464 THE THEORY OF TERRESTRIAL MAGNETISM. 



Hence the constants <* and & will have to be multiplied by - , 



i.e. by the factor 



1.3.5 ... (2n-l) 



(n m)\ 



in order to obtain the corresponding Gaussian constants a n and /3 n . 



Again, let A n , B n be two magnetic constants connected with a n and 

 ft n by the relations 



a n _ & _ n _ 1.3.5 ... (2n-l) 



Then the values of the magnetic constants A n , B n , &c., as determined from 

 the function II n , can be converted into the corresponding Gaussian magnetic 

 constants derived by means of the function H n by multiplying each magnetic 

 constant A n or B n for each value of m by the factor 



n. _ 1 . 3. 5 ... (2n-l) 



Also in the final equations for a,, and /? the coefficients of a n or of 

 /J n will be multiplied by 



( 



\HJ (n TO) ! (n + m)l 



in order to find the coefficients of A v and B n respectively. 



Also the coefficients of a B _., or of /8 B _., in the same equations will be 



^ 

 , H. 



multiplied by " "" 2 to find the coefficients of A n _^ or of B n _v respectively. 



Also the coefficients of a n+2 or of /8 B+2 will be multiplied by " j2 + ~ to 

 find the coefficients of A n+ ^ or of B n+i respectively. 



From the final equations for the determination of the Gaussian constants 

 a n , ft n , &c., and taking the equatorial radius of the Earth as unity, we may 

 write down the final equations for the determination of a n , b n , &c., where 



n n 



