598 FORMATION OF THE FINAL EQUATIONS. 



This will give for g an equation of the type 



(X'ZYwg? + X' m n *'_" wg'\ + X"" n X'* wg + &c. = X'l wx' m ......... (3), 



with similar equations for Y and Z. 



Then integrating or adding together the equations in X or Y or Z 

 for the different latitudes we get the final equations for g of the types 



2 \_(X ';)' w-] g: + S [X f ? X" n w-] g n + 2 [_X f : X', w] gl + &c. = 2 [X': wx' m -\. . . (4), 



2 [( r :) s w] sr; + 2 [ r'r r- i%~ + 2 [ FT y- te^; + & c . = 2 [ Y'? ^ 'j, 



)> W ] sc + 2 [Z'; ZT B u,] <,_" + 2 [Z' H - Z'- *//] ^ + &c. = s [Z'; *']. 



The changes in the values of x' m , y' m and z' m according as (n m) is 

 even or odd have been above explained, and their values in the equations 

 for determining h instead of g have also been given (see p. 594). 



We shall have a separate final equation for each value of n ; thus 

 the final equation for g for a given latitude from the equations for X is 



V'm -V >m m , Y'm Y'm ,,,"' l_ / Y'm\l ,,,,,m , a Ylm n ,.~.t 



& n -A n> W 9n + A -n A , W 9 - + ( A nj ^9 n , + <XC. = A ^WX m , 



where the coefficient of g is the same as the coefficient of g, in equation (3). 



Then adding up, for the constant g, the coefficients in the final 

 equations for all the different belts of latitude we have the final equation 

 from the series (X), which may be represented by the form 



2 [_x r : x':: W } 9 : + 2 [*': x> w\ ^ - + 2 [(xy wj g ~ + & c . = 



where x' m stands for - (a m a' m ) when , m is even and for - (a m + a' m ) 



L Z 



when rij m is odd. 



Equations similar to the above will be derived from the series (Y) 

 and from the series (Z). 



These equations may be solved separately, and the values of the 

 magnetic constants determined from each series ; taking series (X), series ( F) 

 and series (Z) separately. 



7. For another and more satisfactory determination of the magnetic 

 constants the series (X) and the series (F) may also be conveniently 

 combined into one equation in the same way as the above equations for 



