THE THEORY OF TERRESTRIAL MAGNETISM. 447 



If X' be the horizontal force in the meridian towards the north, 



Y' the horizontal force perpendicular to the meridian towards the west, 

 Z' the vertical downward force on the spheroidal surface of the Earth, 

 then X' = X cos \jt + Z sin i/, 



Y'=Y, 

 Z' = X sin \fi + Z cos \jj. 



We may conveniently denote the values of the coefficients of gr cos m\ 

 and A sin m\ in the potential function and in the forces X, Y and Z by 

 the same symbols F, X , Y and Z* respectively as in the case of the 

 sphere, regarding them as functions of H', where H' is the same 

 function of // that H H is of p.. 



The coefficients of g cos m\ and h sin m\ in X' and Z' may be 

 denoted by X n ' and Z n ', where 



and Z n ' = - XI sin t/ + Z^ cos /. 



2. If a be the equatorial radius of the spheroid, then, taking into 

 account only the terms to the order e 2 , 



a 2 l-e 2 sin 2 0' 0/y 



75 = i^r- = l + e 2 cos-0'=l+e>-. 



We have also 



tar a\ e 2 Sin COS n a a n 



sunl = sin (0 0) = - - , = e 2 cos ^ sin = e 2 cos (9 sin 0, 



222 



also pf = cos 0' = cos e'p. (1 p.-^ sin 0, 



or // = /A e 2 /i ( 1 /A"') = p. sin sin i/f ; 



hence -r- = 1 e' 2 (1 3/t 2 ), 



and (l-a' 2 ) 4 -{l-ru- 



Let 6 be the polar axis, and let x and y be rectangular coordinates of 

 the point on the spheroid. Then 



x = r(l-/ 2 ) i = a(l- / L t 2 ) 4 (l+e> 2 ) i , 



y = rn' = biif (1 - e 2 + ey)-* = 6/t [1 - e* ( 1 - /A 2 )]*. 



