DETEKMINATIOX OF THE GAUSSIAN MAGXETIG CONSTANTS. 117 



general terms of the coefficient of cos m\ in the potential function will be 



of the forna f'^^j+ /3„r") H^, 



and the corresponding coefficients in X, Y, and Z will be — 



in X= f ;;;.,+ yS„ 7-"-i j h,{n-m)IL'r' - ],{a + 7n)B.;r' ] ; 

 in (\-^^yY=(^^- +/8,.r"-^) m H^ ; 



If then, as before, we put r= 1, we shall have the final equation for 

 »„ as follows : 



= r X"' x,„ d^c + r Y;r 2/,„ c?^ + (h + 1 )|' H;r ",„ f/;/, 



where the coefficient of ^,i=0. 



And «„ r r (X;;0^- c^iu + r (Yr)2cf^-«(n+l)|' (H;;')-^ rf^'] 

 = r x;;' x„, df^ + r Y;r 2/,, rf/. - J' H™ =;,„ dfi, 



where the coefficient of a„ = 0. 



Hence «„ and /3„ are separately determined from the equations 



. ,, {n-m)\( n + m) ! 

 -">+^)[T:3 .5...(2n- l-)F 



= r X;r a;,,, dfi + {' Y,T 2/„ d^ + (n + 1)|' H^,„ d^i 



(n-m) l{n + Tn)l 



= rX-a;„tf;i+rY-2/„.(;/.-7J'H»s„c?u. 



Thus generally from the values of X and Y we derive 



/ -. x-, / i\ (n — m)\(n + m)\ 



(«,.+/3„)2«(n+l) ^A_^J__V^^__^-- 



= (2n + 1) rj X- x„, dfx + fc 2/,„f?/iJ 



