462 THE THEORY OF TERRESTRIAL MAGNETISM. 



13. Hence when the polar radius of the Earth is taken as the unit of 

 length, the final equation for a H for a given value of m becomes 



n(n+l)-3m 



n(n+l)(2n-3) 

 ~ 



~ a +- 



(n + m + 1 ) (TO + m + 2) (n - m + 1 ) (n - m + 2)} 

 = e " (2n+l) 2 (2n + 3) 2 (2n + 5) j 



= a known quantity of the form 



Similarly the final equation for /3 n becomes 



2(n-m)!(n + m)l f. T 1 2 



[T . 3 . 5 .I. (2n- 1)J ' 



^I) (2n + 1) (2n + "3) 



_ ., 



2 2 J 



= a known quantity of the form 



n 'X'. n (w) dp + y m ' Y'_ n (w) dp + 1 z m 'Z'_ n (w) dp. 



j 



14. If we had expressed the magnetic potential V and the magnetic 

 forces X, Y and Z in terms of the functions Q n , Q ni , &c., instead of in 

 terms of H n , H ni , &c., we should have obtained another series of magnetic 

 constants ; but the two series are related to one another, and the one series 

 may be derived from the other by multiplying each constant in one series 

 by a factor depending on the values of n and m to get the corresponding 

 magnetic constant for the other series. 



