Diurnal Variations of the Magnetic Force at the Earth's Surface. 195 

 But the moment of the earth's magnetism, opposed to this, is 



UAS sin 1', 

 in which U denotes the horizontal component of the earth's 

 magnetic force. Wherefore 



AS = -^qTT • — Til sin ^(2PcosS — QsinXsinS) 

 D-^U sm 1' L ' 



— cos d{2V sin X sin 8 + Q cos 8) + R cos X sin S [ . 



At the equator, this is reduced to 



AS= ^..J . -,, f cos 8(2P sin e- Qcos ^) +R sin B\. 

 D'^V sm 1' t J 



To determine the effect of the magnetic body upon the hori- 

 zontal component of the earth's magnetic force, we must resolve 

 the horizontal parts of the disturbing forces, viz. X sin ^ + Ycos^, 

 acting eastward, and Z cosX + (Xcos ^ — Ysin ^) sinX, acting 

 northward, in the direction of the magnetic meridian. We have 

 thus 



AU= 



(Xsin^ + Ycos^) sinS + {ZcosX + (Xcos^ — Ysin^)sinX}cosS 

 = YT^j sin 6{2V sin S -f Q sin X cos h) + cos ^(2P sin X cos S 



— Qsin S) — RcosXcos Si ■ 

 and at the equator, 



AU=^|sin6(3Psin^-Qcos^)-RcosSJ. 



Lastly, if V" denote the vertical component of the earth's force, 

 we have 



AV= — Z sin X+ (X cos O-Y sin 6) cos X 



= xpj (2P cos ^ + Q sin ^)cos X + R sin xl ; 



a result which, as might have been anticipated, is independent 

 of the magnetic declination. At the equator, 



'AV=^3(2Pcos6' + Qsin^).. 



From the foregoing we learn : — 



1. That the effect of a distant magnetic body on each of the 

 three elements of the earth's magnetic force consists of two parts, 

 one of which is constant throughout the day, while the other 

 varies with the hour-unyle of the luminary. 



2. Each of these parts varies inversely as the cube of the 

 distance of the magnetic body. 



02 



