408-412] 



Magnetic Field of Force 



361 



The analysis here given and the result reached are exactly similar to 

 those already given for an electric doublet in 64. The same result can also 

 be put in a different form. 



Let us put OP ds, and let =- denote differentiation in the direction of 



85 



OP, the axis of the particle. Then equation (336) admits of expression in 

 the form 



(338). 



Let I, ra, n be the direction-cosines of the axis of the particle, then formula 

 (338) can also be written 



where, in differentiation, x, y, z are supposed to be the coordinates of the 

 particle, and not of the point Q. 



411. Resolution of a magnetic particle. Equation (339) shews that the 

 potential of the single particle we have been considering is the same as the 

 potential of three separate particles, of strengths jj,l, fim and /j,n, and axes in 

 the directions Ox, Oy, Oz respectively. Thus a magnetic particle may be 

 resolved into components, and this resolution follows the usual vector law. 



The same result can be seen geometrically. 



Let us start from and move a distance Ids parallel to the axis of x, then 

 a distance mds parallel to the axis of y, and then 

 a distance nds parallel to the axis of z. This 

 series of movements brings us from to P, a 

 distance ds in the direction I, ra, n. Let the 

 path be OqrP in fig. 106. The magnetic particle 

 under consideration has poles m l at and -f m^ 

 at P. Without altering the field we can super- 

 pose two equal and opposite poles + ra x at q, and 

 also two equal and opposite poles + m l at r. 



The six poles now in the field can be taken 

 in three pairs so as to constitute three doublets 

 of strengths m^Oq, m^.qr and rn^rP respec- 

 tively along Oq, qr and rP. These, however, are 

 doublets of strength pi, pm and pn parallel to the coordinate axes. 



FIG. 106. 



Potential of a Magnetised Body. 



412. Let / be the intensity of magnetisation at any point of a mag- 

 netised body, and let I, m, n be the direction-cosines of the direction of 

 magnetisation at this point. 



