390 Permanent Magnetism [CH. xi 



so that n can be put in the form 



m (sin. 1} 



71=00 m=n 



1= 2 2 



4- r n Pjf (sin 1) (A' n j n cos mX 4- B' ntm sin wiX) [ . 



Hence from equations (387) we obtain the values of X, T, Z at any point 

 in terms of the longitude and latitude of the point and the constants such 



as jfLn^m, -t>n,m.) -A- n,m> & n,m* 



By observing the values of X, Y, Z at a great number of points, we 

 obtain a system of equations between the constants A n , m etc., and on 

 solving these we obtain the actual values of the constants, and therefore 

 a knowledge of the potential as expressed by equation (388). 



If the magnetic field arose entirely from magnetism inside the earth, 

 we should of course expect to find $/ = S 2 '= . . . = 0, while if the magnetic 

 field arose from magnetism entirely outside the earth, we should find 



S 1 = > Sf 2 =...=0. 



455. The results actually obtained are of extreme interest. The mag- 

 netic field of the earth, as we have said, is constantly changing. In addition 

 to a slow, irregular, and so-called " secular " change, it is found that there 

 are periodic changes of which the periods are, in general, recognisable as 

 the periods of astronomical phenomena. For instance there is a daily 

 period, a yearly period, a period equal to the lunar month, a period of 

 about 26 J days (the period of rotation of the inner core of the sun *), 

 a period of about 11 years (the period of sun-spot variations), a period of 

 19 years (the period of the motion of the lunar nodes), and so on. Thus 

 the potential can be divided up into a number of periodic parts and a 

 residual constant, or slowly and irregularly changing, part. All the periodic 

 parts are extremely small in comparison with the latter. It is found, on 

 analysing the potentials of these different parts of the field that the constant 

 field arises from magnetisation inside the earth, while the daily variation 

 arises mainly from magnetisation outside the earth. The former result 

 might have been anticipated, but the latter could not have been predicted 

 with any confidence. For the variation might have represented nothing 

 more than a change in the permanent magnetism of the earth due to the 

 cooling and heating of the earth's mass, or to the tides in the solid matter of 

 the earth produced by the sun's attraction. 



This daily variation is not such as could be explained by the magnetism 

 of the sun itself; Chreef has found that it cannot be explained by the 



* The outer surface of the sun is not rigid, and rotates at different rates in different latitudes. 

 Thus it is impossible to discover the actual rate of rotation of the inner core except by such 

 indirect methods as that of observing periods of magnetic variation. 



t Roy. Soc. Phil. Trans., 202, p. 335. 



