454 THE THEORY OF TERRESTRIAL MAGNETISM. 



may express the horizontal and vertical magnetic forces in terms of the 

 magnetic constants as in the case of the sphere. 



We may also analyse the observations of horizontal and vertical forces 

 in the same belt of latitude in a series of the form 



a + ctj cos X + &! sin X + a. 2 cos 2X + 6 2 sin 2 A. + &c. , 

 and equate the coefficients of cosmX and of sin mX in the two series. 



Thus we shall have 



a n x,; + p n x'_ H + a n x n ; + /s n x'_ ni + & c . = x m ', 



, Y n ' + & Y'_ n + a ni Y ni ' + m Y'_, h + &c. = y m ', 

 a n Z n '+ (3 n Z'_ n + a n Z n { + p n Z', H> + &c. =z m ', 



where x m ', y m ' and z,,/ are the coefficients derived from the observations 

 of horizontal and vertical forces. 



dJJ ' 



Substituting the values of X n ', Y n ', Z n ', &c., in terms of H n ', ~, &c., 



in the above equations, we get 



similar terms involving other magnetic constants = x m ', 



+ [ ~lnl a " - n $n ^"" 1 ] Hn cos i/f + similar terms = z m '. 



On multiplying these equations by X n ', Y n ' and Z n ' respectively, 

 i.e. each equation by the coefficient of a n in that equation, and adding them 

 all together, we shall get the partial equation of condition for a n : the 

 coefficient of a n will be 



