188 NOTES ON MAGNETISM. 



The dihedral angle % is then zero, and consequently the 

 equation 



L = -=- (1 + 3 cos 2 - (2 cos cos & - sin sin & cos % ) 2 }* 

 becomes 

 Z= ~ (1 + 3 cos 2 (9 -4 cos 2 (1 -sin 2 0') 



-1- 4 sin 0' cos sin 6 cos 0' sin 2 sin 2 0']* . 

 The quantity between the brackets is equal to 



4 sin 2 & cos 2 6 + 4 sin & cos 0. sin cos 0' + sin 2 cos 2 0'. 

 Consequently 



Z = -=- (sin cos 0' + 2 sin 0' cos 0). 



This result may be readily established by an independent pro- 

 cess, which the reader will find no difficulty in supplying. The 

 last result may be put in the following form : 



Professor Lloyd, in the 19 th volume of the Memoirs of the Eoyal 

 Irish Academy, has investigated this case of the mutual action 

 of two magnets. His result is (mutatis mutandis) 



L = ~ sin e + ~ 3 sin d ~ e ' 



This differs from the last written result, merely because, in the 

 Professor's analysis, and 0' are measured in opposite directions. 

 If we replace in Prof. Lloyd's result by 2?r 0, it becomes 



L = " f3 sin (e + ff] ~ sin (e ~ ff]} ' 



as before. 



The general formula affords a simple solution of the follow- 

 ing problem. The position of a magnet, and that of the centre 

 of a needle being given, to place the needle in the position in 

 which the moment of rotation due to the action of the magnet is 

 a maximum. 



By the formula established in the last number of the Journal, 

 we have 



L = ~ (1 + 3 cos 2 - (3 cos cos 0' - cos 



