FORMATION OF THE FINAL EQUATIONS. 599 



different latitudes in X have been combined, in which case the coefficient 

 of <7* in the final equation for g" will be 



And the coefficient of g in the final equation for g will be 



We have seen above that in the case of a sphere the coefficients of 

 each of the magnetic constants in this final equation except the coefficient 

 of g* will vanish ; but this will only be the case when the summation is 

 taken all over the Earth's surface. The corresponding coefficients on the 

 spheroid will be small quantities depending on the value of the square of 

 the eccentricity. 



The right-hand side of the equation (4) becomes under these conditions 



If for a first approximation small terms be neglected, the value of 

 g" will be given by the equation 



x > m , 



When the belts of latitude which are employed in giving the equations 

 extend over the whole of the Earth's surface, and when the successive 

 belts are sufficiently narrow, the coefficient of (g + g n ) in the final equation 

 for g is approximately 



(n m)l 

 or - x 2 -J- 



and, as before (see p. 439 above), the right-hand side of the equation becomes 



It will be seen that $[X'*X'*w], which is the coefficient of g in 

 the final equation for g [equation (4)], is also the coefficient of g in 

 the final equation for g. 



A similar interchange of coefficients will also hold good in the equa- 

 tions for Y and in the equations for Z. 



This interchange of coefficients will also hold good when the series 

 of equations for (X) and for (Y) are combined. 



