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.

124 MR SCHUSTER, THE PERIODOGRAM OF MAGNETIC DECLINATION
a true periodicity or may be accounted for by purely accidental causes. The principal
results arrived at may be shortly stated here, as far as they concern the present discussion.
The average daily value of magnetic declination, leaving the secular variation out of
account, oscillates round some average value. If / is the difference between any observed
value and its average, there will be some function f (/) such that f(3) d/ will represent
the number of cases in which the value lies between / and / +d/; for instance, if the
ordinary law of errors holds, the number of cases in which the deviation from the average
2hN
lies between f/ and fi + df/ will be -- e-h22 d/, where h is a constant and N the
NV7r
total number of days considered. In this case it is found that the probability that the
Fourier coefficient of any particular period lies between p and p +dp is
Nh2e —Nh2p2 pdp.
This expression holds on the assumption that the values on successive days are
entirely independent of each other.
The expectancy (E) of the square of Fourier's coefficient is in that case
p2.,hVe —^h pdp = N-2,
and the probability that p2 should exceed a value KcE is simply e-K. This latter expression still holds when the law of distribution is not that of errors, and even if the
successive daily values are not independent of each other, as is e.g. the case when
the causes which produce the deviations froin the average persist for several days. In
the last case the expectancy must be obtained by trial, the mean square of the Fourier
coefficients being taken. This expectancy, which according to our definition is the ordinate of the periodograph, should serve as the basis of any attempt to discover real
periodicities, and Table XII. will give at once the probability that a coefficient of the
Fourier series is due to a periodic cause and not to accident. If for instance the
square of a coefficient has been found to be equal to about twice the expectancy, we
obtain by the Table the value of e-' for c = 2 as 135, which means that in one case
out of about seven, accidental circumstances will cause the coefficient to be even greater
than this, and therefore no conclusion can be drawn as to a real periodicity.
When the square of amplitude which for shortness we may call the "intensity"
amounts to about 12 times the expectancy, the probability of mere chance is only one
in 200,000 and we may then begin to be fairly certain of a real effect, or if we are
satisfied with a probability of one in 1000, we may begin to count effects as probably
real when the intensity becomes equal to about 7 times the expectancy.
We may follow the theory of probability a little further in another direction; the
expectancy has in most cases to be determined by trial, and for trial, and for this purpose the mean
of a certain number of calculated intensities is taken. The question arises how many