EARTH’S MAGNETISM AT PAVLOVSK AND POLA (1897-1903). 
301 
phases must, of course, be kept separate, and to illustrate the seasonal changes I have 
divided the twelve months of the year into three groups, summer (May to August), 
the equinoxes (March, April, September, October) and winter (November to 
February)—a plan which seemed better, on the whole, than the adoption of four 
groups of three months each. 
During the seven years there were 28 group sums for each lunar phase during each 
season, and these were added together, the final sum representing about 100 days in 
each case (approximately 7 (years) x 4 (months) x 4 (days)). The monthly sums 
were also added together for the seven years, so as to show the monthly change in 
the semi-diurnal component of the variation ; by grouping these monthfy sums in 
seasons a check on the sum of the eight seasonal sums for the separate phases was 
obtained for verification of the computing. 
The twenty-four sets of inequality sums for the eight lunar phases during three 
seasons were then plotted, and 24 ordinates at equal intervals were read off from each 
of these curves (without smoothing), in order to replace the 25 values at intervals of 
one solar hour by 24 values, starting at lunar hour 0, at intervals of a lunar hour, 
or -ff- solar hours; this was done because the Fourier analysis of 24 values of a 
periodic function is arithmetically much simpler than that of 25. This analysis, as 
far as the first four harmonic components, was then effected. Only at this stage 
were means, and not sums, used, the resulting values of cq, b 1} &c., being divided 
by the number of days corresponding to the sums whose succession of values was 
harmonically analysed. This plan, besides being very economical of time, permits 
of more convenient checks being applied to the routine computations (as already 
mentioned) than would be possible if means were used in an earlier stage of the work. 
The eight values of a, b in the formula 
4 
2 (a n cos nt + b n sin nt) 
71 = 1 
deduced from the magnetic variation for each lunar phase were then all reduced to 
the corresponding values at new moon, by allowing for the progressive change of 
epoch of amount 2 (n — 2)-n- which the n th harmonic undergoes during each lunation.* 
The change of epoch in the interval (one-eighth of a lunation) between one phase and 
the next is \[n — 2) 7r, or —45°, 0, +45°, +90° for the first four harmonics in order. 
We will denote by a', b' the new values of a, b after this change of epoch has been 
allowed for by reducing a, b, to their equivalent values at new moon. Writing Jc for 
1 
v/2’ 
tor 
convenience, the table on p. 302 is useful for the conversion of a, b to a', V. 
Except for accidental error, and a small effect due to the varying distance and 
declination of the moon, the values of a! or b' in any one column should be the same. 
Their mean gives the resultant determination of the various components of the lunar 
* ‘ Phil. Trans.,’ A, vol. 213, p. 287. 
