THEORY OF THE DIPPING-NEEDLE. 87 



Eliminating at the same time Wt and Ha, 



_ (tan 4 + tan a ) (tan t - tan 2 ) 

 ~ (tan 64 tan 3 ) (tan O l + tan # 2 ) 



_ (cot fl 3 + cot 4 ) (cot # 2 - cot t ) 

 ~ (cot 3 cot 4 ) (cot 2 + cot 0J ' 



This expression can be used with accuracy when the 

 needle is greatly out of balance, so that l and 2 differ 

 considerably and 3 and 4 differ considerably : but it is 

 not accurate when th'ese angles approach to equality, 

 because the unavoidable errors of observation then 

 greatly affect the proportionate accuracy of cot 2 - cot t 

 and of cot 3 cot 4 . 



On expanding the numerator and denominator, it 

 will be found that the supposition n = 1 corresponds to 

 this very simple equation : 



tan 0j . tan 3 = tan 2 . tan 4 ; 

 or cot 9 l . cot 3 = cot # 2 . cot 4 . 



But this equation, which is founded on the last, cannot 

 be used with safety when the needle is very nearly 

 balanced. 



From the first pair of the original equations, and 

 from the second pair, we find 





