DETERMINATION OF THE GAUSSIAN MAGNETIC CONSTANTS. 131 



Another method of testing the accuracy of the work of determination 

 of the magnetic constants is to substitute their values in the theoretical 

 expressions for X, Y and Z, and to compare the results with the values of 

 X, Y and Z as taken from the charts. 



For this purpose we have to form for each parallel of latitude the value 

 of the expression 



*K+«'m)=x"^!+x«^»+x»9« 



+ (Xlgl + X]g] + Xy,) COS \ + (Xlhl + XJAJ + XJAJ) sin X 



+ (Xtgt+Xlgl) cos 2X + {Xlhl+Xlht) sin 2 \ 



+ (X^H Xlgl) cos 3\ + (Xlhl + Xlhl) sin 3 \ 



+ X^^^ cos 4\ + X^6 cos 5\ + Xihi sin 4\ + X^hl sin 5 X ; 



and also the value of the expression 



i{a^-a'^)=X?^l+Xy,+Xtgl 



+ {X\g\ + Xlgl + X^O cos A + {Xlh\ + Xm + XWo) sin \ 



+ {X'^l+Xy,+Xlgl) cos 2\ + (Xlhl+Xlhl+Xlhl) sin 2X 



+ (Xlgl + Xlgl) cos 3X + {Xlhl + Xlhf) sin 3 X 



+ (Xtgi+Xtgt) cos i\ + {Xihi+Xthi) sin 4 \ 



+ X^^ cos 5X + Xtgl cos 6X + X^A^ sin 5\ + X«A« sin 6 X ; 



and then to form the sum and difference of these expressions for the 

 values of X in northern or southern latitudes respectively, which may 

 then be directly compared with the charts. 



Similar expressions must be formed in the same way for Y and Z for 

 each parallel of latitude, and their sums and differences taken as in the 

 case of X. 



When the values of the magnetic constants had been determined, they 

 were substituted in the equations of condition for each belt of latitude, 

 the terms of which when added up gave the theoretical value of the abso- 

 lute term for that latitude. This calculated value of the absolute term 

 may then be compared with the value of the corresponding absolute term 

 derived from the observation which has been used in the solution of the 

 equations. 



The following table gives some of these comparisons between the 

 calculated and observed values of the absolute terms of the equations of 

 condition for the period 1880, in the values of X and Z for gl\ for 

 m = and n odd, and for X, Y and Z for both g'jf and h]'^ for to = 1 and 

 n odd. 



The observed values are taken from the Admiralty charts, and are the 

 values used in the solution of the equations, and it will be seen by the 

 comparison of the calculated and observed values that a chart drawn to 

 give the results of the calculations would not differ much from the 

 Admiralty charts. 



As a further test of the accuracy of the work in such laborious and 

 extensive calculations, it is interesting to compare the values of the 

 Gaussian constants as determined by different investigators from the 

 observations for different epochs. 



k2 



