Abstract
An important extension of our ideas regarding periodicity was made in 1927 when Yule pointed out that, instead of regarding a series of annual sunspot numbers as consisting merely of a harmonic series to which a series of random terms were added, we might suppose a certain amount of causal relationship between the successive annual numbers. In that case the system might be regarded as a physical system possessing one or more natural oscillations of its own, all subject to damping; and the effect of annual random disturbances would be to produce a fairly smooth curve with periods varying in amplitude and length, essentially as the sunspot numbers vary. If we call the departures from their mean of our series
u
1
,
u
2
.., Yule showed that the consequence of a single natural period is an equation like
u
x
=
ku
x
-1
-
u
x
-2
+
v
x
, where
v
x
represents the “accidental” external “disturbance”; and if there are two natural periods,
u
x
=
k
1
(
u
x
-1
+
u
x
-3
) -
k
2
u
x
-2
-
u
x
-4
+
v
x
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