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. 109
The integrals A and B are easily calculated and expressed in the form
TA cos (a b)+ cos (a-b)sn gnT,
nTA= 2 L + sin ln. T
y__ c g- +s(
lTB = 2 Lsin (a + b) s sin izgT,
where 2a = ci + Ca, 2b = /3 + 3f2,
and al = T, 1 = -g + s,
2,= (r + fT), /2 = g (r + 2T) +.
Hence nTR = 2 sin gnT[2 (g + K2) + 2 (2 - K2) cos 2b]i.
12 - KC 2 '
If the average of R2 is formed for different values of T, the term containing cos 2b
will disappear, and therefore writing
ry = (g - ),T = T=7r -
it follows that
S = /2 (2+ tK2) sin y
g+K 7r
If?n is large, S will only have appreciable values when g and Kf are very nearly equal,
and in that case we may put with sufficient accuracy
S siny
This is the well-known expression, giving the distribution of amplitude in the
focal plane of the telescope, when a homogeneous vibration is examined by means of
a prism or grating. If we wish to plot down the curve of intensities of vibrations as
analysed by a grating-spectroscope, we may define any direction by the period 27r/Kll which
has its principal maximum in that direction. If the incident light has a period 27r/g
the expression for the distribution of amplitude is
sin [7r Nr (g - K)/K] *
73r (g-K)/K
which is identical with S if V, the number of lines on the grating, is equal to n,; the
number of periods included in the integration. In obtaining the "Periodogram," we have
done by calculation precisely what the spectroscope does mechanically. The analogy is
complete, and just as a ray of homogeneous light does not appear homogeneous in a
spectroscope, there being secondary maxima owing to the finite resolving power, so does a
purely periodic function when analysed by Fourier's series shew apparent periodicities
* This expression may be obtained either from the in my paper "On Interference Phenomena," Phil. Mag.
original papers by Lord Rayleigh on the resolving powers Vol. xxxvnI. p. 509 (1894).
of spectroscopes, or more directly from an expression given