304 
DR. S. CHAPMAN ON THE LUNAR DIURNAL VARIATION OF THE 
§ 9. The next six tables, II. to IY. (a and b) are more important, being derived by 
the separation of the lunar phases, enabling us, as already explained, to determine 
the other harmonic components, of changing epoch, in the lunar variation. The 
values of the coefficients a n and b n in the formula 2 (a n cos nt + b n sin nt), for the first 
four components, are given after reduction to the same lunar phase (new moon). No 
corrections to amplitude or epoch (§§ 5, 6) have been made at this stage, nor are the 
results transformed into the geographical components of force, as the purpose of these 
tables is to show how far the results from the different lunar phases are in agreement. 
Apart from accidental error any eight corresponding values of a or b should be equal 
(except for the small effect of the varying distance and declination of the sun and 
moon). Judged by ordinary standards the agreement of the numbers in any one 
column is in general not at all good, even in the case of the larger amplitudes ; but 
for work of this kind, where the effect sought after is so small in comparison with 
the superposed regular and irregular variations which have to be eliminated in its 
determination, the results may be regarded as satisfactory considering (i) the extreme 
smallness of the unit, and (ii) the short period (on the average equal to 100 days) from 
which each of the values in Tables II. to Y. is determined. Occasionally there are 
largely outstanding values, and it would seem necessary to use a much larger series 
of observations if close accordance of the values for the separate lunar phases were, 
desired. I hope, however, that the mean coefficients deduced from the combination of 
the eight phases are reasonably free from accidental error—the parallelism of the 
results from the two observatories, which will be pointed out later, confirms this to 
some extent. 
§ 10. In order, however, to examine this point more closely, I have derived the 
values of the probable errors of the mean values of a n and b n in each case from the 
discordances from the means in Tables II. to IY. # The results are given in Table Y., 
those for declination (Tables IIIa. and IIIb.) having for convenience of comparison 
been converted into force units. The essential features of the table may be summar¬ 
ized thus : the sum of all the probable errors for all the elements in summer, equinox, 
and winter are 293,276 and 333 respectively, so that the season has no great 
effect upon the magnitude of the probable error ; but as the variations are generally 
greatest in summer, the values of a, b for this season are in general proportionately 
the most accurate. Summing the errors for all seasons and components, but keeping 
the results for each element and observatory separately, we find the following :— 
Horizontal Transverse Vertical 
force. force. force. 
Pola. 171 150 108 
Pavlovsk. 163 191 119 
* The probable error of the mean of n observations is taken to equal 0"845 / Jn- 1 times the mean 
discordance from the mean. 
