MAGNET NEAR THE EARTH'S CENTER. 105 



small) ; and the algebraical difference or numerical sum 

 of these gives the whole horizontal directive force 



= ^Bb . a~ 3 . sin 6. 



The resolved part of the action of the blue pole 

 upon the red end of the needle, in the direction of the 

 vertical at P, is 



B . (a- b cos 0) . (a 2 + b* - 2 ab cos 0)~ : 

 which, expanded to the first power of b, gives 

 B . a~ 3 . (a + 26 cos (9). 



The action of the red pole upon the same red end 

 of the needle is 



-B. o7\ (a -2b cos 0). 



The sum of these gives for the total vertical force 

 downwards upon the red end, 



4>Bb . a\ cos 0. 



As above, there is an opposite force, numerically ad- 

 ditive, upon the blue end : and the whole vertical 

 directive force is 



SBb . oT\ cos B. 



Hence the tangent of dip at P 



vertical force SBb . a~ z . cos 6 



i: T-Tr -- = T~DZ: =3 : ~a = 2 cotan 6 

 horizontal lorce 4tno . a . sin u 



= 2 tan magnetic latitude of P. 

 And the total force at P = 



{(hor. force) 2 + (vert, force) 2 }^ 

 . a' 3 , (sin 2 + 4 cos 2 0)* . 



