![]() ![]() Real times appear to range from 4 to 8 months. It is then readily possible to express in mathematical form the distribution of migration movement in time by adding the first harmonic (sin 2x), when the migrant spends in the north a period appreciably longer or shorter than 6 months. Since it is relatively fixed in time, it forms a convenient zero of coordinates, and, in order so to use it, it is convenient to express the cosine-power curve as a sine-power curve (O ° x ° p). It is found that for most species this bisectrix coincides quite closely with the peak-of-heat (or cold) and comes about a month after the peak-of-light (or darkness). Different species of birds spend very different periods of time in the north, but in general those that go early return late, and the bisectrix of the interval between migration peaks tends to be fairly constant for nearly all species. The cosine-power law is a well-known statistical distribution, being the trigonometrical form of Karl Pearson's Type II statistical distribution curve, which is very similar to the Gaussian for high values of m, but extends only over a finite range (– p 2 ° x ° p 2). ![]() A more suitable theoretical form of graduating curve is the cosine-power distribution, whose parameters can be ascertained from the Gaussian parameters obtained from the probability-paper plots. ![]() Theoretically the distribution cannot be exactly Gaussian, since the Gaussian curve would take an infinity of time to become complete. This can be done, for most passage-migrant species, for both spring and fall migrations, and so the distance or interval between peaks can be ascertained also. This means that the “distribution” of the observations in time is approximately Gaussian, and from the plot the peak of migration and the standard deviation of the timing can be ascertained with some accuracy. If sufficiently systematic numerical observations or trappings of birds or other animals on migration are made, and the accumulating totals plotted on probability paper on a time base or abscissa, they will frequently plot to a fairly good straight line. ![]()
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