DETERMINATION OF THE GAUSSIAN MAGNETIC CONSTANTS. 115 



since we have seen that for any value of m and different values of n and 

 »i, the value of 



[' H™H™c?^t = 0. 

 For the same reason 



j z:z:\df, = 0. 



Now let us consider the application of these formulae to the determination 

 of the numerical values of the magnetic constants of terrestrial magnetism. 

 For a given value of jj. {i.e. for a given latitude) we have a series of terms 

 forming the coefficients of cos mX and sin m,X, in the values of the mag- 

 netic potential and of the magnetic forces X, Y, and Z, which are of the 

 forms 



a„H™ + a„.H«+&C. 



a„X™+c,„,X™+<fec. 

 a„Y™-i-a„Y-+<kc. 

 a„Zr/+a„. Z™+&C. 



where a,„ a„^, &c., are the magnetic constants to be determined. 



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

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

 sponding to the numerical value taken for 8fi, these coefficients must be 

 equated to the values of the forces as derived from the magnetic observa- 

 tions 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 CTq + % cos X + b^ sin X + a., cos 2X -\- h.^ sin 2X -f &c. 



If we take x„^ to represent the coefficient of cos mX in the expansion 

 of the value of the force X for a given belt of latitude corresponding to 

 the colatitude d = cos~^/x : 



then, o^Xy^+a„^ X™ + a„^X™ -f &c.=a;„, 



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

 on one side the magnetic constants a,„ a,, , ifec, and on the other the 

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

 different belts of latitude f>-om the north pole to the south pole — i.e., for 

 all values of fi 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 89 has been taken as 5°, and the area of the belt is pro- 

 portional to 8/j. 



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 8^, 

 and multiplying the equation in X for each value of fx by X™, and 



I 2 



