264 Mr. J. Parker on the Theory of Magnetism and 



Let us now imagine a system identical with the given 

 system before the change in civ, and let this particular element 

 be removed to infinity without causing any other change in 

 the system. Then if w be the work so obtained, we have 

 clearly 



8Y = Sw. 



To find Sw, we may take the volume dv of any form we 

 please. Suppose it is a cylinder with its ends perpendicular 

 to the axis, and the axis parallel to 1. Then, by the principles 

 of the potential, if dco be the section of the cylinder and ds 

 its length, we have 



w = ld(o ds—r~ , 

 ds 



or 



IV — 1 — r- dV. 



as 



dV 

 wbere, in finding — , we travel on the line of magnetization. 



Now since the potential V at any point (x, y, z) is a function 

 only of the three coordinates of that point, we obtain, if 

 (a, /3, 7) be the angles the direction of magnetization at the 

 point (x, y, z) makes with the axes, 



dV_dVdx dVdy dV dz 

 ds dx ds dy ds- dz ds 



and therefore 



dV n dV dV 



= COS OL — + COS B— h COS 7 — r- , 



dx dy dz 



jdY .dV ^dV n dV 

 ds dx dy dz 



Thus, since the potential at any point of the element dv, and 



therefore the values of -— , -^— , — — are independent of the 

 dx dy dz 



magnetization of that particular element when it is small 



enough, we obtain 



S^SA^+SB^ + ScfW 



V dx ay dzJ 



If the element dv be to any extent magnetically " rigid, " 

 its magnetization will not be fully able to obey the directing 

 causes, and there will be relations between SA, 5B, and 8G ; 

 but if the element be " perfectly soft," we may consider 

 $A, 8B, 80 independent. In the latter case, if we put SB 

 and SC both zero, equation (3) gives 



