451-454] Terrestrial Magnetism 389 



With the help of a compass-needle, it will be possible to find the 

 direction of the lines of force of the earth's field at any point. It will 

 also be possible to measure the intensity of this field, by comparing it with 

 known magnetic fields, or by measuring the force with which it acts on 

 a magnet of known strength. 



453. At any point on the earth, let us suppose that the angle between 

 the line of magnetic force and the horizontal is 0, this being reckoned posi- 

 tive if the line of force points down into the earth, and let the horizontal 

 projection of the line of force make an angle 8 with the geographical 

 meridian through the point, this being reckoned positive if this line points 

 west of north. The angle is called the dip at the point, the angle 8 is 

 called the declination. 



Let H be the horizontal component of force, then the total force may be 

 regarded as made up of three components : 



X H cos S, towards the north, 

 F = H sin Sj towards the west, 

 Z=Ht&u&, vertically downwards. 



If H is the potential due to the earth's field at a point of latitude I, longi- 

 tude X, and at distance r from the centre, we have (cf. equations (331)), 



ian i an an 



Analysis of Potential of Earth's field. 



454. Since fl is the potential of a magnetic system, the value of fl in 

 regions in which there is no magnetisation must (by 408) be a solution of 

 Laplace's equation, and must therefore (by 233) be capable of expansion in 

 the form 



(So' + -S 1 'r + ^V+...) (388), 



in which S l} $ 2 , ... $</, $/, S a ' t ... are surface harmonics, of degrees indicated 

 by the subscripts. 



At the earth's surface, the first term is the part of the potential which 

 arises from magnetism inside the earth, while the second term arises from 

 magnetism outside. 



The surface harmonic 8 n can, as in 275, be expanded in the form 



in=n 



S n = S Py (sin I) (A n>m cos m\ + B n>m sin raX), 



