The Rev. H. Lloyd on the mutual Action of pormanent Magnets. 175 



4 — cos 2a . , Pp^ ' , • 



' -o^ Its value ;^3 above deduced. We thus obtain 



Whence 



COS^ I s'"^ f I 1 _ 



, ^ — mn ± V'm^ -\- n^ — I ,^ , 



tanf = -,_r . (39) 



in which we have put, for abridgment, 



— _J_ _ ^ _ _J_ _ Cc' 



This solution becomes impossible when m* -f- ra* < 1, or 



(30) 



The formulae (11) (13) suggest of themselves many other cases of easy 

 solution. Thus, if it be assumed that 7 = 0, a = 90, or the line connecting 

 A and B coincident with the magnetic meridian, and the line connecting b and c 

 perpendicular to it, equation (13) gives ^ = 0. Substituting in (11), it be- 

 comes 3 sin 2/3 = 2 Qo', or, since in this case g — , 



^ cos j3 



sin /3 cos* /3 = ^ Q ; 



from which the angle /3 is determined. This disposition of the magnets is 

 represented in Fig. 6. 



The equilibrium is fulfilled in this case independently of the value of P, or 

 of the relative forces of the magnets a and c : the reason of this is evident. 

 On the other hand, the solution requires that Q shall not exceed a small limit ; 

 for the first member of the preceding equation is a maximum, when tan /3 = ^, 

 and substituting, the greatest possible value of Q is *^ = 0.859 • 



Again, if we have cos 27 = ^, /3 = 0, (11) gives f = , as before ; and 



(13) becomes 3 sin 2a 4- 2 ^^^Pf = 0. But /> = - -^^ = -^. — , 



sm a v'3 sm a 



