Of the Northern results three are exactly on the limit, 0".20, and several others are doubtful, and probably too large. The most likely number for the Northern hemisphere seems to be 12, and if we estimate an equal number for the Southern hemisphere we shall have 24 in all. Adding the four stars within the sphere 3R, we shall then have a total of 28 within the sphere 5R, of which the volume is 125. This gives between 4 and 5 space units to a star.
Let us now consider the space between the spheres 5R and 10R, including all stars whose parallax lies between the limits 0".10 and 0".20. Of these the numbers are:
| Southern | Hemisphere | 6 | (Gill) |
| Northern | " | 15 | (Chase) |
| " | " | 15 | (Others) |
Reasoning as before, we may assume that the number of stars between the assigned limits is 60, making a total of 88 within the sphere 10R. The volume of space enclosed being 1,000 units, this will give one star to 12 units of space.
How far can we rely on this number as an approximation to the actual number of stars within the tenth sphere? The errors in the estimate are of two classes, those affecting the parallax itself and those arising from a failure to include all the stars within the sphere. The very best determinations are liable to errors of two or three hundredths of a second, the inferior ones to still larger errors. Thus, it may happen that there are stars with a real parallax larger than the limit of which the measures fall below it and are not included, and others smaller than the limit which, through the errors of measurement, are made to come within the sphere. As we have seen in the chapter on the parallaxes, it is quite possible that there may be a number of stars with a measurable parallax whose proximity we have never suspected on account of the smallness of the proper motion. We can only say that the nearer a star is to the system the more likely its proximity is to be detected, so that we are much surer of the completeness of our list of large parallaxes than of small ones. Hence, there may well be a number of undetermined parallaxes upon or just above the limit 0".10.
The most likely conclusion we can draw from this examination seems to be that in the region around us there is one star to every 8 units of space; or that a sphere of radius, 2R, equal to 412,500 radii of the earth's orbit, corresponding to a parallax of 0".50, contains one star. This is a distance over which light would pass in 8½ years.
We next see how far a similar result can be derived from statistics of the proper motions. It seems quite likely that nearly all proper motions exceeding 1" annually have been detected. The number known is between 90 and 100, but it can not be more exactly stated because there is some doubt in the case of a number which seem to
