168 NOTES ON MAGNETISM. 



Integrating for /o, we find, for the total moments, 



. 

 (c -07) + 



3MM' 



being for the second magnet what 3/ is for the first. 

 Let L be the resultant of these three moments, then 



Consequently, since a? + /3? + 7*= 1, we shall have 



+ 6 -- {aa (aa + ^ + 07) - 



Now, let ^, ^' be the angles which the axes of the two magnets 

 make respectively with the line joining their centres, and let </> 

 be the angle between the axes themselves. Then 



a = R cos 6 and aa + b/3 + cy = R cos 6', also a = cos $. 

 Consequently 



L* = -{9 cos 2 sin 2 0' + 6 cos (cos & coscjb - cos0) + sin 2 $, 

 or L = 3- {1 + 3 cos 2 9 - (3 cos cos & - cos <) 2 }*. 



Let ^ be the dihedral angle already mentioned, then 



cos <f) = cosO cos & + sin sin 0' cos ^. 

 Consequently the last equation becomes 



L = -^3- 1 1 + 3 cos 2 - (2 cos (9 cos 0' - sin 6 sin ^ cos x) 2 }*, 

 the required expression. 



