200 C. F. GAUSS ON THE GENERAL THEORY OP 



Resolving the horizontal magnetic force into two portions, one of 

 which, X, acts in the direction of the geographical meridian, and 

 the other, Y, perpendicularly to that meridian, — and considering 

 A' as positive when directed towards the north, and Y as posi- 

 tive when directed towards the west, — then 



_ (1 — (2 6 — 6-) COS ?<^) I . d V 

 (1-e)- " Rdu 



r = - v" (1 - (2 e - €') cos ii") • -S-. ^• 



The total horizontal force is then 



= x/'iX-' + Y''), 

 and the tangent of the declination 



_Y. 



~ X 



Neglecting the square of the ellipticity, e, the expressions become 



dV 



X= - {I + (2 - Scosir^) e) . 



Rdu 

 dV 



y= — (1 — ecos U°) . ^fj—. i-r-, 



^ ' R sin u, d A, 



or, setting the ellipticity quite aside, 



Rdu 

 dV 



7? sin u.d\' 



The data furnished by the observations which we possess 

 are much too scanty, and most of them much too rude, to 

 make it advisable at present to take into account the spheroidal 

 form of the earth. It would not be difficult to do so ; but it 

 would complicate the calculations Avithout affording any corre- 

 sponding advantage. We will therefore adhere to the last- 

 mentioned formula, in \a hich the earth is considered as a sphere, 

 whose semi-diameter = R. 



15. 



If X be expressed by a given function of u and X., Y can be 

 be deduced from it a priori. 



Let the integral / X d u = T, considering A, as constant in 



the integration : it is then clear that if we differentiate in a similar 



