The Maximum of a Distribution- or Spectrum-Function


RECENTLY, Bracewell1 published a paper, in which he explained why the position of the maximum of Planck's energy spectrum depends on whether the frequency or the wave-length is plotted on the horizontal axis. I wish to direct attention to the fact that this is true for all distribution- or spectrum-functions. Though it is indeed obvious that the maximum must move, when the horizontal axis is non-linearly transformed, yet errors have been made in this respect. Zernike2 pointed out that, for example, an error of this kind was made when Stern and Gerlach3 described their famous experiments, in which rays of silver atoms were deflected by a magnetic field. They assumed (Fig. 1) that the atoms with the most probable velocity v p would have the most probable deflexion y p (thickest part of the precipitation layer). This is only true if the deflexion y and the velocity v of an atom are related by y = av + b. However, they are related by yv 2 = constant. The true value of y p is one-half of the value derived from the last equation with v = v p. This result is due to Semenoff4, who was apparently the first to recognize the danger, described here. Though geophysicists are particularly vulnerable in this respect, as they have been investigating spectra of all sorts of time-series, yet almost all of them have succeeded in escaping the pitfall.

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  1. 1

    Bracewell, R. N., Nature, 174, 563 (1954).

  2. 2

    Zernike, F., “Handbuch der Physik”, (12), 3, 454 (1928).

  3. 3

    Stern, O., and Gerlach, W., Z. Phys., 2, 49, Abd. 3, 417 (1920).

  4. 4

    Semenoff, N., Z. Physik., 30, 151 (1924).

  5. 5

    Schumann, T. E. W., Notos, 4, 4 (1955).

  6. 6

    For example, Neumann, G., Trans. Amer. Geophys. Union, 36, 985 (1955).

  7. 7

    Panofsky, H. A., and McCormick, R. A., Quart. J. Roy. Meteor. Soc., 80, 546 (1954).

  8. 8

    Swinbank, W. C., and others, Coun. Sci. Indust. Res. Australia, Div. of Meteorological Physics, Tech. Paper No. 2–6 (1955).

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