72 
PROFESSOR A. SCHUSTER ON THE PERIODICITIES OF SUNSPOTS. 
true periodicity. The extent to which this fallacious reasoning has been made use of 
would surprise anyone not familiar with the literature of the subject. 
What distinguishes the method which I am endeavouring to introduce from that of 
others, is the discussion of the natural variability of the Fourier coefficients according 
to the theory of probability, independently of any periodic cause which may have 
influenced the phenomenon. I have shown that where the phenomena are detached, 
and the probability of the occurrence of any one event does not depend on the 
occurrence of a previous one, there is a definite probability for the value of the 
amplitudes of the harmonic terms into which the recurrence of the phenomenon can 
be resolved. In the more complicated and more frequently occurring cases such a 
definite probability cannot be assigned a priori, but must be determined statistically 
from the phenomenon itself. Yet there is even in these cases a definite law which 
defines the manner in which the true periodicity gradually separates itself from 
accidental variations. The periodogram itself therefore furnishes the material for its 
discussion, but it is necessary that the investigation should be carried out systematic¬ 
ally and extend over a large number of periods. If the statistical data spread over a 
sufficient range of time, it will often be found convenient to divide the total interval, 
so as to discover whether the periodogram as determined from one half is similar to 
that as determined by the other half. Other examples are given in the succeeding 
investigation illustrating the manner in which a comparatively scanty material may 
be used so as to give the greatest possible information. 
I have so frequently insisted on the optical analogy that it may be worth while to 
point out that in one respect the periodogram furnishes more definite information 
than the optical instrument can. The spectroscope only determines the average 
intensity,” but the periodogram is also able to fix the phase of a periodic variation. 
Thus if the rainfall were analysed, the periodogram would show a maximum corre¬ 
sponding to the annual variation. With this maximum we may associate the angle 
determining the phase which will give us the date at which the maximum of the 
annual period takes place. 
In fixing the phase of a periodicity, some care is necessary if the trial period is not 
exactly coincident with the true period, as otherwise an appreciable error may be 
introduced. It is necessary to discuss this point in some detail. 
If in the expression of § 3 we write tan (/> for B/A, the angle d> measures the phase 
of the periodicity, supposing there is a true period having a time T = na. But if the 
true period is T v , the angle (j> does not correctly represent the phase. To determine 
the error we may use the equations which I gave in the paper published in the Stokes 
Volume of the ‘ Cambridge Philosophical Transactions,’ but the following short cut 
gives substantially the correct result. If we try to fix a simple periodic curve of 
period T, so as to make it most nearly agree with a periodic curve of period T' during 
an interval varying from r to r+nT, we must adjust their phases so that the extreme 
disagreement is as small as possible, and this is done by bringing the curves to 
