Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

FROM RECORDS OF THE GREENWICH OBSERVATORY, 1871-1895. 117
that such variations are not more than we should expect. Assuming the ordinates of
the Periodograph to vary uniformly between the periods of 24 and 30 days, we obtain,
by taking the mean of the columns of Table V., the ordinate S2 of the Periodograph
corresponding to a period of 27 days. The value of S, or the amplitude of mean
square, i.e. the square root of the expectancy of R,2, is thus found to be 0' 0317 (see
Table V.). This therefore is the order of magnitude we should expect for the amplitude,
TABLE V.
Days in Period 12 l Rs2 R 3 1/42 R5
24 946-9 x 10-6 290-7 x 10-6 176-5 x 10-6 281-9 x 10-6 76-5 x 10-6
25 11-7 205-1 189-6 416-4 146-0
26 1392-4 434-5 125 3 134-2 121-7
27 1099-7 744'3 448-7 204' 2 23'5
28 657-6 396-2 80-2 284-4 89'8
29 225 0 299'9 133-8 39-0 171-1
30 2705-8 293-3 234-0 7 4 143-9
Mean (S2)_ 1005-6 x 10-6 380-6 x 10-6 198-3 x 10-6 195-4 x 10-6 110-4 x 10-6
S= 0'0317 0'0195 0'0141 0'-0140 0'0105
if Fourier's analysis is applied to a record of 25 years of Greenwich declination, the
period being in the neighbourhood of 27 days. As the expectancy of amplitude varies
inversely with the square root of the time-interval, the expectancy of amplitude is as great
as 0'1585 for a single year's record.
The ordinates of the Periodograph may be obtained in another way, agreeing more
closely with the theoretical definition given on page 108. If each of the rows of
Table I. is separately treated by Fourier's analysis, and the coefficients afterwards are
divided by the number of periods included in each row, we obtain the amplitude of the
24 day period for each year; the mean square of this amplitude is the ordinate of the
periodograph for the interval of one year. It was considered sufficient to confine this
method of treatment to the 26 and 27 day periods. If Fourier's series is put into
the form
ri cos (/t - g1) + r cos (2ct - b2) +......,
Table VI. gives the values of r2, r2...... r2 for the 26 day period, Table VII. the same
values for the 27 day period, and Table VIII. the angles 01 and P2 for the same periods.