PROFESSOR A. SCHUSTER ON THE PERIODICITIES OF SUNSPOTS. 
75 
it to Wolf’s scale, and multiplied by the square of 12’53 # or 157, in order finally to 
reduce the result to the unit based on the measurement of areas. The method is 
available for all periodicities containing an integer multiple of the half year. In some 
cases it was advisable to obtain the periodogram for periods lying halfway between 
those to which the above method is directly applicable. The successive rows in the 
original table were then arranged so as alternately to contain a number of figures 
differing by one. Thus, for a period of 7\ years the alternate rows were formed of 
15 and 16 figures. This gives 31 intervals of six months for two complete periods, or 
on the average 7f- years. In the last column alternate numbers were missing, and 
this column was omitted m the calculations for A and B, the number n bemc chosen 
to correspond to the number of columns retained. In some cases it is more convenient 
for the tabulation of the angles and of their trigonometrical functions to include the 
last column, which must in that case be multiplied by a factor correcting for its 
smaller number of entries. When a certain period has been treated, and the half or 
the third of that period is to be introduced, it is not of course necessary to form again 
the system of rows and columns, but we may proceed as in the calculation of the sub¬ 
periods of Fourier’s coefficients. The equations are then replaced by 
.5 = (n — 1) a 
A = 2 <f) s cos 
5 = 0 
2mv 
-5; 
n 
s = (n — 1) a 
B = S <j) s sin 
s = 0 
2m7r 
- 5 , 
n 
where m is two or three, according as the half or the third of the original period 
is required. (See Table IV., p. 97.) 
6. The second column of Table IV. contains the ordinates of the periodogram, 
calculated from the complete record in the manner described, and in fig. 1 the 
periodograph is drawn. The great intensity of the 11-year period is well shown, 
but the shape of the curve for periods between 10 and 11 years and some other 
features seemed to require further elucidation. I therefore divided the whole period 
of 150 years into two portions, which were nearly equal to each other. This could 
be done accurately when the original table contained an even number of rows, but 
when the number was uneven, so that the two portions could not be made exactly 
equal, the first portion was always taken to be the shorter one. In consequence, the 
epoch of the beginning of the second portion varies somewhat; the exact dates are 
given in column 5 of the table, the Roman numeral attached to the year indicating 
whether it is the first or second six-monthly value which forms the beginning of the 
group of periods. Where no such Roman numeral is given, the sunspot numbers for 
the whole year were used in the calculations. 
For some purposes it would have been better to have chosen a fixed epoch for the 
* The general mean during the years 1832-1901 inclusive as given in § 1 is 12'50. When this investi¬ 
gation was commenced the areas for the year 1901 were not available. The number given in the text 
above, which is the mean found when 1901 is excluded, was used in the calculations. The difference 
is of course insignificant. 
L 2 
