THE THEORY OF TERRESTRIAL MAGNETISM. 439 



= f X?x m dp.+ r Y:y m dp + (n+l)\ 

 J-i J-i J 



where the coefficient of /3 n = 0. 



And ajT (XfYdp+l 1 (F.-^-n(n+l) f (H 



U-i J-i J-i 



-i -i 



where the coefficient of a n = 0. 



Hence a,, and /B n are separately determined from the equations 



, . (n m) I (n + m) I 

 -- 



and 



- f 1 ^"x m c^+ f 1 Y 

 J-i J-i 



-i 

 Thus generally from the values of JT and Y we derive 



(ra m) ! (n + m) ! 



= (2n+ 1) 

 and from the values of Z we derive 



[(n + 1) . - n&] 



The above theory assumes that the integration is taken over the whole 

 surface of the Earth, and that the observations are uniformly distributed 



