xxiv PREFACE TO PART II. 



observations of declination, inclination, and horizontal force are analysed for 

 belts of latitude 5 in breadth around the Earth's surface by a formula of 

 the type 



a + ! cos X + b l sin \ + a. 2 cos 2 X + &., sin 2X + , &c. , 



and the coefficients of cos mX, sin mX, in this expansion are equated re- 

 spectively to the coefficients of cos mX and sin mX in the expansion in terms 

 of the potential function and magnetic constants as given above : thus 

 for the force X, if a n , a, (i , a (l ,, &c., stand for the magnetic constants, and if 

 x m ' be the coefficient of cos mX as derived directly from the observations, 

 then 



a. X": + a ni X': + a,, X' + , &C. = x a ', 



and similar equations are obtained from the expressions for the forces 

 Y' and Z'. 



The values -X 7 , Y'" t ', and Z'", taken in these equations, are the values 

 derived for the spheroidal surface of the Earth from the potential function, 

 and these equations include not only the magnetic constants which were 

 determined by Gauss, of the class indicated by a in the above equation, 

 but they also include magnetic constants which may be spoken of for 

 distinction as the ft class (i.e. including, e.y., the class answering to forces 

 of external origin), those forces which depend upon sources outside the 

 surface of the Earth. 



The full values, then, of the coefficients of the magnetic constants will 

 be of this form : 



For the a class 



cos * + '-+. 1 //': sin ,,, 



Z':=~ r (n - m) HT ~(n + m) # T' sin ,/, + H>: cos *. 



For the /8 class, which may be denoted by X' n , Y"" n , and Z' n 

 X'l\ = r^ (n - m) H':* - l - 



cos ^ - nr: sn $, 



Z'1 1 H = - r"- ' g (n - TO) H':* - 1 (w + m) H'^l sin ^ - nr n ~ l H': cos ,/,. 



