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.

130 MR SCHUSTER, THE PERIODOGRAM OF MAGNETIC DECLINATION
or writing = 2 ni
' nT sin 7 {s + K cos 2a
ni7 T sin 7 _t g -
Similarly b' T= in? 5 - sin 2am.
piT' 'y +g
The factor sin ry/y only having appreciable values when y is smali, the value of -g
K +g
will be small compared to unity, hence the sum of s terms containing that factor will
be small compared to s. This reduces the coefficients to
nsT sin y
pT' r
p is defined as the nearest integer to nsT/T', and as ris, the total number of
periods included, was about 350 in the cases to which the above investigation will be
applied, we may with sufficient accuracy write
sin y
The original function investigated cos Kt, having unit amplitude, it is seen that the
approximate method of calculation gives an amplitude which is reduced in the ratio
sin y/y or an intensity reduced in the ratio sin2 7/72.
A Table of sin2 y/y2 is given in Mascart's Optique, Vol. I., p. 324, from which it
appears that as y takes the values 15~, 30~, 45~, 60~, the function becomes -977; -912;
-811; '684. If it is simply desired to decide whether a period is real or accidental,
the intensity need not be accurately known, and we may allow ourselves considerable
latitude therefore in the value of y. If we fix the extreme value of that angle as 45~
which means a reduction of intensity of about 20 ~/, we obtain a relation betxveen T and
T', for in that case
T -T' 7r
y=7rn T < +
i:' 4'
T- T' i
or T' < +4n
If T is 26 days, and n=14, there being 14 periods of 26 days in the year, we
find that by the method indicated all amplitudes may be calculated which lie between
25-54 and 26-47 days. If the coefficients of the 26 day and 27 day periods are
known for each year we shall be able to calculate those of all intermediate periods
with sufficient accuracy, for the extreme reduction in amplitude when T - 7' =-day will
be *789, and it is only when the intensity comes very near the point at which it is
difficult to distinguish between real and accidental periods that this reduction will make
a material difference.