DETERMINATION OF THE GAUSSIAN MAGNETIC CONSTANTS. 121 

 For the a class — 



X': 



1,(71-, n)ii";r' - M« + '")H';;' 



Y'"' = 

 " r"' 



1 



Z'"'= 



sin-^ + 'i>-,?H';rcos;^. 



Tim 

 -Hi 



\ 



For the jS class, which may be denoted by X"i',„ Y'!!,„ and Z'- 

 X'^',— r"-i[l(7i-?u)H'r ^-^(« + «OH'n"'] cos ./.-nr"-i H',? sin;/., 



Z'!!„ =-r"-'[i(«-m)H",r'-K'i + "i)H'r']sin;;/-nr"-iH';;'cos;/.. 



The numerical values of these expressions for all values of m from to 

 10, and for all values of m from 1 to 10 for the spheroidal surface of the 

 Earth, have been calculated from the values of n for every 5° of colatitude, 

 and form the coefficients of the magnetic constants gl\ JC, and (7™,,, h"l„ of 

 the a and ft class respectively in the equations for the determination of 

 these constants. 



The number of magnetic constants contained in these equations which 

 have been completely formed is thus 120 of each class, or 240 magnetic 

 constants in all, in place of the 24 constants of the a class which were 

 previously determined by Gauss. 



Regarding the Earth as a spheroid of revolution, the values of /^'=cos 6\ 



where d' is the geocentric colatitude, have been determined for every 

 5° of geographical colatitude. Also the values of cos \p, sin \b, -— , G',";*, and 



H'lf have been calculated for every 5° of geographical colatitude (i.e. for 

 the above values of /(') for all values of n and m from to 10. 



The weights of the observations of the magnetic elements for these 

 belts of latitude have also been determined on the assumption that the 

 weight is proportional to the area of the corresponding portion of the 

 Earth's surface. 



The values of H'™ as a function of the geocentric colatitude having 

 been determined for every 5° of geographical colatitude on the spheroid, 

 we next proceed to determine from them the values of X';;', X''i',„ Y',',", Y'!!!„, 

 Z'^, andZ"i'„,X'™(=X;;'cos;// + Z;rsin>/.), X"",,, Z'™(=-X;;'sin;// + Z;f cos>|/) 

 and Z"^,„ the resolved parts of the expressions for the horizontal and ver- 

 tical forces in the plane of the meridian on the spheroid. 



These values are required in the formation of the equations of condition, 

 and their numerical values are calculated for every 5° of geographical 

 colatitude as well as for the Equator and the Poles. These values of X',7, 

 Arc, have been calculated and recorded in tables for all values of h and m 

 from to 10, and have been employed as the theoretical coefficients of the 

 magnetic constants g'", li^, »kc., in the equations of condition. 



Formation of the Equations of Condition. 



When ii — m is even, the value of X" contains only odd powers of n, 

 and the values of Y™ and Z",' only even powers, and similarly when n—ni 

 is odd, the value of X;^' contains only even powers of yu, and the values of 



