INTENSITY OF THE EAETH's MAGNETIC FORCE. 247 



respect to the suspended one, the latter is deflected from the 

 meridian, and the amount of this deflection serves to determine 

 the ratio of the deflecting force to the earth's force. The position 

 chosen by Gauss for the deflecting magnet is that in which its 

 axis is in the right line passing through the centre of the 

 suspended magnet, and perpendicular to the magnetic meridian, 

 in which case the tangent of the angle of deflection is equal to the 

 ratio of the two forces. From this ratio it remains to deduce that 

 of the magnetic moment of the deflecting bar to the earth's force. 



The difficulty of this process arises from the form of the 

 expression of the force of the deflecting bar. This force being 

 expressed by a series descending according to the negative odd 

 powers of the distance, with unknown coefficients, it is evident 

 that observation must furnish as many equations of condition, 

 corresponding to different distances, as there are terms of sensible 

 magnitude in the series ; and from these equations the unknown 

 quantities are to be deduced by elimination. Now, the greater 

 the number of unknown quantities thus eliminated, the greater 

 will be the influence of the errors of observation on the final 

 result ; and if, on the other hand, the distance between the 

 magnets be taken so great, that all the terms of the series after 

 the first may be insensible, the angle of deflection becomes very 

 small, and the errors in its observed value bear a large proportion 

 to the whole. 



It fortunately happens, that at moderate distances (distances 

 not less than four times the length of the magnets) all the terms 

 beyond the second may be neglected. The expression for the 

 tangent of the angle of deflection is thus reduced to two terms, 

 one of which contains the inverse cube of the distance, and the 

 other the inverse fifth power ; that is, if u denote the angle of 

 deflection, and D the distance, 



in which Q and # are unknown coefficients, the former of which 

 is double of the ratio sought. Accordingly, the method recom- 

 mended by Gauss consists in observing the angles of deflection, 

 u and u', at two different distances, D and IX, and inferring the 

 coeflicient Q by elimination between the two resulting equations 

 of condition. 



