120 REPORT— 1898. 



the expressions for the magnetic forces become — 



^'= {":U + ftr ') '^^r (1 -l^'f COS'/' 



4.r("±]]fL" - n ft,/'-'"'] H'„ sin \// + similar terms 



Y' = (";,- + /V""') «iH'„(l-;t2) -5 ^_ similar terms, 



+ pi'Jl^]fL" _ n (iy'-^~\ H'„ cos ./; + similar ter; 



ms. 



In these expressions for the magnetic forces the values of H',„ —j—", 



(J IT 



&c., in terms of H„, ,-", 6:c., are substituted for each belt of latitude, and 



af.1 



these theoretical expressions derived from the potential function for a 

 given belt of latitude, and containing the magnetic constants, are equated 

 to the corresponding coefficients derived from the magnetic observations 

 taken in that belt of latitude. 



In the case of the spheroid, as in the case of the sphere, the values of 

 the forces X, Y, and Z derived for every 10° of longitude from the obser- 

 vations 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 — 



«o + ai cos \ + bi sin X-f-ff^ cos 2X + b.2 sin 2/\ + 4fec.j 



and the coeflScients of cos vi\, sin viX, in this expansion are equated 

 respectively to the coefficients of cos m\, and sin ?>i\ in the expansion in 

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

 thus for the force X, if a,„ a„^, a„, <tc., stand for the magnetic constants, 

 and if .x',„ be the coefl&cient of cos m\ as derived directly from the obser- 

 vations, 



then a„ X';;' + o,„ X'- + u„„_ X',™ 4-i'c., =a-„„ 



and similar equations are obtained from the expressions for the forces Y' 

 and Z'. 



The values X',7, Y'ij', and Z'", taken in these equations, are the values 

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

 and these equations not only include 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 /? class, i.e. including those forces which depend upon 

 sourc&s outside the surface of the Earth. 



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

 be of the following form : 



