450 THE THEORY OF TERRESTRIAL MAGNETISM. 



If we resolve the forces X and Z in the horizontal and vertical directions 

 instead of along and perpendicular to the Earth's radius, the change in the 

 value of X is Z sin \\i, and the change in the value of Z is X sin ty, where 



sin t/ = e* sin cos = e> ( 1 - /a') 4 . 



Hence the term ( 1 - f^)~ h H n [(n + 1 ) a,, - nfi n ] e>( 1 - ^) 

 must be added to the coefficient of cosmX in the value of X, and the term 



must be added to the coefficient of cos m\ in the value of Z. Hence taking, 

 as in the case of a sphere, x m , y m and z m for the values of the coefficients of 

 cos mX as derived from observation, and substituting the values just obtained 



ITT I 



for H n ', -v-)- , &c. and collecting terms, we get for the equations of condition 



) - K + A,) ey + [a ?J (n + 2) - A, (n - 1 )] e 



5. Multiplying the equation for x m by (i-^, and the equation 



"~ 

 for y m by 01(1-^)-*^, and adding, we get 



+ terms involving other magnetic constants = X n x m + Y n y m . 

 Then taking the weight of the observations in a belt of latitude as 



