248 ON THE DETERMINATION OF THE HORIZONTAL 



The object of the present Paper is to point out the means by 

 which the quantity sought may be obtained, without elimination, 

 from the results of observation at one distance only ; and thus not 

 only the labour of observation be diminished, but (which is of 

 more importance) the accuracy of the result increased. Before 

 entering on this, however, it will be expedient to ascertain the 

 amount of the probable error in the received method. 



The coefficient of the first term, obtained by elimination 

 between the two equations of condition above alluded to, is 



The distances being greater than four times the length of the 

 magnets, the angles of deflection are small, and there is, approxi- 

 mately, t&mi = ^(tan^, tan*/ = w'tanl', u and u' being expressed 

 in minutes; and making If = qD, the preceding expression 

 becomes 



The probable errors of u and a' are equal ; and, by a well-known 

 theorem of the calculus of probabilities, the probable error of Q is 



^TT AM ' 



In determining the ratio of this error to the quantity itself, we 

 may observe that there is, approximately, <fu' = u, and 



Q = 3 tanl'.; 

 and dividing the formula last found by this, 



It appears, from the preceding theorems, that the value of AQ, 

 corresponding to a given value of Aw, varies with the assumed 

 ratio of the distances, q; and that, in order to apply the method 

 most advantageously, this ratio must be taken in such a manner 

 that the probable error, AQ, shall be the smallest possible. This 

 condition gives 



d /VY+T 



