xii PREFACE TO PART II. 



distance from the Earth's centre, X, Y and Z the magnetic forces in three 

 directions at right angles to one another, X being the force towards the 

 north perpendicular to the Earth's radius, Y the force perpendicular to the 

 geographical meridian towards the west, and Z the force towards the Earth's 

 centre ; also taking cos 6 = p., we have 



d V sin 6 d V _ ( I /A 2 )* d V 

 rdd r dp. r 'dp,' 



dV (I-,*')-* dV 



r sin 6d\ r ' d\ ' 



dV 



if east longitudes be considered positive. 



There are two systems of values of V corresponding to magnetic forces 

 whose origin is situated inside and outside the Earth's surface respectively, 

 and by a convenient notation we may readily distinguish these two systems 

 of values. 



Making use of the functions denoted by H which I have above defined, 

 and taking g and /i to represent the Gaussian magnetic constants, g and 

 h are coefficients of cos m\ and sin mX respectively in the series of terms 

 representing the magnetic potential. 



The value of the magnetic potential V for magnetic forces whose origin 

 is situated in the interior of the Earth is expressed by a series of terms 

 of the form 



: (gr; cos m\ + /C sin mX)]. 



Taking g n and h^ to represent the values of the magnetic constants 

 corresponding to this term of the series for forces situated outside the 

 Earth's surface, the corresponding term in the magnetic potential will be 



'" [_H: (g"l n cos mX + /r )t sin mX)]. 

 Hence 



cos m\ + / sin mX)] + Sr" [H? (g n _ n cos mX + h~ n sin 



In the values of X, Y and Z there will be terms arising from each 

 of these series of terms for V, and we may conveniently express them by 

 modifying the notation in the same sense by using n subscript to refer to 



