444-446] Vector-Potential 383 



Uniform Magnetic Shell. 



446. Next let us suppose that the lines of force proceed from a uniform 

 magnetic shell, supposed for simplicity to be of unit strength. Let I', m, n' 

 be the direction-cosines of the normal to any element dS' of this shell. 

 Then the element dS' will be a magnetic particle of moment dS' and of 

 direction-cosines I', ra', n'. The element accordingly contributes to F a term 

 which, by equations (377), is seen to be 



m 5-7 n' z 



02 C v 



where x ', y', 2' are the coordinates of the element dS f . Thus the whole value 

 of^is 



This surface integral satisfies the condition of 441, so that it must be 

 possible to transform it into a line integral of the form 



The equations giving f, g, h are 



_8__a/; _a/ 



8^ 8y'" 8 

 Clearly a solution is 



/=!, 0=0, 



so that on substitution the value of JF 7 is 



,. 

 r ds 



Similarly 



Thus the number of tubes of induction crossing the circuit s from 

 magnetic shell of unit strength bounded by the circuit s', is given by 



ds ds ds 



f/7( 

 or 



f\/dx dx r dy dy' dz dz'\ 1 , , , 

 =U-- J - 7 +-/-/ 7 +^- T -, - dsds . 

 }] \ds ds ds ds ds ds J r 



