MEAN FIELD OF A ROTATING FRAME. 523 



The differential equation then becomes 



TD 



(A cos x + B sin x) - L (A sin x - B cos x) 

 = HS [cos x + k cos (x - a)] . 



As this equation should be satisfied, whatever be x t it follows 

 that 



RA 

 - + LB = HS(i+cosa), 



TD T> 



-- LA=.HSsina, 



(0 



and therefore 



R (l + k cos a) - La>/ sin a 



R/sin a + Lw(i +/cosa) 



The couple produced by the induced current on the needle is 

 the mean of the actions IGM cos (x - a), relative to the different 

 positions which the frame occupies during a half-turn that is to say, 



GM C* -GM 



[A cos x + B sin x~\ cos (x - a)dx = (A cos a + B sin a) . 



7T J Q 2 



As the needle is at the same time under the action of the field, 

 and, if need be, allowance can be made for the torsion of the wire, 

 the condition of equilibrium is 



G (A cos a + B sin a) = 2H sin a . 

 Replacing the constants A and B by their values, we get 



^ + cosa GSa> 



: + L(0- 



2 sin a 2 



