KARTH'S MAGNTTISM PRODUCED BY THE MOON AND SUN. 291 



!; li!. Following SCHUSTER'S notation and treatment, the earth will be regarded as 

 ;i uniformly magnetized sphere of radius , whose magnetic potential may be resolved 

 into the zonal harmonic of the first degree and the tesseral harmonic of the first degree 

 .ind type. The former harmonic is much the larger of the two, as the inclination (0) 

 of the magnetic to the geographical Jixis is small. The radial force can be expressed as 



V = C cos ti + G tan sin 6 cos X, 



where C is a constant not differing much from $ (the force being measured positive 

 outwards), and X is now the longitude measured from the meridian (68 31' west of 

 Greenwich) containing the magnetic axis. 



The components of electric force, X and Y, measured towards the south and east 

 respectively, are 



Xa = -. 



am 



\js being the velocity potential. 



If we express X and Y in the form 



dS J dli dS 



x = y_ 



~ ' 



dd e sin d\ ' siu 6 d\ e dd ' 



where K is the known resistivity and e the thickness of the conducting atmospheric 

 shell, the function R will be the current function of the electric currents produced by 

 X and Y (neglecting electric inertia). The function S is the potential of a system of 

 electric forces which in the steady state are balanced by a static distribution of 

 electricity revolving round the earth, and causing a variation in the electrostatic 

 potential which is found to be too weak to affect our instruments. 

 To determine R we shall eliminate S, thus obtaining the equation 



dx deV "/-BinedxVdx/ 1 de 



Instead of using the resistivity it', SCHUSTER worked with the conductivity p (using 

 the special form 1 + k cos w), in order to avoid the difficulties introduced by " the high 

 and possibly infinite values which K would take when the conductivity sinks low or 

 vanishes."* These difficulties, however, are found not to be serious, and the work 

 is greatly simplified by the use of K, which enables R to be determined directly, 

 without first evaluating S, as is necessary when p is kept as the variable quantity. 

 The investigation can also be made much more general, without formal complexity, 

 when K' is used. 



* SCHUSTER, 'Phil. Trans.,' A, vol. 208, p. 190. 

 2 P 2 



