412-414] Magnetic Field of Force 363 



Replacing differentiation with respect to x, y, z by differentiation with 

 respect to x, y ', z in this way, we find that equation (342) assumes the form 



fln=- 



ft J) 2 



the quantities A, B, C and the operators ~ , , = , , ^ , being taken outside the 

 sign of integration, since they are not affected by changes in x, y, z. 



If V denote the potential of a uniform distribution of electricity of volume 

 density unity throughout the region occupied by the magnet, we have 



........................ (344), 



so that equation (343) becomes 



O A 9 ^ ftdVQ ridVq 



^ l( *-- A - B ~~ ~ 



or UQ: 



where X, Y, Z are the components of electric intensity at Q produced by 

 this distribution. 



Or again if , denotes differentiation with respect to the coordinates of Q 



vS 



in a direction parallel to that of the magnetisation of the body, namely that 

 of direction-cosines Z, m, n, equation (345) becomes 



ds' 



414. Yet another expression for the potential of a uniformly magnetised 

 body is obtained on transforming equation (342) by Green's Theorem. If 

 V, m', ri are the direction-cosines of the outward-drawn normal to the magnet 

 at any element dS of its surface, the equation obtained after transformation is 



Q = f[(Al' + Em + Cri) * dS. 



By equations (341), 



Al' + Em' +Cn'=I (IP + mm' + nri) 

 = / cos0, 



where 6 is the angle between the direction of magnetisation and the outward 

 normal to the element dS of surface. The equation now becomes 



' Ic -^dS (347), 



T 



shewing that the potential at any external point is the same as that of a 

 surface distribution of magnetic poles of density / cos per unit area, spread 

 over the surface of the magnet. 



