The Frequency Distribution of Some Measured Parallaxes and of the Parallaxes Themselves.

It is well known that the frequency functions of a set of quantities and of their measures are different;' if the error of measurement is comparable in magnitude to the quantity determined, they may be decidedly dissimilar. In the case of good photographic parallaxes the reported probable errors average about eight one-thousandths of a second of arc, making a dispersion of twelve. Moreover, as we have shown,2 there appears to be a Lexian ratio of 1.25 here involved which would increase the true dispersion to o = 15. This might well seriously disturb the frequency distribution of the large number. of parallaxes which lie between 0.'15 and +0. '050. To obtain a fairly comparable set of stars for discussion we restricted our list to a spread of one magnitude in apparent brightness and found 313 determinations3 of parallax by the Allegheny, McCormick and Mt. Wilson observatories of stars between magnitudes 5 and 6. For some stars there were several determinations; it was decided to discuss the frequency distribution of the measures, and thus each measure was counted separately. To have combined by averaging, the plural measurements would have introduced a variety of different weights; to have chosen one of the measures and discarded the rest would have involved a more or less arbitrary choice.4 The frequency curve obtained is very skew, but looks like a logarithmic transform, and is, in fact, normal in x = logio (C + 20) the parallax co being taken in thousandths of a second of arc. Plotted on probability paper, the distribution nowhere departs from a straight line more than 2 or 3 per cent, which is about the magnitude of the errrors of sampling. The fit appears better than would be expected on chance (figure 1, line with points). It may therefore be assumed that the frequency distribution of these parallax-measurements is: