DETERMINATION OF THE GAUSSIAN MAGNETIC CONSTANTS. 119 



ternal and external sources of magnetic force respectively, the coefficient 

 of cos m\ in the general term of the potential function V is 



The coefficient of cos »?2X in the general terms of the forces X, Y, and Z are — 



for Y, (^, -f- A. r'^-') '«(1 -t^")-' H';r, 



forZ, (^iB(^)-/3„.,^.r»-l)H'^ 



In the following investigation of the coefficients of cosmA, &c., in 

 which m remains the same, whilst n may have different values, it will be 

 convenient to denote H;;' by H„, X™ by X,„ <fec. We will denote the 

 corresponding quantities on the spheroid by H/,„ X'„, &,c., and regard them 

 as functions of ^', 6' being the geocentric colatitude. 



Taking the equatorial radius =1, 8S an element of the Earth's surface 

 and e the eccentricity, and taking into account only the terms to the 



order e^, we have — = 1 + eV^ sin (/^=:e7j(l— /j^)- to the order e"^, 

 r- 



fi' = cos 0— sin 9 sin 1/^ = ^i— 6^(1— P') -;"^ = 1 — e-(l — S/a^), 

 and -f^ = -'i^{\-eY-); 



also -l,^ = 1 + ^^ e>2, and r'-^ = 1 - -~i eV'- 



Regarding H'„ and "■ , kc, as functions of /./, we have by Taylor's 



theorem — 



H'„=H„-eV(l-/^'') ^^ to the order e\ 



djx' d^i ^v A^ ^ (7^2 



from which we derive the value of X„ for the spheroidal surface — 



and ^ -I' = " - ey(l - ^2) 



X„-(l-M } -^^ 



If now we substitute the values of X, Y, and Z in terms of H'„, 



d H' 



'-j-fi ^^-i in the equations — 



X'=X cos i/^ + Z sin ^, 

 Y'=Y, 



Z' =-X sin »// + Z cos ^Z', 



