404 SEE— THE NEW THEORY OF EARTHQUAKES [November 15, 



If F be the magnetic potential of the earth, / the latitude, and A the 

 longitude of any point on its surface, and if a be the radius of the 

 earth assumed to be spherical, we shall have 



I dV ^^ I dV ^ dV ^ 



a ol a cos I d\ or ^ ^ 



where r is the distance of any point from the center of the earth. 



If 5'i + vSg + 6^3 -] 6^1 be a convergent series of spherical sur- 

 face harmonics defining for every point of its surface the potential 

 of all the magnetized molten or electric currents within the earth,, 

 the potential at all external points will be given by the series 



r=.,(^)%..(^)'.....s,(^)", ,.) 



The functions S^, S^, S^, . . . are functions of known form, con- 

 taining 3, 5, • • • 21 -\- I constants ; and if we neglect terms beyond the 

 ith order, there will remain in the expression for V r + 2i arbitrary 

 constants. These constants may be determined by observation, and 

 then the magnetic action at all points on the surface or outside the 

 earth becomes known irrespective of the internal distribution of the 

 magnetic causes which are inaccessible to observation. 



§21. The Mutual Potential Energy and Mutual Action of Two 

 Magnetic Systems. — If any portion of the earth's crust or an atmos- 

 pheric current above it should be suddenly magnetized by an instan- 

 taneous charge of electricity or otherwise, during the violent com- 

 motion of an earthquake or volcanic outburst, we should have at 

 least a temporary magnet suddenly formed in the field of the earth's 

 magnetism, and the result would be the disturbance of the magnetic 

 needle. Whether the magnetism of the earth be distributed accord- 

 ing to Poisson's theory, with a certain volume distribution of density 

 V and a surface distribution of density o-; or according to Gauss'" 

 theory, with a distribution wholly on the surface, this result is 

 equally true. 



To determine the mutual action of two magnets on each other 

 requires the evaluation of a sextuple integral, every point in one 

 field of space acting upon the corresponding points of the other. If 

 W be the potential energy of the whole magnetic system, R and R^ 



