297 



after tlie first may be insensible, the angle of deflection be- 

 comes very small, and the errors in its observed value bear a 

 large proportion to the whole. 



It fortunately happens, that at moderate distances (dis- 

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

 the terms beyond the second may be neglected. The ex- 

 pression 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, 



Q , Q' 

 tan M = — , H : ; 



in which q and q' are unknown coefficients, the former of 

 which is double of the ratio sought. Accordingly, the me- 

 thod recommended by Gauss consists in observing the angles 

 of deflection, u and ?/, at two different distances, d and d', 

 and inferring the coefficient Q by elimination between the 

 two resulting equations of condition. 



It is evident, however, that if the coefficient of the in- 

 verse fifth power of the distance be evanescent, — or, more 

 generally, if the ratio of the two coefficients be known a 

 priori, — the quantity sought may be obtained, without elimi- 

 nation, from the results of observation at one distance only. 

 For if q' = /jQ, h being a known quantity, the preceding ex- 

 pression becomes 



tan« = ^3(l-l-^,); 



and accordingly the value of q is obtained, from the result of 

 observation at a single distance, by the formula 



D^tanw 



Q = 



1 -f hi>- 



And, not only is the labour of observation thus diminished, 



but (which is of more importance) the accuracy of the re- 



2 B 2 



