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. 11
so that the periodograph is a straight line, parallel to the axis of T, the distance
between the two lines being inversely proportional to the nurnber of ordinates.
CASE 5. The function is given in the form of an irregular curve which satisfies
the condition that there is a definite law of probability that the quantity R should
lie within assigned limits; this probability being independent of the initial time r. If
we consider for instance the curve representing the height of the barometer, excluding
lunar and solar periodicities, the changes in the curve will apparently be quite irregular
but will satisfy the above conditions. Let A1 and B1 be taken to be components of
a vector defined by the equations
r~+inT T+ÏrnT
InTA f = f(t) cos ictdt, ~ nTBî = f(t) sin /ctdt.
7'+2mT T+2nmT
Similarly nTA2 = f(t) cos Ktdt, rnTB, = f(t) sin Ktdt,
T+îmT T+mT
T+smT rT+snmT
and so on until nTA, = f(t) cos ctdt, ~, nTBs = f f(t) sin ctdt,
r+ (s-1) mT r+(s-1)mT
with the condition that smn=n, m not being necessarily an integer number.
We may choose mT sufficiently large to secure complete independence of successive
vectors, all directions of the vectors being equally probable. In that case the vector R
which is the resultant of the separate vectors A, B, etc., will, as shewn by Lord Rayleigh*,
have a value such that the expectancy of R2 is proportional to the number S of vectors;
hence keeping mn constant and increasing S, the ordinates of the periodograph will vary
inversely with nT. This is the only general conclusion we can draw in this case.
CASE 6. The function f(t) is formed by the superposition of one or more simple
periodicities superposed on the irregular curve of case (5). This includes the important
cases of barometric, thermometric or magnetic changes. The Periodogram may in all these
instances be used to separate the real from the accidental periodicities. For the value of
the ordinates of the Periodogram has been shewn to be independent of the range of time
over which the integration is performed when the periodicities are real (Case 1), but to
vary inversely with the time when they are accidental (Case 5). Hence we may obtain
a conclusive criterion to distinguish between the two cases. The fundamental proposition
on which the separation depends may be stated thus:
T
The value of f (t) cos Ktdt fluctuates for the functions under consideration about
soie value which is proportional to T when f(t) = cos /t and proportional to i/T when
f(t) contains no real periodicity of periodic time 27r/1c.
* PF1il. lllag., Vol. x. p. 73 (1880).