158 Mr May all, On Current Sheets, especially [Feb. 26, 



(1) It is first of all necessary to find the equations connecting 

 the vector potential due to a magnetic field of force with the 

 scalar potential. With fixed rectangular axes, these are of 



where F, G, H are the components of the vector potential along 

 the co-ordinate axes, and fl is the scalar potential ; but their 

 form is altered if F, G, H are the components along the normals 

 to the mutually orthogonal systems of surfaces a — const., h = const., 

 c = const, instead of three fixed directions. 



Let OBPG be an element of the surface a = const, bounded 

 by two pairs of surfaces 



b = const., c = const., b + db — const., c + dc = const. ; 



OB being normal to 6 = const., OG to c = const.; and let F, G, H 

 be the components of the vector potential at along the normals 

 to the surfaces a, b, c respectively. 



Then the line integral of the vector potential round OBPG 

 is equal to the magnetic induction through the same area, which 

 gives 



GBdb + ( RGdc + ^ HGdc db) 



-(OBdb + ^GBdbdcj 



dn 



- HGdc = -^ BdbG dc, 

 da 



the orthogonal surfaces being given as before mentioned by the 

 equation 



ds^ = A''da^ + B'dF + G'dc\ 



