59 



their values, R cos0 and R sin 9 {6 being the inclination), the 

 preceding equation becomes 



mRs{n{e-r,)=^Wr; (2) 



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

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

 fVand r are known, and the angles 6 and »j given by obser- 

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



mY cos II = 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 Z), the distance of the centres of the two 

 needles, of the form 



U'W'{'^w*h} 



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 r{ for m and r\. Hence the equation of equili- 

 brium is 



y cosrj' - Xsin T)' cos a=mU. (4) 



When the plane of motion of the needle coincides with the 

 magnetic meridian, or a = 0, this becomes 



R^m{Q-ri') = mU\ (5) 



which gives the ratio of the earth's magnetic force to the mag- 

 netic moment of the needle, when U is known, and the angles 

 Q and y\' given by observation. The coefficients p and q, in 

 the value of U, may be obtained (as in the ordinary method) 



