To give numerical precision to this result, let us take as unity the total amount of light received from the stars of the first magnitude. The sum total for this and the other magnitudes, up to the tenth, will then be:
| Mag. | 1 | Light | = | 1.0 | |
| " | 2 | " | = | 1.4 | |
| " | 3 | " | = | 2.0 | |
| " | 4 | " | = | 2.8 | |
| " | 5 | " | = | 4.0 | |
| " | 6 | " | = | 5.7 | |
| " | 7 | " | = | 8.0 | |
| " | 8 | " | = | 11.3 | |
| " | 9 | " | = | 16.0 | |
| " | 10 | " | = | 22.6 | |
| Total | 74 8 | ||||
That is, from all the stars to the tenth magnitude combined, we have more than seventy times as much light as from those of the first magnitude.
There must, evidently, be an end to this series, for, were this not the case, the result would be that which we have shown to follow if the universe were infinite; the whole heavens would shine with a blaze of light like the sun. At what point does the rate of increase begin to fall off?
We are as yet unable to answer this question, because we have nothing like an accurate count of stars above the ninth, or at most, the tenth magnitude. All we can do is to examine the data which we have and see what evidence can be found from them of a diminution of the ratio.
It must be pointed out, at the outset, that the ratio must be greater in the galactic region than it is in other regions. This follows from the fact that the proportion of small stars increases at a more rapid rate in the galaxy than elsewhere. This is shown by the comparisons we have already made of the Herschelian gauges with the counts of the brighter stars. While the galactic region is less than twice as dense as the remaining regions for the brighter stars, it seems to be ten times as dense for the Herschelian stars. If we knew the limiting magnitude of the latter, we could at once draw some numerical conclusion. But unfortunately, this is quite unknown. All we know is that they were the smallest stars that Herschel could see with his telescope.
The ratio in various regions of the heavens has been very exhaustively investigated by Seeliger, in the work already quoted. The bases of his investigations are the counts of stars in the Durchmusterung. Instead of taking the ratio for stars differing by units of magnitude, as we have done, Seeliger divides them according to half magnitudes. The reproduction of his numbers in detail would take more space than we can here devote to the subject and would not be of special interest
