SECTION VII. 



NUMERICAL CALCULATION OF THE MAGNETIC FORCES ON THE 

 EARTH'S SURFACE REGARDED AS A SPHEROID. 



1. Expressions for the Magnetic Forces at the. Equator (/j. 0). 



SINCE 



m-2) 



'-' - / _, 1 \ 



1.3.5 ...(2n-l) 



nd = / _ j fT 1 1 -3 .5...(n-m-2)1.3.5... 



1.3.5... (2?i-l) 



we have (n - m) & n m+1 = - (n + m) G^ = X" = X 



(n + \}G'; t l =Z and -nG? = Z^. 



If n m is odd, the value of 6r = 0, and the forces Y n , Y m n , Z n and 

 Z n vanish. 



If n m is even and = 2r, 



n~ - 1 _ i y 1.3.5...(2r-l) 1 .3.5... 



1.3.5...(2n-l 



and the forces JT,"' and A'"',, vanish. 

 For m and n = 2r, 



{l.3.5...(2r-l)} 8 _ (2r-l? 



'l.3.5...(4r-l) (4r-3)(4r-l) n ~* 



For m I and ?i = 2r + 1 , 



1.3.5...(2r-l)l.3.5...(2r+l)_ (2r-l)(2r+l) 

 ^""^ 1.3.5...(4r+l) (4r-l)(4r+l) 



For m = 2 and n = 2r + 2, 



_ (2r-l)(2r + 3) 



- 



_ 



T7 3 . 5 . . . (4r + 3) (4r + 1 ) (4r + 3) "- 



592 



