xxvi PREFACE TO PART II. 



Section VIII. treats of the mode of formation of the Equations of 

 Condition and the Final Equations for determining the magnetic constants, 

 the solution of the equations and the discussion of the results. 



Formation of the Equations of Condition. 



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

 and the values of F and Z only even powers, and similarly when n m 

 is odd, the value of X contains only even powers of p., and the values of 

 F and Z only odd powers. Hence, if the coefficient of cosmA. in either 

 of the quantities X, Y, Z be denoted by a m and the coefficient of sin m\ 

 by b m for a given north latitude, and if a m ', b m ' denote the similar 

 quantities for the corresponding south latitude, then we have, when n m 

 is even, 



s(x;+x^)= K-<O and 2 (xm+xzW'-l&.-W. 



2 ( YZ& + Y^gty = \ (b m + b m ') and 2 ( FT/C + Y? n h^) = - \ (a m + a,,/), 

 *(Zg~ + Z>g^)~(a m + aJ and 2 (Z^ + Z' n ti n ) = 1 (b m + b m ') 



-- Zj 



and when n m is odd, 



2 (X: 9 : + *<,) = 1 (a m + a m ') and 2 (X?K + X>L\h_ n ) = \(b m + b m '), 



2 ( Y: 9 : + Yl^ n ) = \(b m - bj) and 2 ( Y'K + Y n A_- ) - - 1 (., - <), 



* z 



2(Z^: + Z^ B ) = ^(a m -a m ') and 2 (Z:*:+ Z-.A!?.) = ^ (6. -b m '). 



Hence the equations for the quantities h and h m _ n will be found from the 

 equations for g and g n , when n-m is even, by substituting 



^(b m -b m r ) for -(a m -a m ') in the equations for X, 



-^(a m + a m ') for - (6 m + 6 m ') in the equations for F, 



and 2^ m + &m/ ) for 9( a m + a m') m tne equations for Z. 



