134 REPORT— 1898. 



then we may find the final equations for a„ and 5„ from the final equations 

 for a„ and B„ respectively by multiplying the final equations by ^, and then 



substitute the values of a„ and y3„ in terms of a„ and b„ respectively. 



Hence, in the final equations for a„ and /3„, the coefficient of a„ or of fi„ 

 will be multiplied by 



(^Yor ( l-3.5----(2.- l)y 

 VhJ V (n-m)! ; 



in order to find the coefficients of a„ and 6„ respectively. 



Also the coefficients of a„_2 or of )8„_2 in the same equations will be 

 multiplied by 



Q,^:Q^ -r 1.3.5 ••••(2n-l).1.3.5--- -(2^-5) 

 H„.H„_2 (n-m) ! ' (n-m-2) ! 



to find the coefficients of a„_c, and 6„_2 respectively. And the coefficients 

 of a„^2 and ^,,+2 will be multiplied by 



Q.Qn.2 „^ 1-3.5 • • • • (2n-l).1.3.5- • • •{2n + 3) 

 H„H„,2 («-m)! {n-m+2)l 



to find the coefficients of a„+2 ^.nd b„^.„. 



Or generally the coefficients of a„_ and ;8„^ in the final equations for 



a„ and )8„ will be multiplied by ^"' Ji"' to find the coefficients of a„ and 



6„^ in the corresponding final equations. 



Hence the constants a™ and 6" will have to be multiplied by ^'^, i.e. 



by the factor " ^ ^— -, in order to obtain the corresponding 



Gaussian constants n„ and /3„. 



Again, let A„, B„ be two magnetic constants connected with a„ and /3„ 

 by the relations 



"j. _/^«_n„_ 1.3.5 .... (2.1-1 ) 

 a: B„ H„ [{n-m) ! (n + m) if 



Then the values of the magnetic constants A,„ B,„ <fcc., as determined 

 from the function IT,, can be converted into the corresponding Gaussian 

 magnetic constants derived by means of the function H„ by multiplying 

 each magnetic constant A^ or B™ for each value of 7?i by the factor 



n:;'_i .3.5 • • • •_(2?i - 1) 

 H™ [{n—7n)\{n + myY 



In his paper in Vol. I., No. 1 of ' Terrestrial Magnetism,' published 

 January, 1896, at the Chicago University Press, Dr. Ad. Schmidt has 

 introduced .-i symbol R;;,, which is connected with the symbol IT™ employed 

 above by the relation 



R;' = s/(27j + l)en™ 



where £=1 when )/i=0, and €=2 when m >0. 



