Examples 395 



19. If a small spherical cavity be made within a magnetised body, prove that the 

 components of magnetic force within the cavity are 



20. If the earth were a uniformly magnetised sphere, shew that the tangent of the 

 dip at any point would be equal to twice the tangent of the magnetic latitude. 



21. Prove that if the horizontal component, in the direction of the meridian, of the 

 earth's magnetic force were known all over its surface, all the other elements of its 

 magnetic force might be theoretically deduced. 



22. From the principle that the line integral of the magnetic force round any circuit 

 ordinarily vanishes, shew that the two horizontal components of the magnetic force at 

 any station may be deduced approximately from the known values for three other stations 

 which lie around it. Shew that these six known elements are not independent, but must 

 satisfy one equation of condition. 



23. If the earth were a sphere, and its magnetism due to two small straight bar 

 magnets of the same strength situated at the poles, with their axes in the same direction 

 along the earth's axis, prove that the dip d in latitude X would be given by 



24. Assuming that the earth is a sphere of radius a, and that the magnetic potential 

 Q is represented by 



shew that Q, is completely determined by observations of horizontal intensity, declination 

 and dip at four stations, and of dip at four more. 



25. Assuming that in the expansion of the earth's magnetic potential the fifth and 

 higher harmonics may be neglected, shew that observations of the resultant magnetic 

 force at eight points are sufficient to determine the potential everywhere. 



26. Assuming that the earth's magnetism is entirely due to internal causes, and that 

 in latitude X the northerly component of the horizontal force is A cos X + B cos 3 X, prove 

 that in this latitude, the vertical component reckoned downwards, is 



