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. 123
the proper factor, give the Fourier coefficients. The average square of amplitude for
the year was found to be *003460 and this has to be divided by 25 to get the
ordinate of the periodograph for the 25 years interval. The number 1384 x l0-6 so
obtained is almost identical with that previously found for the 5'4 day period, which
tends to shew that for short periods the expectancy of a Fourier coefficient is independent of the period. Fig. 1 gives the shape of the Periodogram for periods up to 30
days. The vertical ordinates give the heights actually determined, while the curve is
drawn continuously so as to pass nearly through these points.
For longer periods the monthly averages, as published in the Greenwich records,
served as basis of calculation. To obtain the coefficient of the annual period, the
interval of 25 years was divided into 5 groups of 5 years, and the harmonic analysis
was applied to each of these 5 groups. The average square of amplitude then gave the
ordinate of the Periodograph for a range of 5 years, which has to be divided by 5 in
order to reduce it to our normal interval of 25 years.
Periods of 11 and 13 months were treated similarly and the coefficients obtained
for 5 groups of 55 months and 4 groups of 65 months. The average squares of amplitude have in these cases to be divided by 60/11 and 60/13 to reduce to the normal
interval. The results are given in Table XI., and it will be noticed that the PeriodTABLE XI.
Period in Months S12 S22 S2 S42 S52
11 -04591 -00475 -00158 -00079 -00054
12 -08828 -01610 -00842 -00287 '00218
13 -09344 -01082 -00891 -00237 00196
Average -07588 -01055 -00630 -00201 -00156
Period in Months 12 6 4 3 24
ogram continues to increase rapidly with increasing lengths of period. The conclusion
we must draw from the curve in Fig. 1 and the figures of Table XI. is, that the
causes which produce the variations of declination are on the whole persistent in
character, so that the variations of short periods have on the average a much smaller
amplitude than those of longer periods.
IV. APPLICATION OF THE THEORY OF PROBABILITY.
In a previous paper* I have applied the theory of probability to the solution of
the question whether the value of any particular coefficient of Fourier's series indicates
* Terrestrial Magnetism, Vol. III. p. 13.
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