THE THEORY OF TERRESTRIAL MAGNETISM. 463 



Thus let a n and b n be two magnetic constants derived from the function 

 Q n (as defined above), and let a n and (5 n be the corresponding magnetic 

 Gaussian constants as derived from the function H n . Then these magnetic 

 constants a n and b n are connected with a n and ft n by the relations 



fi f) 1*3*1 /9_i\ 



O-n _ Pn _ Vn _ O J . \4H' if 



a n -b n -ff n = (n-m}\ 



for a given value of m, and similarly 



_= 

 a., & H ni 



and in particular 



_ _ 



&.-, #-, (n-m-2)! 



, a n+ , _ /3 n+2 _ Q n+2 _ 1 . 3 . 5 . . . (2n 



IHQ. ^ -j=j 



+i &+ a H n + * (n-m + 2)l 



We may find the final equations for a n and b n from the final 

 equations for a n and fi n respectively by multiplying them by -~ , and 

 then substituting the values of a n and /3 n in terms of a n and b n respectively. 



Hence in the final equations for a n and /8, t the coefficient of a n or of 

 /? will be multiplied by 



W } or 

 xi. 



in order to find the coefficients of a n and & respectively. Also the 

 coefficient of a n _ 2 or of ^ H _ 1 in the same equations will be multiplied by 





1.3.5...(2n-l)1.3.5...(2n-5) 



(n-m)\(n-m-2)\ 



to find the coefficients of a n _ 2 and 6 n _ 2 respectively. And the coefficients of 

 a B+a and n+2 will be multiplied by 



. 



' 



to find the coefficients of a n+2 and & re+2 . Or generally the coefficients of a Bi 

 and /8 Bi in the final equations for a m and /? will be multiplied by jf-^ff- 



n T,' . 



to find the coefficients of a ni and 6 Mi in the corresponding final equations. 



