DiurnalVariatiortsoff/ie Magnetic Forceaf ike Earth's Surface. 193 



Let the magnitudes of the lines connecting the points {a, b, c) 



and {x, y, z), respectively, with the origin be denoted by u and s, 



and let the angle contained by their directions be w. Then 



e^' — ii^ — 2ms cos o) + s^ ; 



and if s be so small in comparison with u that the squares and 



v, 

 higher powers of — may be neglected, 



35 



e ^=.u-'^l\-\ cos oj ). 



Again, if a, /S, 7 denote the angles contained by the axis of the 



magnet with the three axes of' coordinates, 



x=-sco?,u, y=5cos/3, z=sco%y. 



Substituting these values in the expressions for the components 



of the force above given, integrating, and observing that ^/jids = 0, 



we have, for the components of the total force exerted by the 



magnet on the magnetic element. 



Mm /- a \ 



— 5- I o- cos (o — cos « I , 

 vy \ u /' 



Mm(J} \ 



^(^3-cosa)-cos^j, 



Mm/ c \ 



-^(^3-cosa,-cos7J; 



in which we have put, for abridgement, M= ^/Msds. The angle 

 ft) is connected with a, /3, 7 by the relation 



u cos ft)= « cos a + b cos /8 + c cos y. 

 Now let the point {a, b, c) be on the eai'th's surface, and let 

 us suppose, for simplicity, that the acting magnet is in the plane 

 of the equator. Let that plane be taken as the plane of {x, y), 

 and the line connecting the centre of the magnet and that of the 

 earth as the axis of x. Then, if the distance of the acting mag- 

 net be considerable, relatively to the earth's radius, b and c are 

 small in comparison with a, and we may neglect the small quan- 



^2 c^ be 

 titles of the second order, -o, -5, -3-. Wherefore, substituting for 



a'- a^ a^ 



cos ft) its value^ the components of the acting force become 



Mm/„ 36 ^ 3c \ 

 —^1 3 cos « H cos p H cos 7 1, 



Mm/ ,, Zb \ 



_(^-cos^+-cos«j, 



M?rt/ 3c \ 



-^(^-cosy+-cosaJ. 



Phil. Mag. S. 4. Vol. 15. No. 99. March 1858. O 



