378 Astronomical and Nautical Collections. 



and at one third of the distance of the compass from the sphere. 

 The direction of the force referred to the horizontal plane will 

 be the projection of this direction, and its magnitude may be 

 found from the relative latitude of the compass,, with regard to 

 the axis of the sphere (N. 6), requiring also to be reduced to the 

 horizontal plane. 



18, But, for an easy and useful example of the result of 

 such a calculation, it will be sufficient to take a case in which 

 the primitive directive force of the sphere remains always hori- 

 zontal, and the reductions are avoided. This will happen 

 when the distance of the sphere before the compass, a, is to b, 

 its depth below the compass, as v' 2 to 1 ; and when the ship'.s 

 head is at the same time E. or W. Now, the dip being D, 

 the distance of the intersection of the axis with the plane 

 already mentioned, and with the horizontal plane, from the 

 middle of the ship's breadth, will be b cot D, and the cotan- 

 gent of the spontaneous deviation, — cot D — V a cot D, the 



tangent */ 2 ta D, the sine ir v/ ■: and the cosine v^ . 



^ ^ ^ 1-1- ss ^ 1 +SS 



The magnetic latitude A, with respect to the sphere, will be such 



that sin \ — ^'" , c being ^/ (a« + 6*) ir v' (2 is + 6«) =: ^/3A 



and sin ^ ::z >/ A sin D = v' J- s ; consequently the sine of the 



dip t with respect to the sphere is found ■ , /' ^ =: J 



^ Vl^ + ss) 



2 s 



: and the magnetic force of the sphere, which varies 



V (3 + 3 s s) 



sin L ... „, , sin X , i i / 



as —. — r- (IN.o) or here as-: may be represented by J 



sm D ^ sm £ •' ' *' 



(1 -f ss), considering the magnetism of the sphere as constant. 



But the magnetism of the sphere is proportional to that of 



the earth itself at the place of observation, so that the law of 



the composition of these forces is not affected by the change 



of their magnitude : the direct force of the earth, however, 



requires to be reduced to the horizontal plane, while that of 



the sphere is already exerted in that plane. The direct force 



therefore may be called •>/ (1 — ss) and the disturbance 



