THE THEORY OF TERRESTRIAL MAGNETISM. 403 



in the value of Z 



= 2'r"- 1 [ - nH?] (g"l n cos mX + / sin mX), 



where g m _ n , h m _ n are the magnetic constants for positive integral values of m 

 and n. 



2. Let V" ~r-H!!' and V=r n H, 



also let F; = --, '#? (1 - /*')" and F = r-' m 



and let Z; - ? -^ +2 (w + 1) H? and Z!' n - r"- ( - n#;). 



Then taking both classes of terms together we have 



V= 2 { F; (rjH cos HiA. + /C sin mX)} + 2 { Fr B (/_',, cos mX + # sin mX)}, 

 X = 2 {Z: (^' cos mX + /i, 1 ; 1 sin mX)} + 2 {^ (.9"_' K cos mX + ^ sin mX)}, 

 y = S { 17 (fl' sin mX - /i^ cos mX)} + 2 { F*. (^ sin mX - A*, cos mX)}, 

 Z = S {Z (#;;' cos mX + ^ sin mX)} + 2 {Z M (g H cos wiX + A, sin mX)}. 



Collecting coefficients of cos m\ and sin m\, 



the coefficient of cosmX in V is 2 (P7gr+ F,^,,), 

 .................................... ^ is 2,Y 



.................................... Z is 2(3T< 



the coefficient of sin mX in F is 2 ( F;;'/C + 



.................................... X is SA 



in which n takes all integral values for a given value of m. 



The relations between the functions when the suffix is changed from 

 n to n are 



n + 1 

 On the surface of a sphere of radius unity F and F"' w are each of 



5 



them equal to H, i.e. to G (1 /* 2 )% and it will be convenient to express 

 their values in terms of p. the cosine of the colatitude of a point on 

 the surface of the sphere. 



512 



