250 ON THE DETERMINATION OF THE HORIZONTAL 



in which we may take tan u = u tan 1', as before. Hence there is 



AQ Aw 

 -Q- '^' 



and the probable error is less than in the usual method in the 

 ratio of 1 to 5*563, even when the latter is employed in the 

 manner most conducive to accuracy. Accordingly, if by any 

 means the coefficient of the inverse fifth power of the distance can 

 be annihilated, or rendered so small that the term shall have no 

 sensible influence, the accuracy of the results will be increased 

 more than five-fold, and, at the same time, the observations being 

 taken at one distance only, the labour of observation will be 

 halved. 



The same advantages will be gained if, the coefficient of the 

 inverse fifth power of the distance retaining a sensible value, the 

 ratio of the two coefficients may be known d priori. Let 



h being a known quantity. In this case the expression for the 

 tangent of the angle of deflection becomes 



and the coefficient sought is obtained, from the result of obser- 

 vation at a single distance, by the formula 



_ J 3 tan u 

 ^ ~ 1 + hD~* * 



It is evident that the probable error of Q thus obtained, arising 

 from an error in the observed deflection, is the same as in the case 

 last considered, and therefore between five and six times less than 

 in the ordinary method. 



The object of the following investigation is to point out the 

 means of attaining these advantages. 



Let the axis of the deflecting magnet be supposed to lie in the 

 right line joining the centres of the two magnets, and let the axis 

 of the suspended magnet make the angle $ with that line. Then, 

 if X and Y denote the forces exerted by the deflecting magnet 

 upon any element of free magnetism, m, of the suspended magnet, 

 in the direction of the line connecting it with the centre of the 



