THE THEORY OF TERRESTRIAL MAGNETISM. 453 





' m n , and 



or 



These formulae may be simplified in the cases when m=n, and when 

 n I. 



When m = n, we have 



F: = L(l-^)^( S T, and F-=r"(sin0'r, 



_ _ 



** n+2 > anu ' J - n nr (sin v ) , 



and F- = m*- (sin tf')"-, 



X: + ?! ': = 0, and *_". + / F_" - 0. 



When m = u 1 , we have 



'.7 =/', and H'r=/(l-/'F=/( ')"", 



*T- (n+1 > "-?. and ^-=- B ^y (s in^-., 



y ._(.-l)/(.in) 



_ n (sin y)" - (n - 1) (si 



" 



sn 



and JT. m n = r n ~ l [n (sin ^)- - (n - 1 ) (sin 0')"- 2 ]. 



The above formulae have been employed to determine the numerical 

 values of X?, Y and Z n m and of JT_, Y^ and Z_ m n , and also the values 

 of X^, Z^ and of X'_ n , Z'_ n , for every degree of the geographical colatitude 

 over the surface of the Earth. 



8. We will now give a more complete investigation of the case of 

 the spheroid. 



For a given value of //, i.e. for a given narrow belt of latitude, we 



