PROFESSOR A. SCHUSTER ON THE PERIODICITIES OF SUNSPOTS. 
71 
counting, the sunspot activity is over-estimated when the activity is small, or under¬ 
estimated when it is large. The direct measurement of areas derived from photo¬ 
graphic records gives therefore a greater range of variability between the minimum 
and the maximum. A further systematic difference will be pointed out in § 13. As 
it was only necessary for my purpose to bring the different series into approximate 
coincidence, I have used the mean value of the ratios and multiplied Wolf’s numbers 
throughout by 12A. This factor, at any rate as regards order of magnitude, reduces 
them to the scale of sunspot areas. 
3. The mam object of this investigation was to determine the periodogram of 
sunspot variability. Let a function 6 ( t) of the time t take the values </> 0 , <f> a , &c., 
at equidistant values of the time t 0 , t 0 + a, t 0 + 2a, &c., and let it be required to 
calculate the periodogram of </> (t) by means of the separate equidistant values 
Put 
s = (n-l)a o_ 
A = v ^ 
V 
s = 0 
(b s cos — s ; 
n 
277 
s = (n — 1) a 
B = 2 <j> s sin —s 
s =0 n 
where n and s are integer numbers, and let 
S - (A 2 +B 2 ) a/p. 
i hen by definition the average value of S in the neighbourhood of a particular 
period na gives the value of the periodogram for that period as derived from the 
time interval pa. The different periods obtained by varying na should be chosen 
near together, but there is a limit beyond which it would be useless to go. This 
limit is leached when the values of A and B for two closely adjoining values n x and 
n 2 are no longer independent of each other. The theory of vibration shows that 
independence begins when there is an ultimate disagreement of phase amounting to 
about one quarter of a period, so that if r Ij and T 2 are the times of two periods and 
the total number of periods is N, independence begins when 
N (Ti-Tg) = TT , 
or when T : - T 2 is equal to T/4N, T being the approximate value of d\ and T 2 . 
4. The periodogram may be said to put the statistical material in a form in which 
it may be most readily discussed, but there may always be cases in which the inter¬ 
pretation is difficult. A few words on this point may therefore prove useful. I do 
not, of course, claim to have first introduced the application of Fourier’s Theorem to 
the discovery of hidden periodicities. Hornstein among many others made use oi 
the harmonic analysis, and obtained for the elements of magnetic variations the 
Fourier coefficients for periods in the neighbourhood of 26 days. The process is 
sufficiently obvious to have been frequently introduced, but it has generally been 
assumed that each maximum in the amplitude of a harmonic term corresponded to a 
* ‘Roy. Soc. Proc.,’ A, vol. 77, p. 13G (1905). 
