402 THE THEORY OF TERRESTRIAL MAGNETISM. 



Then V= 2 ~ {H? (gl cos mX + h: sin mX)}, 



for the first class of terms : 



and V= Sr" \H? (g m _ n cos m\ + h"\ sin mX)], 



for the second class of terms. 



and T'" - r"~ l (I u 2 V J 

 , ^ 8 L -A_.- /A; , . 



Then by equation (22) of Section I. (see p. 257) 



X: = ~ [(n - m) H;r -mp(l- ^ H^ , 



or by equation (16), 



X: = r L [I (n - m) Hr -\(n + m] J7.-'] ; 



hence for the first class of terms, i.e. for forces whose origin is situated in 

 the interior of the Earth, 



X = 2 [X? ( cos mX + h? sin mX)] 



= S ~ [^ (n - m ) H: +} -\(n + m) H'^ (^ cos mX + ^ sin mX), 



sn m - cos 



__ 

 = 2 - H: (flC cos mX + A: sin mX), 



where ^r, A are the Gaussian magnetic constants for positive integral values 

 of m and n. 



And for the second class of terms, i.e. for forces whose origin is 

 outside the Earth, the corresponding terms are : 



in the value of X 



= Sr"- 1 Hj (n - m) H +1 -~(n + m) Hf*] (g m _ n cos mX + h"l n sin mX), 



in the value of Y 



= Sr"- 1 [mZT; (1 - ^ 2 )- J ] (g m _ n sin mX - h m _ n cos mX), 



