418-421] Potential Energy 367 



421. Equation (353) assumes a special form if the magnetic field is due 

 solely to the presence of a second magnetic particle. Let this be of moment 

 ft, its axis having direction cosines I', ra', ri, and its centre having coordinates 

 x', y', z . Then we have as the value of H, from 410, 



, 3 /1\ , /,, 3 ,3 , 3\ /1\ 

 n = uf - = p. '(l f + m ^ ; + ri ^-, }(- . 

 8s \r) \ dx dy dz J \rj 



Substituting these values for 1 in the formulae just obtained, we have as 

 the mutual potential energy of the two magnets, 



dsds' 



id 3 3 3W 7 , 3 ,3 , 3 N /1\ 

 = LULL l^- + m~-+n~-][l = , + m 5-> + 5-7 ] { - 1 . 



V 3# 3y 3-sy V dx dy dz J \rj 



This is symmetrical with respect to the two magnets, as of course it ought to be it is 

 immaterial whether we bring the first magnet into the field of the second, or the second 

 into the field of the first. 



If we now put 



1 1 



we obtain on differentiation, 



3/l\_ x x _ x ~~ 



M W ~ {(x - X J + (y- yj + (,- /)f ~ ~* 

 so that 



3 2 /IN _ 1. _ 3 (x - xJ 



r j ^3 y,5 ' 



l\ 3(x-x')(y-y f ) 



r) r 5 



Hence we obtain as the value of W, 



W = - (ll f + mm' 4- nn 1 } 

 r 3 



Let us now denote the angle between the axes of the two magnets by e, 

 and the angles between the line joining the two magnets and the axes of the 

 first and second magnets respectively by and #'. Then 

 cos e = IV + mm' + nn', 



cos 6 =- [I (x x'} + m(y y'} + n(z z')}, 



cos & = - [l f (x - x') + m'(y- y') + ri(z- z')}, 



so that W can be expressed in the form 



/ 



W = (cose -3 cos cos 6') (354). 



