GAUSS'S THEORY OF TERRESTRIAL MAGNETISM. Ill 



equator towards the spectator: and z towards the north 



pole. Then 



x = r . sin u . cos X, 



y = r . sin u . sin X, 

 z = r . cos u. 

 For changing our coordinates, we must put 



dV = dVdu dV d\ dVdr 

 dx ~ du ' dx d\ ' dx dr'dx' 



and similar equations for y and z\ where u, X, and r, are 

 supposed to be explicitly expressed (as was V) in terms 



/v. 2 I aj% 



of a?, y, z. Now tan 2 u = - f ; from which, after due 



reductions, 



du 1 



= -cosw.cosX, 

 dx r 



du 1 



-y- = - cos u . sin X, 

 dy r 



du 1 . 

 -r --- sin u. 

 dz r 



And tan X = - ; from which 

 x 



d\ 1 sin X 

 dx r ' sin u' 

 d\ _ 1 cos X 

 dy r ' sin u' 



And r 2 = a; 2 + 2/ 2 + ^ 2 ; whence 



<2r . 



T- = sm it . cos 



