TRANSACTIONS OF THE SECTIONS. 559 



needle coincides with the Hne of the dip, while thatof the/orce 

 is least when the needle is at right angles to that line. These, 

 then, are the most advantageous positions for the determina- 

 tion of the two elements ; and accordingly the best mode of ap- 

 plying the preceding method consists in observing, 1st, the 

 position of the needle when unloaded ; and, 2ndly, when loaded 

 with a weight sufficient to render it nearly perpendicular to the 

 line of the dip. As the inclination of the needle in the first 

 position is nearly equal to the dip, and would be accurately so 

 if the centre of gravity of the needle perfectly coincided with 

 the axle, it is convenient to consider this first angle as the ap- 

 proximate value of the dip, and to seek the correction reqviired 

 in order to reduce it to its true value. If e denote this correc- 

 tion, it can be readily shown from the formulae already given, 

 that 



8 = ? + a, sin. = p^sin(?-6), 

 cos 9 



p denoting the ratio of the moment of the needle itself to that of 

 the weight afterwards added. When the needle is well con- 

 structed, this ratio is very small, and the correction itself may 

 be disregarded. The force is deduced from the second posi- 

 tion of the needle when loaded, and is given by the formula 



_ jScosfl 



~" sin (S - fl)' 

 (8 being a constant, which is determined from the values of 8 

 and 9 at the place where the force is taken as unit. 



In the usual method the horizontal force is determined by 

 the rate of vibration of a horizontal needle, and the actual force 

 deduced by multiplying it by the secant of the dip. The in-- 

 strumental errors, therefore, are of two kinds, and as these may 

 have the same sign, the limit of error is thereby increased. But 

 even supposing the determination of the horizontal force to be 

 perfect, the limit of error in the actual force, arising from the 

 error of the dipping-needle, is to that in the method now pro- 

 posed, in the ratio of the tangent of the dip to unity ; so that 

 the latter method is more accurate whenever the dip exceeds 

 45°, and in our latitudes its accuracy is nearly three times 

 greater than that of the received method. This result has been 

 verified by observation, and it has been found that, with a small 

 circle 4|^ inches in diameter, the value of the force deduced from 

 the mean of two or three observations may be depended on 

 with certainty, to the third place of decimals inclusive. 



