THE THEORY OF TERRESTRIAL MAGNETISM. 435 



becomes 



_ , n m (n m) (n m 1) (n m) (n m 1) (n m 2) . 



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



n m (n m)(n m 1) (n m)(n m 1) (n m 2) . 



n + ml (n + m-l)(n + m 2) ( 



Hence we see that the sum of this series = 1 . 



Also by a similar arrangement of the terms we have 



1 n m 



In -I)- -+('2n-'6)r- . . 



n + m l (n + m l)(n + m 2) 



_ , N 1 n m n 



= 1 + 2 (n-w)x -- = ! + - - = . 



2m m m 



Hence the odd terms of this series = - 



2 m 



and the even terms of this series = - 



2 m 



[By means of these series the simple values for I (Xydp. and 



i 



-i 



^)' 2 dp., as given above, were first obtained.] 



23. Now let us consider the application of the above investigations 

 to the determination of the numerical values of the magnetic constants 

 of terrestrial magnetism. For a given value of p. (i.e. for a given latitude) 

 we have a series of terms forming the coefficients of cos m\ and sin m\, 

 in the values of the magnetic potential and of the magnetic forces X, Y f 

 and Z, which are of the forms 



+a n ,z;;+&c., 



where a,,, a n] , &c., are the magnetic constants to be determined. 



552 



