Popular Science Monthly/Volume 58/February 1901/Chapters on the Stars VIII

by Simon Newcomb

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CHAPTERS ON THE STARS.

By Professor SIMON NEWCOMB, U. S. N.

THE CLUSTERING OF THE STARS.

A STUDY of Schiaparelli's planispheres, which we gave in the last chapter, shows that some regions of the heavens are especially rich in lucid stars and others especially poor.

Neither telescope nor planisphere is necessary to show that many of those stars are collected in clusters. That the Pleiades form a group of stars by itself is clear from the consideration that six stars so bright would not fall so close together by accident. This conclusion is confirmed by their common proper motion, different from that of the stars around them. The singular collection of bright stars which form Orion, the most brilliant constellation in the heavens, and the little group called Coma Berenices—the Hair of Berenice—also suggest the problem of the possible connection of the stars which form them.

The question we now propose to consider is whether these clusters include within their limits an important number of the small stars seen in the same direction. If they and all the small stars which they contain were within their actual limits removed from the sky, would important gaps be left? The significance of this question will be readily seen. If important gaps would be left, it would follow that a large proportion of the stars which we see in the direction of the clusters really belong to the latter, and that, therefore, most of the stars would be contained within a limited region. The clusters which we shall especially study from this point of view are the Pleiades, Coma Berenices, Præsepe and Orion.

The Pleiades.—In the case of this cluster the question was investigated by Professor Bailey, by means of a Harvard photograph 2° square, having Alcyone near its center. It was divided into 144 squares, each 10' on a side. The brighter stars of the cluster were included within 42 of these squares. The count of stars gave the results:

Within cluster: 1,012 stars, or 24 per square.
Without cluster: 2,960 stars, or 29 per square.

It, therefore, seems that the portion of the heavens covered by the cluster is actually poorer in stars than the region around it.

Two opposite conclusions might be drawn from this fact. Assuming that the difference is due to the presence of the cluster, we might suppose that the latter was formed of material that otherwise would have gone into numerous smaller stars. Accepting this view, it would follow that the material in question was a sheet so thin that the thickness of the space filled by the cluster was an appreciable fraction of that occupied

 

Fig. 1. Photograph showing Structure of the Milky Way, by Barnard.

 

by the stars. In other words, one-fifth of the stars of the region would be contained in a thin sheet. This result seems too improbable to be accepted.

The other and more likely conclusion is that the number of very minute stars included in the cluster is no greater than that in the surrounding regions, and that the lesser number in the region is to be regarded as accidental.

Fig. 2. Rifts in the Milky Way, Photographed by Barnard.

Coma Berenices.—This cluster, which may be seen east, south or west of the zenith on a spring or summer evening, contains seven stars visible to the naked eye, each of the fifth magnitude. It may be considered as comprised within the limits 12h. 13m. and 12h. 25m. of R. A., and 25° to 29° of declination, an area of 10°.5. This existence of seven lucid stars within so small an area suggests that they belong together, and may have smaller stars belonging to the group, and making the star-density of this area greater than that of the sky in general.

The question whether there is any corresponding excess of richness in the fainter stars will be decided by a count of those contained in Graham's section of the A. G. Catalogue, which extends to the ninth magnitude. With the area above defined this catalogue gives seventy-one stars. Subtracting the seven lucid stars, we have sixty-four small stars left within the area. To the same belt of declination 336 stars are listed in the twelfth hour of R. A., giving an average of sixty-seven stars to an area equal to that of the cluster. The small stars are, therefore, no thicker within the area of the cluster than around it. It may be added that the seven lucid stars do not seem to have any common proper motion, so that their proximity is probably an accident.

Præsepe.—This object, situate in the constellation Cancer, appears to the naked eye as a patch of nebulous light. It is actually a condensed group of stars, of which the brightest are of the seventh magnitude. The stars of the ninth magnitude included within the area of the group probably belong, for the most part, to it, but they are too few to serve as the base for any positive conclusion.

Orion.—I find by measurement and count that a circle 20° in diameter, comprising the brightest stars of this constellation, contains eighty stars to magnitude 6.3. Of these six are of the first or second, leaving seventy-four from the third to the sixth. The resulting richness is 24 to 100 square degrees, about the average richness along the borders of the galaxy. It follows that this remarkable collection of bright stars has no unusual collection of faint stars associated with it.

A very natural inquiry is whether the bright stars in Orion have any common proper motion, indicating that they form a system by themselves. The answer is shown in the following statement of the proper motions in a century:

				Proper Motions.
	Star.		Mag.		R. A. "		Dec. "
	Rigel		1		+0.1		0.0
η	Orionis		3		+0.1		-0.3
γ	Orionis		2		-0.6		-1.7
δ	Orionis		2		0.0		-0.2
ε	Orionis		2		0.0		+0.1
ζ	Orionis		2		0.0		-1.4
κ	Orionis		2		+0.1		-0.3
α	Orionis		1		+3 0		+0.9

For the most part these motions are too small to be placed beyond doubt, even by all the observations hitherto made. In the case of α Orionis the motion is established; in those of γ and ζ, it is more or less probable, but not at all certain; in all the other cases it is too small to be measured.

This minuteness of the motion makes it probable that these stars are very distant from us, an inference which is confirmed by the smallness of their parallaxes. The careful and long-continued measures of Gill show no parallax to Rigel, while Elkin finds one of only 0".02 to α Orionis.

The general conclusion from our examination is this: The agglomeration of the lucid stars into clusters does not, in the cases where it is noticeable to the eye, extend to the fainter stars.

Let us now study the question on the opposite side. The planispheres show regions of great paucity in lucid stars; is there here any paucity of telescopic stars?

The two regions of greatest paucity are near the equator; one extends through the hour of 0 of R. A.; the other from 12h. 20m. to 12h. 40m. The richness of these and of the adjoining regions may be inferred from Boss's zone of the A. G. Catalogue, including a belt from 1° to 5° of declination. The number of stars in each hour from 23h. to 3h. is as follows:

In 23h.:		271 stars.
In  0h.:		293 stars.
In  1h.:		299 stars.
In  2h.:		295 stars.

These numbers show no paucity in the hour 0, and no excess in the hour 2, which is much richer in lucid stars than the hour 0.

In the strip from 12h. 20m. to 12h. 40m. the catalogue contains seventy-eight stars, a richness of 234 to the hour. In the hour preceding there are 211 stars; in that following, 225. There is, therefore, no paucity in the strip in question.

 

THE STRUCTURE OF THE MILKY WAY.

The most salient problems suggested by the appearance of the Milky Way are to be approached on lines quite similar to those followed in the last chapter. We begin with a description of this wonderful object as it appears to the observer. We recall that it can be seen through some part of its course on any clear night of the year, and in the evening of any season except that of early summer. We begin with the portion which will be visible in the late summer or early autumn. We can then trace its course southward from Cassiopeia in the northwest. It passes a little east of the zenith down to Sagittarius, near the south horizon. This portion of the belt is remarkable for its diversity of structure and the intensity of the brighter regions.

In Cassiopeia it shows nothing remarkable, but above this tion, in Cepheus, we notice in the midst of the brighter region a nearly-circular patch several degrees in diameter, in which little light is seen. A little farther along we notice a similar elongated patch in Cygnus lying across the course of the belt. In this region the brighter portions are of great breadth, more than 20°.

In Cygnus begins the most remarkable feature of the Milky Way, the great bifurcation. Faintly visible near the zenith, as we trace it towards the south, we see it grow more and more distinct, until we reach the constellation Aquila, near the equator. Between Cygnus and Aquila the western branch seems to be the brighter and better marked of the two, and might, therefore, be taken for the main branch. About Aquila the two appear equal, but south of this constellation we see the western branch diverge yet farther toward the west, leaving the gap between it and the eastern yet broader and more distinct than before. This branch finally terminates in the constellation Ophiuchus, while the eastern branch, growing narrower, can still be followed toward the south.

Between the equator and the southern horizon we have the brightest and most irregular regions of all. Several round, bright patches of greater or less intensity are projected on a background sometimes moderately bright and sometimes quite dark. If the night is quite clear and moonless we shall see that, after a vacant stretch, the western branch seems to recommence just about the constellation Scorpius. In this constellation we have again a bifurcation, a dark region between two bright ones.

This is about as far as the object can be well traced in our middle latitudes. From a point of view nearer to the equator it can be traced through its whole extent. South of Scorpius and Sagittarius it becomes broad, faint and diffused through the constellations of Norma and Circinus. It reaches its farthest southern limit in the Southern Cross, where it becomes narrower and better defined. The most remarkable feature here is the 'coal sack,' a dark opening of elliptical shape in the central line of the stream. West and north of this, in the constellation Argo, is the broadest and most diffused part of the whole stream, the breadth reaching fully 30°. Here we again reach the portion which rises above our horizon.

Returning now to our starting point, we shall notice that, as we make our observations later and later in the autumn, the southern part, which we have been mostly studying, is seen night by night lower down in the west, while new regions are coming into view in the northeast and east. These regions rise earlier every evening, and, if we continue our observations to a later hour, we shall see more and more of them above the eastern or southeastern horizon. By midwinter Cassiopeia will be seen in the northwest, and we can readily trace the course of the galaxy from that constellation in the opposite direction from that which we have been following. South of Cassiopeia we see, near the central line, the well-known cluster forming the sword handle of Perseus. Farther south the belt grows narrower and fainter; although the irregularities of structure continue, they are far less striking than on the other side. On a moonlight evening it will scarcely be visible at all. If the moon is absent and the air clear we shall see that it grows slightly brighter toward the southern horizon, near which will be the narrowest part of its entire course. Below is the broad and diffused region in Argo.

One conclusion from the inequalities of structure which we have described will be quite obvious. The Milky Way is something more than the result of the general tendency of the stars to increase in number as we approach its central line. There must be large local aggregations of stars, because, as we have already pointed out, there cannot be such diversity of structure shown in a view of a very widely stretched stratum of stars. When, instead of a naked eye view of the belt, we study the photographs of the Milky Way, we find this evidence of clustering to grow still stronger. It is shown very strikingly in the photograph by Barnard, showing the singular rifts in the Milky Way in the constellation Ophiuchus. Yet more singular are three-minute openings in the constellation Aquila, the positions of which are:

(1).	R. A.	= 19h. 35.0m.;	Dec.	= $+$ 10° 17'
	"	= 19h. 36.5m.;	"	= $+$ 10° 37'
	"	= 19h. 37.2m.;	"	= $+$ 11° 2'

The fundamental question which we meet in our farther study of this subject is: At what magnitude do these agglomerations of stars begin? Admitting, as we must, that they are local, are they composed altogether of stars so distant as to be faint, or do they include stars of considerable brightness? We consider this question in a way quite similar to that in which we discussed the clustering of the stars in the last chapter. We mark out on a map of the Milky Way the brightest regions—that is, those which include the densest agglomeration of very faint stars. We also mark out the darkest regions, including the coal sack. For this purpose I have taken the maps found in Heis's Atlas Cælestis for the northern portion of the Milky Way and the Atlas of Gould's Uranometria Argentina for the southern portion. In order to enable any one to repeat and verify the work I give the position of the central part of each patch or region studied. This serves simply for the purpose of identification. The outlines can be drawn by any one when the patch is identified. The third column of the table is given, approximately, the number of square degrees in the patch as outlined. Then follows the number of stars as found on the map. Here are included stars somewhat fainter than those regarded as lucid. Heis maps all stars down to about magnitude 6.2 or 6.3. Gould gives the places of all stars to the seventh magnitude.

 

A.—Number of lucid stars in certain bright regions or patches of the Milky Way.

I.—Northern portion, from Heis.

Position of patch.				Square degrees.		Number of stars.
R. A.		Dec.
19h. 10m.		+35°		60		21
20h. 0m.		+37°		25		11
20h. 20m.		+47°		20		11
21h. 5m.		+45°		12		4
0h. 20m.		+60°		25		9
2h. 20m.		+55°		60		16
3h. 30m.		+36°		32		7
3h. 40m.		+44°		43		12
		Sums		277		91

 

II.—Southern portion, from Gould:

Position				Area		Stars.
8h 4m		-47°		10		14
2h. 24m.		-44°		9		7
10h. 35m.		-58°		12		19
11h. 40m.		-62°		10		11
16h. 10m.		-53°		7		7
18h. 0m.		-28°		25		9
18h. 10m.		-18°		8		5
18h. 42m.		-8°		16		5
		Sums		97		77

 

B.—Number of lucid stars in the darker regions or patches of the Milky Way.

I.—Northern part, from Heis.

Position.				Area.
H. A.		Dec.				Sq. Deg.		Stars.
21h. 0m.		+50		26		10
22h. 0m.		+67		33		7
22h. 25m.		+60		30		12
0h. 0m.		+69		56		10
4h. 0m.		+55		98		19
4h. 20m.		+35		98		13
6h. 15m.		+18		86		17
6h. 12m.		+ 4		48		9
		Sums		475		97

 

II.—Southern part, from Gould.

Position.
7h. 22m.		-38°		18		8
7h. 28m.		-38°		12		5
8h. 0m.		-22°		11		4
8h. 40m.		-50°		30		16
9h. 0m.		-45°		12		6
10h. 0m.		-52°		11		5
12h. 40m.		-63°		18		2
15h. 10m.		-56°		31		3
17h. 30m.		-27°		18		3
18h. 10m.		-35°		18		7
18h. 0m.		-22°	${\begin{matrix}{\big \}}\end{matrix}}$	24		10	[1]
18h. 30m.		-8°
18h. 50m.		-5°		16		5
Sums				219		74

To derive the best conclusions from these numbers we must compare them with the mean star-density for the sky in general, and for the regions near the galactic plane. Heis has 3,903 stars north of the equator; Gould, 6,755 south of it. The area of each hemisphere is 20,626 square degrees. It will be convenient to express the various star densities in terms of 100 square degrees as the unit of area. Thus we have the following star-densities according to the two authorities:

	Heis.	Gould.
Star-density of the entire hemisphere	19.0	32.7
Star-density of the darker galactic regions	20.4	33.8
Star-density of the bright-galactic regions	32.9	79.4

The first two pairs of numbers lead to the remarkable and unexpected conclusion that the darker regions of the Milky Way are but slightly richer in lucid stars than the average of the whole sky; certainly no richer than is due to the general tendency of all the stars to crowd toward the galactic plane. On the other hand, the bright areas are 60 per cent, richer according to Heis, and more than 100 per cent, richer according to Gould, than the darker areas seen among and around them. The conclusion is that an important fraction of the lucid stars which we see in the same areas with the agglomerations of the Milky Way is really in those agglomerations and form part of them.

A study quite similar to this has been made by Easton for the portions of the Milky Way between Cygnus and Aquila, where the diversities of brightness are greatest. His count of the stars in the bright and dark regions differs from that made above, principally by including all the stars of the Durchmusterung, which we may suppose to extend to about the ninth magnitude.^[2]

He divides the regions studied into six degrees of brightness. For our present purpose it is only necessary to consider three regions, the brightest, the faintest and those intermediate between the two. Besides the count from the Durchmusterung he made a count of the same sort from Dr. Wolf's photographs and from Herschel's gauges of the heavens. In the following table I have reduced all his results, so as to express the number of stars in a square degree in the three separate regions. At the top of each column is given the authority, whether Argelander, Wolf or Herschel. Wolf had two sets of photographs, one supposed to include all the stars to the eleventh, the other to the twelfth magnitude. The magnitudes included are given in the second line. That Herschel's count extends to the fifteenth magnitude is by no means certain; but we can judge from the great number of his stars that it goes considerably beyond Wolf's in the faintness of the stars included. Below this we give, in the regions A, B and C, which are, respectively, those of least, of medium and of greatest brightness, the number of stars per square degree according to each of the authorities:

Authority		Arg.		Wolf (A).		Wolf (B).		Hersch.
Magnitude		1—9		1—11		1—12		1- -15 (?)
Region A		23		72		224		405
Region B		33		134		764		4114
Region C		48		217		1,266		6,920
C—A		25		145		1,042		6,425
Ratio C:A		2.1		3.0		5.7		14.0

 

The vastly greater number of individual stars per square degree in the brighter regions is what we should expect from the studies we have made of the lucid stars. But what is of most interest in the table is the continual increase in the proportion of faint stars in the separate regions. We notice that, when we consider only the stars of the ninth magnitude, there are twice as many in the brightest as in the darkest portions. When we go to the eleventh magnitude, as shown by Wolf's photograph A, we find the number of stars in the brighter regions to be threefold. When the twelfth magnitude is included we find that there are between five and six times as many stars in the bright regions as in the dark ones. Finally, when we come to stars from Herschel's gauges there are fourteen times as many stars per square degree in the brighter regions as in the dark.

At first sight this result seems to show a great difference between the clusters of stars described in the last chapter, and the collections of the Milky Way, in that the former include few or no faint stars, while the latter include a greater and greater number as we ascend in the scale of magnitude. This difference is important as showing a vastly greater range of actual brightness among the galactic stars than among those which form the scattered clusters. Allowing for this difference, the results from the two classes of objects can be brought to converge harmoniously toward the same conclusion.

We have collected abundant evidence that, separate from the accumulations of stars in the Milky Way, perhaps extending beyond them, there is a vast collection of scattered stars, spread out in the direction of the galactic plane, as already described, which fill the celestial spaces in every direction. We have shown that when, from any one area of the sky, we abstract the stars contained in clusters, this great mass is not seriously diminished. We have also collected abundant evidence that the distances of this great mass are very unequal; in other words, there is no great accumulation, in a superficial layer, at some one distance. The question which now arises is whether the darker areas which we see in the Milky Way are vacancies in this mass. Although some of the counts seem to show that they are, yet a general comparison leads to the contrary conclusion. In the darkest areas of the Milky Way, when of great extent, the stars are as numerous as on each side of the galactic zone. Our general conclusion is this:

If we should remove from the sky all the local aggregations of stars, and also the entire collection which forms the Milky Way, we should have left a scattered collection, constantly increasing in density toward the galactic belt.

 

THE INCREASING NUMBER OF STARS WITH DIMINISHING BRIGHTNESS.

We mentioned in an earlier chapter that, when we compare the number of stars of each successive order of magnitude with the number of the order next lower, we find it to be, in a general way, between three and four times as great. The ratio in question is so important that a special name must be devised for it. For want of a better term, we shall call it the star ratio. It may easily be shown that there must be some limit of magnitude at which the ratio falls off. For, a remarkable conclusion from the observed ratio for the stars of the lower order of magnitude is, that the totality of light received from each successive order goes on increasing. Photometric measures show, as we have seen, that a star of magnitude m gives very nearly 2.5 times as much light as one of magnitude m+1. The number of stars of magnitude m+1 being, approximately, from 3 to 3.75 times as great as those of magnitude m, it follows that the total amount of light which they give us is some 40 or 50 per cent, greater than that received from magnitude m. Using only rough approximations, the amount of light will be about doubled by a change of two units of magnitude; thus the totality of stars of the sixth magnitude gives twice as much light as that of the fourth; that of the eighth twice as much light as that of the sixth; that of the tenth twice as much again as of the eighth, and so on as far as accurate observations and count have been made.

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 to our readers. I have, therefore, derived their general mean results for different parts of the sky with reference to the Milky Way and for stars of the various orders of magnitude. The following table shows the conclusions:

 

		Ratio of						Concluded
Zone.		increase.						result.
		D. M.		S. D.		Diff.
I.		2.99		—		—		3.24
II.		3.00		3.49		0.49		3.25
III.		3.07		3.72		0.65		3.37
IV.		3.32		3.85		0.53		3.58
V.		3.55		4.15		0.60		3.85
VI.		3.28		3.68		0.40		3.48
VII.		3.23		3.55		0.32		3.37
VIII.		3.44		3.56		0.12		3.40
IX.		—		3.49		—		3.24

 

In the first column we have the designation of the zone or region of the sky, as already given.

In the second and third columns we have the mean ratio of increase for whole magnitudes as derived from the Durchmusterung and the southern Durchmusterung, respectively. It will be recalled that region I., around the north galactic pole, is entirely wanting in the S. D., while the adjoining regions, II. and III., are only partially found, and that, in like manner, the D. M. includes none of region IX. around the south galactic pole, and but little of the adjoining region.

It will be seen that there is a very remarkable systematic difference between the two lists, the ratio of the number of faint to that of bright stars being much greater in the S. D. This difference is shown in the fourth column. I have assumed that the two systems are equally good, and there diminished all the ratios of the S. D. by 0.25, and increased those of the D. M. by the same amount. The mean of the two corrected results was then taken, giving the principal weight to the one or the other, according to the number of stars on which they depend.

It will be seen that the increase of the ratio from either galactic pole to the Milky Way itself is as well marked as in the case of the richness of the respective regions in stars. We may condense the results in this way:

In the galactic zone,	ratio	= 3.85
In zones IV. and VI.,	"	= 3.53
In polar zones I., II., VIII. and IX.,	"	= 3.28

It will be recalled that zone V. is a central belt 20° broad, including the Milky Way in its limits. But the latter, as seen by the eye, especially its brightest portions, does not fill this zone. These portions, as we know, comprise the irregular collection of cloud-like masses described in the last chapter. Seeliger has investigated the ratio within these masses, and compared it with, the stellar density, or the number of stars per square degree. The mean results are:

In that portion of the galaxy extending from Cassiopeia to the equator near 6" of R. A., ratio = 4.02.

In that portion from Cassiopeia in the opposite direction to near 19" of R. A. in Aquila, ratio = 3.70.

These remarkable results are derived from the D. M., and will be yet more striking if corrected by half the difference between it and the S. D., as we have done for the sky generally. They will then be 4.27 and 3.95, respectively.

As might be expected, the regions of greater star density have generally, though not always, the higher ratio. The highest of all is in a patch south of Gemini, between 6^h and 7^h of E. A., and about 5° of declination. Here it amounts to 5.94, showing that there are eighty-six stars of magnitude 9.0 to every one of magnitude 6.5.

The D. M. does not stop at magnitude 9, as the above numbers do, but extends to 9.5, while the S. D. extends to magnitude 10. For these magnitudes Seeliger finds a yet higher ratio. This is, however, to be attributed to the personal equation of the observers, and need not be further considered.

The only available material for finding the ratio of increase above the ninth magnitude is found in the Potsdam photographs for the international chart of the heavens, which extend to magnitude 11. These are published only for a few special regions. Five of the published plates fall in regions not far from the galactic pole. I have made a count by magnitudes of the 312 stars contained in these plates. An adjustment is, however, necessary from the fact that the minuter fractions of a magnitude could not be precisely determined from the photographed images. The results are practically given to fourths of a magnitude, although expressed in tenths. But it is found that the numbers corresponding to round magnitudes and their halves are disproportionately more frequent than those corresponding to the intermediate fourths. For example, there are only nineteen stars of magnitude 10.7 and 10.8 taken together; while there are forty-nine of 10.5. Under these circumstances I have made an adjustment to half magnitudes by taking the stars of quarter magnitudes, and dividing them between half magnitudes next higher and next lower. The result is as follows:

Mag.		Stars.
6.5		2
7.0		2
7.5		4
8.0		11
8.5		15
9.0		29
9.5		33
10.0		39
10.5		64
11.0		115

It is difficult to derive a precise value of the star ratio from this table, owing to the small number of stars of the brighter magnitudes which are insufficient to form the first term of the ratio. Assuming, however, that the ratio is otherwise satisfactorily determined up to the ninth magnitude, we find that there is but a slight increase from the ninth up to the tenth. The number of the eleventh magnitude is, however, nearly three times that of the tenth and nearly double that of 10.5.

Another way to consider the subject is to compare the total number of stars of the fainter magnitude with the number of lucid stars corresponding, which, in the general average, will be found in the same space. We may assume that near the poles of the galaxy there is about one lucid star to every ten square degrees. The five belts included in the above statement cover about thirteen square degrees. The region is, therefore, that which would contain about one star of the sixth magnitude. An increase of this number by somewhat more than 100 times in the five steps from the sixth magnitude to the eleventh, would indicate a ratio somewhat less than 3; about 2.5. But the comparison of the photographic and visual magnitudes renders this estimate somewhat doubtful. Besides this, it is questionable whether we should not reckon among stars of the eleventh magnitude those up to 11.5, which would greatly increase the number. It is a little uncertain whether we should regard the limit of magnitude on the Potsdam plates as 11.0 or 11 plus some fraction near to one-half.

Altogether, our general conclusion must be that up to the eleventh magnitude there is no marked falling off in the ratio of increase, even near the poles of the galaxy.

I have not made a corresponding count for the galactic region, but the great number of stars given on the plate show, as we might expect, that there is no diminution in the ratio of increase.

The question where the series begins to fall away is, therefore, still an undecided one, and must remain so until a very exact count is made of the photographs taken by the international photographic chart of the heavens, or of the Harvard photographs.

There is also a possibility of applying a photometric study of the sky to the question. From what has already been shown of the total amount of light received from stars of the smaller magnitudes, it would seem certain that a considerable fraction of the apparently smooth and uniform light of the nightly sky may come from these countless telescopic stars, even perhaps from those which are not found on the most delicate photographs. It is certain that the background of the sky itself is by no means black. The only question is, whether the light from this background is mostly reflected by our atmosphere from the stars. It may seem questionable whether such is the case, because the fraction reflected in a clear atmosphere is not supposed to exceed one-tenth the total amount of light of the stars themselves. On the other hand, the seemingly blue color of the sky might seem to militate against this view, since the average color of all the stars is white rather than blue. The subject is an extremely interesting one and requires further investigation before a definitive conclusion can be reached.

1. A long narrow region between these limits.
2. Easton's work is given in detail in the 'Astronomische Nachrichten,' Vol. 137, and the ’Astrophysical Journal’ Vol. I, No. 3.