In a paper contributed to the Hann celebration volume of the
Meteorologische Zeitschrift
, I endeavoured to account for a curious relation between the yields of wheat for successive years in the eastern district of England by referring the variation to periodic components. The relation referred to will be understood from an inspection of the table of yields on p. 75, or the diagram which illustrates it. The yields for the years 1896, 1897, and so on, taken in chronological order, apparently “compensate” respectively those for 1895, 1894, and so on, taken in reversed chronological order. Each pair of years at equal intervals, one before and one after 1895—6, gives a mean value approximating very closely to 30.8 bushels per acre, the average yield for the 20 years 1885 to 1904. This relation is further illustrated by the yield for 1905, which was not ascertained when the paper referred to was written (November, 1905). The data have been issued since then by the Board of Agriculture, and the yield for 1905 comes out at 32.0 bushels per acre, which “compensates” the yield for 1886, 29·2 bushels per acre, the mean being 30·6, surprisingly near to 30·8. The work of the paper referred to was based upon the supposition that a reversal of the kind indicated points to the yield being represented by the combination of a number of simple harmonic components, each having a node, ascending or descending, at the epoch 1895—6. At first I did not suppose that the components belonged to a harmonic series, and I tried to evaluate them by eliminating components of specified period, two years, four years, etc., by numerical process. The simplification produced by taking the mean of consecutive years, and thus eliminating the simple periodic term of two years’ period was considerable, and, in my contribution to the “Hann” volume, I dealt with the curve thus simplified. When I had selected periodic terms to give the best representation I could make of the curve of two-year means, one of the terms had a period of 11 years, and I could draw no practical distinction between the others and the harmonic components of a fundamental period of 11 years. In the result it was shown that the average yield for two consecutive year' yield of wheat in the eastern counties of England between 1885 and 1904 was represented with remarkable fidelity by the equation
y
= 31+2·8 sin 2π/11
n
+ 0·4 sin 4π/11
n
- 1·2 sin 6π/11
n
+ 1·2 sin 8π/11
n
, where
n
is the number of years, counting backwards or forwards, from the point representing the mean for 1895—6.
A New Way of Understanding why the Harmonic Series is Divergent
