Of late years astronomers have given greatly increased attention to the question of the distances of the stars, and systematic campaigns of the most laborious kind have been carried on by Gill; Elkin and Chase, of Yale; Kapteyn, of Groningen; and Schlesinger, at the Yerkes Observatory, Chicago. Some 350 stars have now been studied by the standard method of parallaxes, and for most of these objects, perhaps about 200 in number, fairly satisfactory data have been deduced; but the method can be extended only to stars within less than 100 light-years of our sun, and is therefore very limited in its applicability, owing to the small diameter of the earth's orbit, and the insensible effects of the annual displacements resulting from the orbital motion of our planet. As nature herself has fixed the limits of this method, astronomers have naturally cast about for other methods of greater generality and have finally developed processes of surprising power, of which an account will be given in the present paper.

§ 1. OUTLINE OF THE METHODS ADOPTED.

Among previous investigators who have occupied themselves. with the difficult problem of the profundity of the Milky Way the first place will be universally assigned to the incomparable Sir William Herschel, who extended his researches over many years, and reached results which were for a time accepted, but have been rejected for three quarters of a century, and yet are now proved to be essentially correct. It is very remarkable and exceedingly unfortunate that Herschel's conclusions have been generally rejected by his son, Sir John Herschel, and other astronomers during the past seventy-five years. But before discussing the circumstances which. led to this outcome I shall recall the modern attempts at the solution of the problem of determining distances in the Milky Way.

After the spectroscope came into use and Huggins had applied Doeppler's principle to the motion in the line of sight (1868) it was pointed out by Fox Talbot in 1871 (Brit. Assoc. Report, 1871, p. 34, Pt. II.) that the possibility existed of determining the absolute dimensions of the orbit of a pair of binary stars which had a known

angular dimension in the sky, and thus parallaxes might be found of systems very remote from the earth. In 1890, while a postgraduate student at the University of Berlin, I developed this method still further, and showed how it could be used also to test the accuracy of the law of universal gravitation in the stellar systems. The spectroscopic method then outlined was brought to more general form in 1895, and it at once occurred to me to point out its use for measuring the distance of clusters in the Milky Way (A. N. 3,323), as more certain than Herschel's method of star gauges.

Our age is one of rapid improvement in all scientific processes, and during the past sixteen years naturally much progress has been made in double-star astronomy, as well as in our knowledge of nebulæ and clusters. On looking more closely into the spectroscopic method, which in 1895 had been shown to be applicable to objects 1,000 light-years from the sun, and might thus include all suitable double stars within this sphere, I became convinced that while it is a great theoretical advance over the old method of parallaxes, it still is quite inadequate for finding the distances of the most remote objects in the sidereal universe. Accordingly in 1909 I returned to the improvement of Herschel's method as the most promising, for the determination of the distances of the most remote objects. Here are the grounds for this decision:

I. It was noticed, as remarked by Burnham, that revolving double stars are rare, if not unknown, in clusters, and among the star-clouds of the Milky Way-not because such systems are not present in these masses of stars, but because they cannot be separated, owing to the great distances at which these masses of stars are removed from us.

2. When double stars cannot be clearly separated in the telescope they cannot be used for parallax by the spectroscopic method; and thus the spectroscopic method, while having a wider range of application than the method of parallaxes, in something like the ratio of the size of the double star orbit to that of the orbit of the earth, is yet applicable only to stars within about 1,000 light-years of our sun.

3. It will be shown below that the most remote stars are separated from us by a distance of at least 1,000,000 light-years, and as this space is a thousand times that to which the spectroscopic method may be applied, it follows that there is no way of fathoming these immense distances except by the improvement of the method of Herschel.

And just as in my Researches on the Evolution of the Stellar Systems," Vol. II., 1910, p. 638, I had been able to adduce substantial grounds for returning to the vast distances calculated by Herschel, so also during the past year I have been able to add to the proof there brought forward, and will proceed to develop it in the present paper.

§2. HERSCHEL's Method Depending on the Space PenetrATING POWER OF TELESCOPES.

In his celebrated star gauges Herschel employed a twenty-foot reflector of 18 inches aperture, and calculated the space-penetrating power of such an instrument from the ratio of the aperture of the telescope to that of the pupil of the eye. The comparative distance to which a star would have to be removed in order that it may appear of the same brightness through the telescope as it did before to the naked eye may thus be calculated. Herschel found the power of this 20-foot reflector to be 75; so that a star of 6th magnitude removed to 75 times its present distance would therefore still be visible, as a star, in the instrument.

Admitting such a 6th magnitude star to give only a hundredth part of the light of the standard first magnitude star, it will follow that the standard star could be seen as a sixth magnitude star at ten times its present distance; and if we then multiply by the space penetrating power, we get 750 as the distance to which the standard star could be removed and still excite in the eye, when viewed through the telescope, the same impression as a star of 6th magnitude does to the naked eye. Thus if Alpha Centauri be distant 4.5 light-years, it would be visible in Herschel's telescope at a distance of 3,375 light-years. This is about the distance ascribed to the

remoter stars of the Milky Way by Newcomb and many other modern writers; but of course it is much too small, for the following reasons:

(a) Newcomb and other astronomers cite the possibility of some of the stars being as much as 1,000,000 times brighter than the average solar star, and in that case the star might be seen at V1,000,000 1,000 times that distance, or 3,375,000 light-years, with an instrument having a space penetrating power no greater than that employed by Herschel, provided that no light is extinguished in its passage through space.

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(b) If the telescope be more powerful than Herschel's 20-foot reflector, the light gathered will be increased in the ratio of r2/(18)2, where r diameter of mirror; and for the 60-inch reflector at Pasadena, 60, over nine times as much light could be gathered, or stars seen over three times as far away. Thus if the stars have only about 10,000 times the luminosity of the sun, they could still be seen with the Pasadena reflector at a distance of over a million light-years. For 3,375 1.-y. X 3 X 100=1,012,500 light-years.

(c) The sensitiveness and accumulative effects of the photographic plate, will enable us to extend our sounding line still further out into space by some three magnitudes, or four times the distance; and thus with a modern 60-inch reflector we could photograph stars at a distance of about four million light-years, if they have 10,000 times the standard solar luminosity, and no light is lost in space. How much light is really lost in space will be considered later, but it may be stated here that it probably is decidedly less than was concluded by Struve.

83. INDEPENDENT CALCULATION OF THE DISTANCE OF THE REMOTEST STARS OF THE HELIUM TYPE.

From the data given in Lick Observatory Bulletin No. 195, we find that 225 helium stars employed by Campbell in his line of sight work have an average visual magnitude of 4.14. Of the four variables given in this Bulletin, we have used the maximum brightness in three cases, because they are of the algol type. In the case

of u Herculis, we have used the mean magnitude, because the type of variable does not appear to be as yet well established.

Here then we have 225 helium stars at an average distance of about 540 light-years. For in Lick Observatory Bulletin No. 195, p. 121, Campbell finds the 180 class B, or helium, stars to have an average distance of 543 light-years, while in Publications of the Astronomical Society of the Pacific for June-August, 1911, p. 159, Professor Curtis gives 534 light-years as the average distance of 312 helium stars. The former distance for 180 stars being greater than the latter distance for 312 stars, we may take 540 light-years as the distance of the 225 helium stars here under discussion, the average magnitude of which is 4.14.

If the average magnitude were decreased to 21.14, by removal to 2,512 times their present distance, which would reduce the average brightness by 17 magnitudes, the distance of the stars would be multiplied by 2,512, and become 1,356,480 light years. This is for the helium stars as they are, without any hypothesis as to brightness, or as to the extinction of light in space, which will be considered later.

The question will naturally be asked whether helium stars really exist at these great distances. We may unhesitatingly affirm that they do, because of the well-known whiteness of the small stars of the Milky Way. It is true that Pickering has investigated the distribution of the helium stars in the Harvard Annals, Vol. 56, No. II., and Campbell quotes these data in Lick Observatory Bulletin No. 195 as showing that the helium stars are all bright objects. Pickering believed his tabulations to indicate "that of the bright stars, one out of four belongs to this class (B), while of the stars of the sixth magnitude there is only one out of twenty; and that few if any would be found fainter than the seventh or eighth magnitude." The implication here is that no helium stars exist at very great distances corresponding to small magnitudes; but of course. such a view is untenable.

It probably is true that the group of helium stars at a distance of some 540 light-years from our sun, and thus comparatively near