THE THEORY OF TERRESTRIAL MAGNETISM. 449 



3. In the following investigation of the coefficients of cos mX, &c., in 

 which TO remains the same, while n may have different values, it will be 

 convenient to denote H by H n , H' by H n ', X by X n , &c., regarding 

 H n ', X n , &c. as functions of /// or cos 9', where ff is the geocentric 

 colatitude. 



JTT I 



Regarding H n ' and . " , &c. as functions of p!, we have by Taylor's 



CtLL 



theorem 



to the order e, and 



.-- 



dp! ' dp > dp? 



Hence 



4. Expressions for the Magnetic Forces on the Earth's Surface. 



If a n and /S n be taken to represent magnetic constants depending on 

 the internal and external sources of magnetic force respectively, the co- 

 efficient of cosmX in the general term of the potential function V is 



The coefficients of cos m\ in the general terms of the values of the forces 

 X, Y and Z are : 



for 



for 7 2 r/ a "( n+1 ) _ a nr -A JT /] 



lor A, 2, i ^^ p n nr \t n . 



A. II. 57 



