Boitdifch on computing the Dip of the Magnetic J^'eedle, 59 



may also be reduced to the vertical force in the direction C E, rep- 

 resented hy F (1 — 2x sin. x) nearly, and the horizontal force in 

 the direction sC or C w, equal to F. ori. cos. k. The sum of these 

 vertical forces which act in opposite directions is F. (1 + S a', sin. h) 

 F (1 — S a;, sin. x) =« 4 F. ^. sin. x ; and the sum of the hori- 



zontal forces is F 



F. X. cos» X. H 



senting the first of these forces by the line E C, and the last by the 

 line C A, taken in the direction C tif the line E A will represent 

 the direction of tlie magnetic needle, and the angle C E A will be 

 the complement of the dip ; so that if we represent the dip by /, we 



CFi 



shall have C AE = z j and since tang. C AE=rrj^,we shall have 



by using the preceding values, tan. i — ., / ^J.^ , or, by reduction, 

 this very simple formula published by me in the year 1807, in the 



*^ Practical Navigator^ without a demonstration 



7 



tan. i = 2tanx. (l) 



That is, the tangent of the dip is equal to tirice the tangent of 

 the magnetic latitude. This latitude being found by the usual 



J 



rules of spherics, assuming the latitudes and longitudes of the 



magnetic poles beforfe given. 



Instead of this simple method, Mr. Biot (in Vol. fin. Tilloch) 

 uses the following. He first computes the angle E B P = ^, by this 



formula 



g = .^'""". (3) 



1 -j- COS. 2 u 



3 



aud then i by the followir^ 



o 



^. 



I 



90^ — ^ + u. ^ (3) 



Substituting in this last, the value of 21= 90 — X; we get C 

 180^ — {i + h)f whence tan. ^ = — tan. (J + x). and then by means 

 of the equation (3) we get 



tan. (i+K} = — — r. 



(4) 



