THE THEORY OF TERRESTRIAL MAGNETISM. 465 



Hence the final equation for a n becomes 



2 



, 2 n(/i+l)-3m 2 1 (n + m)! 

 ' C 



(n + m)\ 

 (2n-l)(2n + t)(n-m-2)I 



n 



+ n+i 



(n-m)\ 



= 1 ' 3 '(n "-m}\ ~ 



Also the final equation for b n will be 



K + ( 



s n (n + I ) - 3m" ] (n + m) ! 

 ' a " e (2w-l)(2+l)(2n+3)J(n-m)! 



+ o f 2 - - - fe 2 (n-2)(,-l) "1 (/* + ?)! 



L a "" ae (2w-3)"(2w-l)(2w+l) "~ 2e (2n-3)(2n-l)J (n-m-2)'. 



_ 9 i, 2 *i(n+l) _(w + m + 2)! 

 n+ - e (2n-f l)(2n + 3) ~(^- m) f 



In these equations x m ', y m ' and z m ' are quantities derived from the 

 magnetic observations. 



Ai a n b n U n 



Also since --? - = ~ = = 



-4. 5 M ^ n 



U n U n _, [(n-m)\ (n-m-2)n 4 

 we have n 7T = / VT/ ^\, 



Q,, Qn-, \-(n + m) I (n + m - 2) ! J 





A. II. 59 



