on the Mariner's Compass. 45 1 



In this instance it will be convenient to transform the latter of 

 these expressions into another which shall represent the force in 

 the direction O N : this will readily be effected by adding to it 



a force equal to — ^— ) acting in the direction a C, and then sub- . 



tracting the same quantity from the first expression, that the ef- 

 fect upon the whole may not be altered. We shall then have 

 for the forces in the directions C O and O N respectively, 

 3e. cos ip ,_, 



~^~ " •' ^^^ 



i (^) 



To estimate the effect of these forces upon the compass, it will 

 be necessary to reduce them to the horizontal plane ; for which 

 purpose put GH = i; HP = A; and CN = d. Then the forces 

 in the respective directions G O, D O will be represented by 



— ^ 3 cos k cos i cos 1^ — cos dl . . . . (5) 



-— . sin i cos ^ cos ip . , . , (6) 



Where, if i and k are known, (p may be found by the formula 

 Cos (p = sin d. sin k + cos d. cos k, cos i. 

 Let S be the deflection caused in the direction of the needle 

 by the action of these forces: then since the force by which the 

 terrestrial magnetism endeavours to restore it to the meridian 

 varies as sin Sj we shall have the following equation: 



m,sin8=-^|3cos A. sin i. cos <p. cos 5— (3 cos k. cosj. cos(p— 

 Whence by reduction ^°^ ^^ sin S J 



^ ^ A sin i. COS /:. COS *. ,_. 



tan S = A. ^ . . . . (7) 



iP -{- a < 3 cos k. cos i. cos f — cos J V 



This is the rigorous formula that expresses the relation be- 

 tween the place of the compass v^ith regard to the sphere, and 

 the consequent deviation produced in the direction of its needle : 

 when however o is small, an approximate expression may be found 

 hy neglecting, in the right hand number of the first of the above 

 equations, the coefficient of sin 8, and substituting in the place 

 of 6in i. cos k, their equals sin (f . cos / ; where / is the distance 

 of the meridian N P measured from D, on a great circle at right 

 angles to N S. Our expression (7) will then become 



. ^ • cos I. sin2p /c\ 



tan6 = A— ^-^^ (8) 



which is the formula derived by Mr. Barlow in his Essay on 

 MagnPtic Attractions. 



T t 2 To 



