On periodicity in series of related terms

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 seriesu 1 ,u 2 .., Yule showed that the consequence of a single natural period is an equation likeux =ku x -1-u x -2+vx , wherevx represents the “accidental” external “disturbance”; and if there are two natural periods,ux =k 1 (u x -1+u x -3) -k 2 u x -2-u x -4+vx