DETERMINATION OF THE GAUSSIAN MAGNETIC CONSTANTS. Ill 



geographical meridian towards the west, and Z the force towards the 

 Earth's centre ; also taking cos O^jj., we have 



X = - — = ^^ '^ = fc^- ^ 

 rdO r dfi. r dft. 



Y = _ "^ - - (l-/^')^-- (^ 

 Z = - 



r sin 6 dX. r d\ ' 



dV 



dr' 



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 de- 

 fined, and taking gr™ and 7C to represent the Gaussian magnetic constants, 

 <7™ and Ajf are coefficients of cos mX 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 



^^fiffl' cos mX+h^ sin mX)!. 



Taking g^^ and A,™„ 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 wiU be 



r"[H;;'(i^™„ cos mX + h'H:,, sin mX)]. 

 Hence 



V=S J-frH,7(^™ cos m\+/C sin mX)1 + S AW;^{ff^!!„ cos 7«X + 7i':!„sinmX)1. 



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 

 internal forces, and — n subscript to refer to external magnetic forces, or 

 forces whose origin is outside the Earth's surface, i.e. corresponding to 



negative powers of (- J. 



The corresponding terms are 



in the value of X, 



J_(l_/,2)!. ^ (^;. COS mX + K sin mX) 



and r'-^l— ft2)J rf^ (^n',. cos mX+h'\ sin mX) ; 

 dfi 



