OP ARTS AND SCIENCES. 359 



If now one of the coils is free to move, so that there will be a 

 variation in 6, the angle between the axes of the two coils, it is 

 evident that the moment of the force tending to increase 6 or i^ is 

 „ dM 



^ = -^/,/sin.-6'/,,'sin.^-^-f;_^/,>in.^^ 



F = - smd(G>ff> + G,'ff,' ^^ + &c.). 



Here 6 is the angle between the axes of the coil ; if we call 6 the 

 angle between the planes of the coils, we have 



i^ = cos ^ (G.'ff,' + G,'g,' '^£^ + &c.). [XXX.] 



In all these equations G' is dependent upon the larger coil, and g' 

 upon the smaller This same expression holds good if the smaller 

 coil is replaced by a magnet, provided suitable values be given to </'. 

 Let the magnet be long and uniformly magnetized. Let 2 ^ be the 

 length ; then 



g, = 2l; 5-2 = 0; cf, = —GP. 



Take the case of a single coil galvanometer with a small magnetic 

 needle ; when the needle is at rest, the two couple are 



FCm — JIMsmB; 

 substituting i^'from equation [XXX.], we have 



Cm cos d ((7/y/ + ajff.J '^ ^J^^^ + &c.) = HMsind; 



or C= tan^; 



m(G,V + G'/5r,'li^ + &c.) 



