PREFACE TO PART II. xxill 



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 Regarding H n ' and j-j- , &c., as functions of //, we have by Taylor's 



theorem 



H n ' = H n e*p.(l p?) -T-" to the order e", 



, dH n ' dH , <d-H n 



and ~-rr ~j -- e > I 1 ~I J - ) ~T~r> 



d\i! dp. dp.- 



from which we derive the value of X n for the spheroidal surface 



Y /i 



-^ 



p. 



If now we substitute the values of X, Y, and Z in terms of H n ', 



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" , &c., in the equations for X', Y', Z', the expressions for the magnetic 



forces become 



+ ""+ - nft n r n l H n f sin t// + similar terms, 



I \ 



Y' = ( ri + /8 B r"' 1 ) mH H ' ( I - uJ-)~ * + similar terms, 



* !, "" - w A,'' 1 "" 1 ] H cos ^ + similar terms. 



dH' 



In these expressions for the magnetic forces the values of H n ', , " , &c., 



Ct Li 



cJTT 



in terms of H n , -y-"- , &c., are substituted for each belt of latitude, and 



these theoretical expressions derived from the potential function for a given 

 belt of latitude, and containing the magnetic constants, are equated to the 

 corresponding coefficients derived from the magnetic observations taken in 

 that belt of latitude. 



In the case of the spheroid, as in the case of the sphere, the values 

 of the forces X, Y, and Z derived for every 10 of longitude from the 



