The Distribution of Absolute Magnitudes among K and M Stars Brighter than the Sixth Apparent Magnitude as Determined from Peculiar Velocities

i. The statistical method of determining the distribution of abso'ute magnitudes from the reduced r-components and peculiar radial velocities given in the preceding Contribu- lion has been applied to the K and M stars in Boss's Catalogue brighter than 6 mag. The stars were divided into three spectral groups, the number in each group being 659, 330, and 253, respectively. Table I shows the distribution of x=log r+o.2m, as observed and also after correc- tion for the effect of errors. It also shows the distribution of x computed from the adopted distribution of absolute magnitudes and of peculiar radial velocities V', and the differences between the observed and the computed distributions. The distributions of x are shown graphically in Figs. i, 2, and 3. The distribution of y =log V' - 1.6756, as observed and after correction for the effect of errors in the radial velocities, is given in Table II for different absolute magnitudes. The final distributions of absolute magnitudes are shown in Table IV and in Figs. 4, 5, and 6 with a class interval of 0.2 mag. 2. Among stars of spectral classes Ko-K2 the absolute-magnitude distribution shows four distinct maxima, occurring at absolute magnitudes - i.8, +0.2, + 2.9, and +6.7. The proportions of stars in the four groups are 9, 78, 12, and i per cent, respec- tively; these stars are designated by the terms "supergiants," "ordinary giants," "sub- giants," and "dwarfs." In the spectral interval K3-K9 three distinct frequency maxima occur, at -2.4, -0.1, and between 4 and 9, with relative proportions of 19, 78, and 3 per cent. These groups are identified as supergiants, ordinary giants, and dwarfs. No subgiants exist in this spectral interval. Among the M stars only supergiants and ordinary giants appear. The frequency maxima occur at - i.6 and -0.1, and the relative proportions are 44 and 56 per cent. 3. Stable conditions in the stars and relatively rapid transitions from one stable state to another are suggested as the probable cause of the frequency maxima and minima