74 
PROFESSOR A. SCHUSTER ON THE PERIODICITIES OF SUNSPOTS. 
The table shows in a remarkable manner how, quite independently of the intensity, 
the 11-year period ruled the periodogram during the interval considered. The 
phases <£ are in regular progression, the intervals between two succeeding ones being 
almost equal. The last column gives the phase of the true period. As the 
calculation is only strictly applicable to periods near the real one, we may take 10° 
as the phase of the period as determined from the table. This gives for the 
time ot the maximum 11T25 x -^j = 0'31 years, after the epoch which was 
1749'25. The times of the other maxima are therefore included in the expression 
1749’56±llT25?i, or, as we may also put it, 1905'31 ± llT25n, where n is an 
integer number. 
o 
5. In order to obtain the periodogram for periods having a length of several years, 
it is obviously necessary to use as long a series as possible. I have consequently had 
recourse to Wolf’s and Wolfer’s monthly numbers, beginning with the year 1749. 
I have used the numbers given by Wolfek as “ observed,” and not those obtained by 
a process of smoothing, which are given in a separate table under the heading of 
“ compensated numbers.” The process of smoothing I consider to be a harmful 
process, and quite unnecessary when periodicities are to be examined by a scientific 
process. 
Each year was divided into two equal parts, and the sums of the numbers for each 
half-y ear were calculated. These sums represented therefore six times the mean 
sunspot activity for the first and second halves of the year respectively. The values 
of A and B in the above expressions were then found. For this purpose the 
six-monthly values were, as is customary, arranged in rows containing as many 
figures as there are intervals of six months in the selected period. Thus, to get the 
value for twelve years, each row contained 24 numbers. The number of rows was 
arranged so as to get in as many complete periods as possible, beginning always with 
the total sunspot number for the first six months of 1749. We may therefore 
consider April 1 , 1749, to be the epoch which determines the phase of all the 
periodicities derived from the series (a). 
As twelve periods of twelve years bring us to 1894, there was not sufficient material 
to include a further complete period, and the calculations had to be stopped at that 
point. The sum of the twelve rows of 24 columns, in the example chosen, having- 
been formed, the expressions for A and B are easily calculated by substituting 
p = 288 (which is twice the total number of years included). The year being the 
unit, and the intervals between the times to which successive numbers refer beiim- 
six months, we must put a = 0 - 5. The ordinate of the periodogram will then 
be (A 2 +B 2 )/144. The factor in the denominator varies slightly for different 
periods, and sufficient accuracy is obtained by taking the divisor to be 150 in 
all cases. 
As the sum ot six successive sunspot numbers was used in place ot the average 
during the corresponding interval, the result must be divided by 36 in order to bring 
