xxii PREFACE TO PART II. 



The values of the coefficients of g cos m\ and h sin m\ in the potential 

 function and in the forces X', Y', and Z' are denoted by the symbols V, 

 X'^, Y'Z, and Z'f respectively. 



If r be the radius vector, //. = cos 6 and pf = cos 6'. 

 Then V": = ^H':, and F'_" = r" #'!' , 



-H 7 being the same function of // that // is of p. 



The expressions for the magnetic forces on the spheroidal surface of the 

 Earth will be as follows : 



Taking a n and /3 n to represent magnetic constants depending on internal 

 and external sources of magnetic foi-ce respectively, the coefficient of cos m\ 

 in the general term of the potential function V is 



-t-fi r n )H' m 



.ji + i Pn ' I ** n 



The coefficients of cos m\ in the general terms of the forces X, Y, and Z 

 are 



For F, 



Taking the equatorial radius =1, SS an element of the Earth's surface 

 and e the eccentricity, and taking into account only the terms to the 



order e; we have -=l+ey, sini// = e> (1 -/r)^ to the order e 2 , 



/*' = cos 6- sin sin i = *- 

 and 



also r 



and r"-' = i_!^: - 



