194 Dr. Lloyd on the Influence of a distant Luminary upon the 



Now, if D denote the distance of the centre of the magnet from 



the centre of the earth, r the earth's radius, X the latitude of the 



point {a, b, c) on its surface, and 6 the angle contained by the 



meridian passing through it with that containing the acting 



magnet, 



a=D — ?'COsA,cos^, b = rcosXsin6, c = ?'sin\. 



be T . 



Hence the maximum values of -, — are equal to - : and if we 



a a ^ a 



disregard the terms containing them in comparison with the rest, 



the preceding values are reduced to 



_ Mm Mm ^ Mm 



2^3- cos «, -^ cos/3, -^3-0037. 



Now, in place of a single magnet, let there be an indefinite 

 number distributed in any manner throughout the entire mag- 

 netic body ; and let us make, for abridgement, 



S(Mcos«)=P, S(Mcos/S) = Q, S(Mcos7) = R. ' 



Then, if the radius of this body be small in comparison with its 

 distance, we may neglect the variations of D, and we shall have 

 for the three components of the acting forces, 



y_ 2mP ^_ — niQ ^ _ — »iR 



In order to determine the effect of these forces upon a freely 

 suspended horizontal magnet at the earth's surface, we must 

 resolve X and Y in the direction of the tangent and of the radius 

 of the parallel of latitude. The resolved forces are, respectively, 



Xsin^ + Ycos^, Xcos^— Ysin^. 

 Again, resolving the forces Z and X cos Q — X sin Q in the direc- 

 tion of tlie tangent to the meridian, and in the direction of the 

 radius of the earth, we have finally the three components, viz. 



X sin ^ + Y cos Q, horizontal, and directed eastward ; 

 Z cos X + (X cos ^ — Y sin Q) sin X, horizontal, and directed 



northward ; 

 — Z sin X+ (X cos — Y sin 6) cos X, vertical, towards centre. . 

 Of these, the latter has no effect upon the horizontal magnet ; 

 the moment of the two former to turn it is 



(Xsin ^ + Ycos^)cos8— {Z cosX + (Xcos^— Ysin^)sinX} sin 8, 



S denoting the magnetic declination ; or, substituting for X, Y, Z 

 their values, 



■^ { cos S(2P sin 0- Q cos &) - sin S(2P cos ^ -H Q sin 6) sin \ 



— R cos Xsin 8 1 ■ 



