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.

110 MR SCHUSTER, THE PERIODOGRAM OF MAGNETIC DECLINATION
having secondary maxima near the principal one. These secondary maxima I have termed
" spurious " periods.
Their intensity remains the same when the "resolving power" n is increased, but
they approach nearer and nearer to the principal maximum. They are therefore distinguished from the true periodicity by the fact that their position changes with n.
CASE 2. The function to be analysed consists of two overlapping simple periodicities.
The integrals A and B will split up into two parts which we may call AI, A2,
B1, B2 respectively. Hence
R2= (A, + A2)2 + (B1 + B2)2.
The products A1A2 and B1B2 will vanish in the expression for S2 when the
average is formed for varying values of r. Hence
S2 = A 2 + B12 + A,'2 + B,22 = R12 + R22,
or the Periodogram of two simple periodicities may be formed by the superposition of
the separate periodograms *.
CASE 3. The function varies uniformly with the time. Putting f(t) = t, and performing the integrations, it is found that
2c 2c
A = sinr c; B = - - cos cr,
K Kf
R2 = S2 = 4c2/K2 = 2T2/r2.
Hence the Periodograph is a Parabola.
The consideration of this case, which has no analogy in the analysis of luminous
disturbances, is of importance in the treatment of secular variations, such as that of the
magnetic elements.
CASE 4. So far the function f(t) has been taken to be continuous; but cases
arise, where f(t) is given numerically for a number of values of t, which we may for
the sake of simplicity assume to be equidistant. As Fourier's analysis applies also to
discontinuous functions, we may include cases of this kind. Let the different detached
values of f(t) follow the law of errors so that, NT being the total number of ordinates, the
2hN
number having a value intermediate between / and /3 + d is e-hI82 d8. I have
27r
shewn t that in this case
S h 2
Nh2 '
* In my paper " On hidden periodicities " (Terrestrial change to the latter form is apparent from the above.
Mag'netisnm, Vol. III. p. 13) I defined the ordinate of the t On the investigation of hidden periodicities, loc. cit.
Periodogram to be S instead of S2. The advantage of the