320 



MAGNETIC METHOD 



[Chap. 8 



of the forces on a magnetic needle free to move in space. Assume a Car- 

 tesian system of coordinates oriented in the astronomic directions, north 

 (x'), east (?/'), and nadir (2'). Let the components of the earth's magnetic 

 force in these directions be X', Y', Z'. Assume the x axis of another 

 system to coincide with the magnetic axis of an imaginary needle free to 

 move in all directions. Then y represents the component acting at right 

 angles to it and z the component in the axis of revolution if the movement 

 of the needle is confined to the plane ABC. The inclination of this plane is 

 given by the angle i, its intersection with the horizon by the angle (azi- 

 muth) a, and the position of the needle on the plane by the angle rj (Fig. 

 8-17). The force acting upon the needle may be expressed by its three 

 components x, y, and z as function of X', Y', and Z' : 



X = X' (cos a cos t; + sin a sin 77 cos t) 



-|- Y' (sin <x cos T] — cos a sin 77 cos t) 



-}- Z' sin 7; sin i. 

 y = X' (cos OL sin 17 — sin a cos 77 cos i) }■ (8-14) 



-f- Y' (sin a sin 77 -|- cos a cos 77 cos t) 



— Z' cos 77 sin I. 

 z = —X' sin a sin t + Y' cos a sin i -f- Z' cos i. 



Since all magnetic matter is polarized, the forces acting upon north and 

 south pole balance each other so that x = 0. The three terms in each of 

 the above equations may be reduced to two by revolving the system of 

 coordinates through the angle of declination, so that X' = H, Y' = 0, 

 Z' = Z, and a = magnetic azimuth. The total force acting upon the needle 

 is F = m' • v'y^ + Z', where m' is the magnetic pole strength of the needle, 

 and 



F = m' 



[H(cos a sin 77 — sin a cos 77 cos i) — Z cos 77 sin t]^ ,q. e\ 



+ [Z cos I — H sin a sin t]^ 



This represents the force acting upon a needle free to move in any direction 

 (Swedish mining compass). In all other magnetic instruments the move- 

 ment of the needle is confined to the xy plane and the z component is neu- 

 tralized by the pressure of the pivots in the bearings. Then only the y 

 component remains. Its couple is Z>i = My, where M is the magnetic 

 moment of the magnetic needle. Thus, 



Di = Af [H(cos a sin 77 —sin a cos 77 cos i) — Z cos 77 sin t]. (8-16) 



