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. 133
The amplitude was next calculated for this corrected period and its square entered
into the second section of Table XIV. The intensity now exceeds the expectancy, being
5'86 as great. There appeared also to be a minor tendency of groupings about the lines
B1B2 and (GC, and to bring the phases along these lines into agreement the corrected
periods were calculated to be 26'255 and 26'181. Table XIV. however shews that the
intensities corresponding to these times barely exceed the expectancy.
Plate II. gives similarly the distribution of phases for the 27 day period, the
straight lines along which there seems a possibility of clustering are marked on the
Plate, the corresponding periodic times being 27'061, 26'814, 27-327 days. The intensities of the two first of these periods are entered into Table XIV. It will be noticed
that the two periods which shew the greatest amplitudes are those of 26'814 and 25 809
days. As regards the latter, reference to Table XII. or independent calculation shews
that it will happen about once in every 350 trials that, owing to accidental circumstances, the square of a Fourier coefficient exceeds 5'86 times the expectancy. It
will of course be noticed that the period which gives the high value for the
amplitude has been selected with that special object in view, and regard must be had
to the fact that it represents the greatest intensity that can be obtained within the
range of periods extending from 25.5 to 27'5 days. The question how many independent
trial periods that range may be considered to contain may be answered by our previous
investigation (p. 130) from which it appears that two periods T and T' may be considered as independent when
T-T' 1
T' >4n'
n being the total number of periods included in T. For T= 27, n was 338, and hence
T - T' is almost exactly '02 day. As our range covered all periods between 25*5
and 27'5 days, we must consider that we have dealt with 100 independent periods and
found the two greatest intensities to be respectively 5'64 and 5'86 times the expectancy. What it comes to therefore is this, that 100 trials have given us one intensity
5'86 times the expectancy, while on the average this should only happen once in 350
trials. Or taking the two greatest amplitudes into consideration, it ought according to
chance to happen once in every 150 trials that an intensity of 5 times the expectancy
is found, while in the actual case this happened twice in 100 trials. It is obvious
that no conclusions as to the reality of the periodicity can be drawn from this argument. There are however two considerations which lead me to pause before finally rejecting the 25 809 period; the high amplitude is accompanied also by a considerable amplitude
of the half period, and if these half periods are plotted in a manner illustrated in Plates
III. and IV., it is found that a somewhat greater value is obtained if the time were altered
to 25'825 days. This however gives a decidedly smaller value for the main period (see
Table XIV.). The coincidence of two high intensities for a period and its semi-period
much increases of course the probability of its reality, but even if this is taken into
account, the excess of intensity over the expectancy is insufficient to establish the period.
The second consideration lies in the fact that the most definite result so far in the