20 
DR. S. CHAPMAN ON THE SOLAR AND LUNAR 
by the same potential function if more harmonics of higher degree were included. But 
I doubt whether any improvement thus made would be very substantial or of real 
value, and I have therefore judged it best not to discard the above simple and success¬ 
ful representation based on the West force variations alone. It may be added that if 
an independent attempt were made to determine, from the North force variations 
alone, the values of the harmonic functions present in them, of the type which 
represent the West force data, the results would differ little from those actually 
calculated from the latter. 
The notation of the harmonic functions used in this analysis is described in § 9. In 
that notation a function 
( 2 ) (A m "cos nt + B„,” sin w£) Q m ”(cos 0 ) (n = I, 2 , 3. 4) 
was used in Tables III. (a) and III. (/3) to represent the variations (a n cos nt + b n sinnt) 
at the individual stations of various co-latitudes 6. The value of m in each case was 
found to be n+1 . In Tables III. (y) and III. ($) two such functions, corresponding to 
m = n and m = n + 2 , were used in each instance. The constants A m n and B, rt “ are set 
out in Table C § (9). 
As previous investigations have indicated, both inside and outside causes contribute 
to the magnetic field at the earth’s surface, so that the vertical force variations cannot 
be deduced theoretically from the horizontal force variations. On the contrary, the 
potential function (if such exists), which represents and is calculated from the vertical 
force data alone, affords the means of separation of the respectively external and 
internal parts of the whole variation field. It proved on examination that 
functions ( 2 ), of precisely the same type as were used for the horizontal-force data, 
serve likewise for the vertical-force variations, only the numerical coefficients (denoted 
in this case by A,/ and B,„") being different. These also are given in ’fable C, along¬ 
side the values of A m n and B m ". The corresponding calculated values of a n and b n are 
given in Table III. for comparison with the observed data. 
For the purpose of the subsequent discussion it was clearly advisable that A m n 
and B,„", A ((1 ” and B, ;1 ” should be determined on some definite plan which would at 
least give results which were comparable in the different cases. Where only one 
harmonic function was involved in the representation of a given set of Fourier 
coefficients, as in Tables III. (a) and III. (/3), the course adopted was very simple. 
The weighted mean of the various values of the function Q m ", corresponding to the 
mean latitude of each of the nine groups of observatories, was taken numerically , 
i.e., negative values being treated as positive ; the similarly weighted sum of the 
group mean values of a n (or b n ) was also taken, the signs being reversed where this 
had been done for the calculated (negative) values. A simple division then gave the 
required coefficient A„”, B m ”, A,/, or B,»". As regards the weighting, the Northern 
group means were each given unit weight, and the Southern means each half a unit 
