r 



60, Boicditch on computing the Dip of the Magnetic JSTeedle. 



L 



which is the form finally assumed by Mr. Biot, m the 49th vol. of 

 lilloch's Magazine; and it is evidently much more complex than 

 the formula (1) mentioned above; but we may deduce formula (-l) 

 from formula (1), in the following manner. By a well l^nown thee- 



rem tan. (i + k)= T'lt-Tr.\ 5 substituting the value of tan. i 



1 — • lan. u tan, A 



J, 



3 tan. A 



% tan. A, it becomes tan. (i + a) = , ' ^, . Multiplying tlienume- 



. 1 ^ 1 ± j^* \ 3- sin. A COS. A 



rator and denominator by cos. k\ we get tan. (i + a}= co s A^- FaTuA^ 



r 



3. sin A COS. A gjiijstitutins in this the well known values a. sin a 



1—3. sin. A» '^ 



, . . , |. sin.SA 



cos A= Sin. 3 a, ana sin. a^ = 1 — 4. cos. 2 a, it becomes — r—-^ — -,,. 

 and dividing the numerator and denominator by |, tan. (»+ '^) 



— , which is the same as Mr. Biotas formula (4) 



COS. a A- A 



As the tangents of small angles are very nearly proportional to 

 the angles themselves^ the formula (1) will become^ when the lati- 



h 



tudes are small, 



i = 2A. 



that isj the dip is then very nearly equal to tiince the magnetic lati- 

 tude,. This was observed by Mr. Biot, in his paper published in 

 Tilloch's Magazine, vol. 49. 



Notwithstanding the great elegance and originality observable 

 in the various publications of Mr. Biot ; it is not uncommon to 

 find his expressions not reduced to their most simple form, as in 

 the instance just noticed. For example, we may mention that 

 which occurs in the "Memoires de la Classe des Sciences Math- 

 emati^ues et Physiques de I'lnstitut de France/' 1809, p. 80, where 



1 



fl I Z 



the value of ^ is computed by this complicated expression ne§. 

 lecting a factor — m A 



/ 



r*" 



