78 PROFESSOR A. SCHUSTER ON THE PERIODICITIES OF SUNSPOTS. 
but long periods of 17'1, 11 ' 2 , and 13’5 years, respectively, and these afterwards 
settled down to a fairly regular periodicity of ll'l years. It will appear that the 
periodicities in question are probably never quite extinct, but that the chief part 
m ruling the sunspot phenomena is sometimes taken up by one and sometimes by 
another period. 
During the 75 years to which the curve A applies the shape of the periodogram 
is very nearly that which a true and, in the optical sense, homogeneous period would 
give. In order to show this, I have marked in fig. 2 by means of crosses the points 
through which the periodograph would pass if the period was a simple* one of 
11125 years. These crosses will be found to lie close to the actual curve. A curve 
drawn through the crosses, marking the periodograph of a homogeneous period, would 
show a small but appreciable rise corresponding to the periods of 9^ and 14^ years, 
and rather less marked elevations for shorter and longer periods. These secondary 
elevations correspond to the diffraction bands seen on the two sides of the central 
maximum of a spectroscopic line. It has already been pointed out in § 4 that the 
phases of the periodicities of nearly 11 years are in arithmetical progression such as 
we should expect if the main period were homogeneous. 
Reference to Table IV. shows a decided maximum corresponding to a period of 
5‘625 years in the second half of the complete time interval. This is undoubtedly 
to a great extent due to the first sub-period of the llT25-year cycle, though the 
displacement in time is greater than it should be. This maximum seems to exist to 
a much smaller degree in the first half of the interval. Further features of the table 
and curves will be referred to later. 
The periodicity corresponding to the time of revolution of Jupiter (11 ‘86 years) 
has been examined, but reference to Table IV. shows that there is no evidence of a 
periodicity connected with the orbital revolution of that planet. In fact, we may 
definitely assert that no influence directly traceable to Jupiter exists. 
7. It has been stated that in the absence of definite periods the expectancy of the 
intensity of the periodogram must be obtained from the periodogram itself in all 
cases where the events to be analysed are not, as regards their succession, independent 
of each other. The expectancy not depending on the period, we may select for the 
purpose any portion of the curve in which we have no reason to suspect any 
periodicities. 1 he portion most suitable for this purpose in our case is that lying 
between 54 days and U5 years. Shorter periods must be avoided, because if the 
length is only a few days the intensity of the periodogram is depressed, owing to the 
fact that sunspots as a rule last several days. That this is so may easily be 
recognised by imagining all the sunspots to have the same duration, each spot 
also keeping the same area during the whole course of its life. It is obvious that 
in this case the period having a length equal to the life of the spots would be 
totally absent. 
* A simple period is one represented by a circular function. 
