436 THE THEORY OF TERRESTRIAL MAGNETISM. 



The numerical values of H, X, Y, and Z B m for different values of 

 n and m must be calculated, and in any belt of latitude of breadth 

 corresponding to the numerical value taken for 8/a,, these coefficients must 

 be equated to the values of the forces as derived from the magnetic 

 observations taken in that belt of latitude. 



The values of the magnetic forces X, Y, and Z are derived for every 

 10 of longitude and every 5 of latitude from the declination (8), the 

 dip (i), and the horizontal force (w), as given in the charts from which 

 the observations are obtained. These values of the forces X, Y, and Z 

 are analysed for belts of latitude 5 in breadth around the Earth's surface 

 by a formula of the type + a, cos X + 6, sin X + a, cos 2X + 6 2 sin 2X. + &C. 



If we take x m to represent the coefficient of cos raA. in the expansion 

 of the value of the force ^Y for a given belt of latitude corresponding to 

 the colatitude 



then a n X; + a^X^ + a n X + &c. = x m , 



where x m is derived from the observations. Similar equations, involving 

 on one side the magnetic constants a n , a ni , &c., and on the other the 

 values derived from the observations, must be formed for all the successive 

 different belts of latitude from the north pole to the south pole i.e. for 

 all values of p. between 1 and - 1 . 



The numerical values of X,"', X, &c., as well as the values of H (as 

 above defined), have been determined for every degree of latitude and 

 recorded for future use, but, in the actual determinations of the magnetic 

 constants which have been made, belts of latitude 5 in breadth have been 

 taken, or 80 has been taken as 5, and the area of the belt is propor- 

 tional to S/x. 



Supposing the observations equally distributed over the surface of the 

 globe, or supposing the weight of any determination proportional to the 

 surface of the corresponding element about the point of observation, then 

 the weight of each of the above equations is proportional to S/A, and 

 multiplying the equation in X for each value of p. by X, and summing 

 up the separate equations for the whole surface of the Earth, we get the 

 final equation 



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