all exterior to it, we must first determine 

 <>, the value of < at its inner surface, making = %U} i} r~ i ~ l since 

 there are no interior bodies, and thence deduce the value of (<). 

 Substituting for < (i) and <, (i) their values before given, making 



= and r = a, we obtain 



and the corresponding value of (</>) is (Mec. CeL Liv, 3) 



