On the Intensity of the Earth's Magnetic Force. 213 



netic forces are mX cos a, mY ; and their moment to turn tlie 

 needle is 



m{ Y cos ij — X cos « sin »j) ; 



in which ij denotes the actual inclination of the needle to the 

 horizon. This moment is opposed by that of the weight. Let 

 this be applied in the manner adopted by Mr. Fox, namely, 

 at the circumference of a light pulley, whose centre is on the 

 axis of the cylindrical axle. Its moment is in this case inde- 

 pendent of the position of the needle, and is equal to the 

 weight W, multiplied by the radius r, of the pulley at whose 

 circumference it is applied. Accordingly, the equation of 

 equilibrium is 



tw(Ycos>j — X sin>jcosa) = Wr. . . . (1.) 



There are two cases which deserve consideration, namely, 

 that in which the plane of motion of the needle coincides with 

 . the magnetic meridian, and that in which it is perpendicular 

 to it. In the former case «=0 ; and substituting for X and Y 

 their values, R cos and R sin (0 being the inclination), the 

 preceding equation becomes 



mRsin (0— >j) = Wr; (2.) 



from which we obtain ?wR, the product of the earth's magnetic 

 force into the moment of free magnetism of the needle, when 

 W and r are known, and the angles and »j given by obser- 

 vation. In the latter case, a = 90°, and (1.) becomes 



mY cos y} = Wr ; (3.) 



which gives the similar product in the case of the vertical 

 component of the force. 



Now let the needle be removed, and applied to deflect 

 another which is substituted in its place; and let the deflect- 

 ing needle be placed so that its axis passes through the centre 

 of the supported needle, and is perpendicular to its axis. Then 

 the moment of its force to turn the needle is mm'U, in which 

 m' is the moment of free magnetism of the second needle, and 

 U a function of D, the distance of the centres of the two 

 needles, of the form 



D 3 \ D 2 + DV* 



The moment of the earth's magnetic force, opposed to this, is 

 of the form already assigned, in which we have only to sub- 

 stitute m' and »/ for m and >j. Hence the equation of equili- 

 brium is 



Ycos >)' — X sin ij'cos «=mU. . . . (4.) 



