594 FORMATION OF THE EQUATIONS OF CONDITION. 



The values of the coefficients X', X' n , Y' &c. in these equations 

 are the values derived for the spheroidal surface of the Earth from the 

 formulae given in Section VI. (p. 452) ; their logarithms are recorded in 

 tables in Section VII. (see pp. 482519). 



When n m is even, the value of X' contains only odd powers of 

 /A, and the values of Y' and Z','" contain only even powers of p.. 



Similarly, when n m is odd, the value of X' contains only even 

 powers of p., and the values of Y' and Z' contain only odd powers of p.. 



Hence if the coefficient of cosmX in either of the quantities X, Y or 

 Z be denoted by a m and the coefficient of sinmA. by b m for a given 

 north latitude, and if a' m , b' m denote the similar quantities for the cor- 

 responding south latitude, then combining the equations for these two belts 

 together we have, when n m is even, 



2 ( X': g: + X' n sC.) = \ (a m - a' m ), and 2 (X": % + X'~_ n h - ) = I (6 m - b' m ) 



...(1), 



Z(Y':g: + Y'-n9-n) = l(b m +b' m ), and 2(7':*:+ 7'-*-)= -\(a m + a' m ], 



Z(Z':g: + Z' n g n ) = (a m + a' m ), and 2 (Z': h: + Z'" n A") = \(b m + V m ); 

 and, when n m is odd, we have 



2 (X': g: + *'_- g n ) = 1 (a m + <), and 2 (X'- ^ + X' n A" ) = \ (b m + b' m ), 



*(Y':g:+ Y' n g n ) = l -(b m -b' m ), and 2(77 /C+ Y'^h^) = - l -(a m -a' m ], 

 - g n ) =(a m - a'.), and 2 (Z': A^ + Z'_-. A_ M n ) = (6. - b' m ). 



Thus the equations for the magnetic constants, when n m is even, 

 are separated from the equations for the constants when n-m is odd, 

 and each equation contains only half the number of unknown magnetic 

 constants to be determined. 



Also the equations for the quantities h will be found from the 

 equations for g" by substituting, 



