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.

108 MR SCHUSTER, THE PERIODOGRAM OF MAGNETIC DECLINATION
The method has been suggested by the analogy between the variable quantities we
are here concerned with, and the disturbance in the luminous vibrations. If we could
follow the displacements in a ray of light, we should find them to present characteristic
properties not unlike those of meteorological variables. There is the same irregular fluctuation combined with a certain regularity of succession, which becomes revealed to us by
prismatic analysis, and shews itself in the distribution of energy in the spectrum. Absolute
irregularity would shew itself by an energy-curve which is independent of the wave-length,
i.e. a straight line when the energy and wave-length or period are taken as rectangular
coordinates, while the perfect regularity of homogeneous vibrations would shew itself as a
discontinuity in the energy-curve.
Fourier's analysis gives us a means of doing by calculation for any variation what
the spectroscope does experimentally for the luminous vibrations, and if we construct a
curve which represents the relation between the coefficient of Fourier's series for a given
period and that period, we have a simple way of representing the regularities of the
quantities to be investigated. We shall also incidentally gain the great advantage of
separating in a clear and definite way the fluctuations which take place in definite
periods, such as the lunar and solar variations, froin the more complex changes on which
they are superposed.
II. THE PERIODOGRAM.
Let f(t) be any function of t, and consider the quantity R determined by the
equations,r+nT 7+nT
-nTA f(t)costd, = nTB f(t) sin sctdt... (1),
R =A + B2.................................................(2),
where K = 27r/T and n is an integer. In these equations T represents a certain interval,
and, a time which can be varied. In the class of functions f(t) to which this paper
refers, a change in T with a constant value of n and T will cause R to fluctuate round
some mean value. Let S2 be the mean value of R2 which, still keeping n constant, will in
general depend on T. With T as abscissa and S2 as ordinate, draw a curve, which may be
called the "Periodograph." I define the "Periodogramn" as the surface included between
this curve and the axis of T. It will be seen that the "Periodograph" corresponds
exactly to the curve which represents the distribution of energy in the spectrum. The
treatment of a few special cases will render this clear, and lead gradually up to the
complex phenomena which form the chief subject of this investigation.
CASE 1. Let f(t) be a simply-periodic function, so that we may put
f(t) = cos (gt + S).