116 HEPORT— 1898. 



summing up the separate equations for the whole surface of the Earth, 

 we get the final equation — 



«„ r (X;';)H'+"».r xr,x:dfi+&c.=? X';;..v,„diu 



Similarly, the final equation for «„_ is found by multiplying the above 

 equations by X;", Y"', and Z[\\ respectively, and we get 



of' x;rx™cz^i +a,{' {X';,\yd^+S:c.= f' x:,\r,„df,. 



In the same Avay, if //„, denote the coefficient of sin m\ or —cos m\ 

 in the value of the force Y as derived from observations, we have 



5KY„).= ?/„„ 



and the final equations for finding a„ and a,,^ respectively will be 



a„ r {Y'::y-d^i + a„. T y™ y^. rf^ + .tc. = [' y;;' y,„ d^., 



and o„ r Y;;' Y;;; d,i + a„, P (Y^y d^i + &c. = P Y;;; y„. d^,. 



Combining the final equations for a„ from X and Y together, we have 

 a„ r [(X;;-)-^ + (Y;;')^J dfi={' X« x,„ du + [' Y;r y,„ d^., 



since the coefiicients of f>„^ and all the other terms on the left-hand 

 side of this equation vanish when the integration is taken all over the 

 Earth's surface. 



Hence a„ . n{u + 1 ) f (H;r)- du. = f X™ x-„. djx + f Y;;' y„. d,, ; 



T / , i\ (n — m)\(n + m)': 



"" ^ -"(" + ') [I.3.5.. ii v:^n2n + 1)- 



= rX,Ta;„.c?/x+rY;,"2/„.c?;/. 



Similarly, by putting 7i^ for ti, we may get the value of a,,^. 

 In the same way the final equation for finding o„ from the equations 

 for Z would give us 



. o„ r (Z;r)2 dfc + aS Z™ Z- c?;* + &c.= [' Z"' -,„ .Z/< ; 

 or a„{n + 1 y r (H;r)-' cZ/i = T Z;r ;:,, rf/<, since P Z;; Z;» cZ;* = ; 



" ^ ^ [1. 3. 5. ..(2tc-1)]2(27i+1) J_, " " '^ 



If we take into account separately the parts of the magnetic force at a 

 point due to the internal and external centres of magnetic force, the 



