94 
PROFESSOR A. SCHUSTER ON THE PERIODICITIES OF SUNSPOTS. 
years in the two portions of the time interval (see Table III.). This period wants 
farther careful study. The rise and fall of the sunspot curve during the 8-year 
cycle is shown in fig. 7, which is based on Wolf’s numbers, making use of all 
observations since 1749. 
16. We have now ascertained the existence of at least four periods, three of which 
have been determined with considerable accuracy. Their periodic times are 4‘80, 
8 '36, and 11*125. Taking frequencies into consideration in place of periodic times, 
we are led to consider reciprocals and thus find 
(11T25) -1 = 0*08989 
(8-36)" 1 = 0-11962 
Adding up, we find (4"77) _1 = 0"20951 
Hence the sum of the frequencies of two of the periods agrees within the limits of 
possible errors with the frequency of the third period. This relationship suggests an 
origin of the 4‘8 years’ variation, which may be similar to the origin of the vibrations 
of sound which go by the name of combinatioji tones. Two causes varying as cos n x t 
and cos n 2 t respectively give rise, if the effect depends partly on the higher powers 
of the cause, to oscillations varying as cos ( n 1 + n 2 )t and cos (n x —n^)t. The corres¬ 
ponding time of the second combination tone would be 33"38 years. 
But it is also found that the two first numbers are very nearly in the ratio of three 
to four, so that we may also express the three periodic times as sub-periods of 
33'375 years, thus: 
^x 33"375 = 11-125 
£ x 33-375 = 8-344 
}x 33-375 = 4-768 
How far this connexion is accurate or approximate is impossible to say at present, 
but the fact that the three periods which have been traced with certainty should also 
bear a remarkably simple relationship to each other is worthy of note. 
The question naturally occurs whether the great difference in sunspot activity 
observed when different maxima are compared with each other may be accounted 
for by the overlapping of different periods which sometimes will strengthen and 
sometimes weaken each other’s effects. That the difference is not exclusively due 
to such overlapping is proved by the evidence of the periodogram, which for instance 
teaches us that in the second part of the eighteenth century the 11-year period was 
reduced to a fraction of its normal value. 
Nevertheless it is true that in the exceptionally great outburst in 1870 nearly all 
the periods found seem to have been at their maximum phase. Similarly in the 
great outbreak of 1837 all but the 13"5-year period agreed in phase. This is shown 
by Table XVI. 
