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ing to 14.6% of the sky, there are 117 regions, or 26% of the regions used, which are situated in this area, and these 117 regions contain 2997 spirals, or about 48% of the whole.

To avoid the effect of this concentration in the galactic north polar area, it will be advisable to consider this area separately. Assuming that the 117 regions in this area, amounting to 87.75 square degrees, are sufficiently uniform in their distribution to represent adequately the occurrence of nebulae in the galactic north polar zone, this zone would contain 205,000 nebulae to correspond with the 2997 found in the 117 regions. Assuming likewise, in the remaining 0.854 of the sky, that the proportion of nebulae is adequately represented by the 3214 nebulae found in 322 regions, or 241.5 square degrees, these would correspond to 469,000 nebulae, a total for the entire sky of 674,000.

Any objection depending upon the effects of a marked concentration of the small nebulae in the north galactic area may be further met by considering separately the north galactic and south galactic polar zones, the two zones 15° wide extending from —30° to —45° and from -4- 30° to + 45° galactic latitude, and a zone 60° wide, 30° on each side of the galactic plane. Such a subdivision gives the following results:


The concentration of small nebulae is seen from the above tabulation to be greatest in the area about the north galactic pole, just as is the case with the larger spirals. While only 43 regions are available in the area about the south galactic pole, it will be seen that they indicate a similar, but less marked, concentration of the small nebulae in the southern polar area. The results given in the third line of the table for the zones 15° wide between the galactic zone and the two polar zones are of great interest. While the number of regions is only 62 they give the valuable indication that the density of the small nebulae persists to at least 60° from the galactic poles, with only a comparatively small diminution of the frequency of distribution which obtains about the two galactic poles.

I regard the estimate of 722,000 nebulae made by the foregoing subdivision of the observed data, as more reliable than the two estimates, 778,000, and 674,000, made earlier in the discussion.

As this revised estimate is equal to that of Fath plus that of Perrine, with several thousand to spare, a discussion of possible reasons for the discrepancy becomes imperative. The following points may be considered:

(a) It may be urged that my count has not been sufficiently conservative, and that I have possibly included many spurious objects. The detection of the faintest and smallest nebulae is very largely a matter of experience; all who have worked with photographic plates soon learn, by hard necessity, to recognize the average flaw at a glance. A very large proportion of the objects counted are unmistakably nebulae; as to the faintest nebulae, it is astonishing how faint and small are the nebulae which two "good" duplicate plates will reproduce. For a large proportion of my regions no duplicate plates exist, and I have necessarily been guided by the experience derived from regions taken in duplicate. I am unwilling at present to admit that as many as 5% of the nebulae counted by me are spurious. If 20% were spurious, we should still have to account for a total of over half a million nebulae.

(b) The theory may be advanced that the small spirals occur in greatest profusion in the regions immediately contiguous to the larger members of the class, which would explain why fewer nebulae were found by Fath, inasmuch as his plates were taken at the centers of Kapteyn areas where the larger nebulae would be included only by chance, whereas, from the purpose of the Crossley nebular programme, nearly all the plates have some N. G. C. object central. This point is difficult to prove or disprove without a special investigation based upon many plates taken at random in the galactic north polar region. It is certain that the small nebulae frequently show a gregarious tendency; occasionally one-half of a plate will record many small spirals while the other half records very few; the remarkable region at 12*1 55ra, -(-28° 30' shows 304 nebulae, while the region about N. G. C. 4826, less than 7° distant, shows only 2. While the small nebulae are evidently quite irregular in their distribution, it would seem that the large number of regions included in this discussion is sufficient to afford a true representation of their average frequency.

(c) I have already pointed out that sharp focus and perfect images are essential for the detection of the smallest and faintest spirals. On plates where great numbers of small nebulae are found, the majority are, as a rule, detected in the area 20' in radius about the optical axis as center, comprising only 0.35 of a square degree. At distances of 30' from the optical axis the parabolic images are very poor, and only the brighter of the small nebulae can be detected in these regions. Almost stellar and moderately bright nebulae are easily detected at such distances from the optical axis, as they will show fan-shaped images which are as large as those of brighter stars, but much fainter, and of nebular texture, but the very faint nebulae are obliterated by the blurring and spreading of the image. These "blurring factors" and the form of the extra-axial image in optical systems have been investigated by Schwarzschild6 in three papers of great power, the work of a master. For the convenience of those to whom the original papers may not be accessible the formulae are given here for the only two aberrations which affect the form of the image of an infinitely distant object in the focal plane of a single, perfect, parabolic mirror.

Radial distortion from astigmatism and field curvature (Bildwolbung) =5876 g"v.

Radial extension of coma = 2073 g v2. In the above, g is the diameter of the field, regarding a field diameter of 6° as the unit, while v is the ratio of aperture to focal length, taking the ratio 1:10 as the unit. It will be seen that the distortion due to coma increases as the square of the focal ratio, so that this quickly becomes very large for reflectors of large focal ratio, even at small angular distances from the optical axis. Dr. Fath used very large plates, &y» x 8V2 inches in size, in his work with the 60-inch reflector on the number of the small nebulae, and it appears that he used nearly the full area, inasmuch as he states that the area of his plates was 1.88 square degrees. As the full exposed area of the Crossley plate is 0.9 square degrees (and the outer regions of this are so poor because of the parabolic distortions that the effective area used in the counts is believed to be less than 0.75 square degrees) this would mean, if the two reflectors were of the same focal ratio, that the images on one-half of the angular area of the large plates used by Fath were worse than on the rejected edge strips of the Crossley plates. But the greater focal ratio of the 60-inch would increase this disadvantage, even allowing for certain advantages which would partially counterbalance this due to the greater linear scale of the plates. The following comparisons, computed from Schwarzschild's formulae, will illustrate this point.

The Crossley Reflector; Focal Ratio =1:5.8

Distance from Distortion by Distortion by

optical axis field curvature coma

30' 2'.'8 1071

41'(corner of plate) 5.2 13.7

The 60-inch Reflector; Focal Ratio =1:5.0.
30' 373 1375

41' 6.1 18.5

61' (corner of plate) 13.5 27.5

0 Untersuchungen zur geometrischen Optik, Abh. Kon. Ges. d. Wiss. zu Gottingen, Maih.-phys. El, N. F. 4, 1, 2, and 3, 1905.

One inch on the 60-inch plates equals ll15 as against 1614 on the Crossley plates, but it would not appear that this increase in scale could counterbalance the effect of the larger distortions in the outer regions of the plate. From my own experience in counting these minute objects on the Crossley plates it would appear to me that the actual affective area used by Fath must have been very much less than the 1.88 square degrees assumed in his calculations.7

It is my opinion also that the Lumiere Sigma plates which Fath used are not the best for the end in view. These plates are of very great speed, and are invaluable for some purposes. I have long since ceased to use them for nebular work, however, believing that the slightly slower, but beautifully "clean" Seed 27 and Seed 23 plates really show faint details better. With their smaller grain and clear background, very small and faint nebulae "stand out" on these plates much more plainly than on the more rapid Sigma plates.

(d) It is not impossible that a considerable proportion of the thirty or so plates which Fath took within 45° of the north galactic pole, happened to strike regions of few small nebulae. Had he changed to include four such regions as the following:



his estimate would have been increased by at least 60 per cent (he found 864 new nebulae in all, and the larger plates he used would have added a number to those counted on the smaller Crossley plates).

Perhaps all the reasons outlined above may be regarded as contributing to an explanation of the difference between Fath's estimate, and the larger ones due to Perrine and to the present investigation. Of these, the possibility noted under (b), that the small nebulae may conceivably occur in greatest profusion just where the visually discovered objects of the N.G.C. exist, would appear to be the only reason for changing the larger estimates, and evidence on this point is only to be secured by taking many additional plates at random.

In conclusion, I see no reason, at present existing, for changing the estimate made in this paper, that at least 700,000, and very probably 1,000,000 small spirals are within reach of large reflecting telescopes.

i In order that those who are not familiar with the character of the image in the field of a parabolic reflector may not misunderstand the data given above, the point must be strongly emphasized that this distortion of the image at points outside the optical axis is inherent in all reflectors, without exception, though their optical figure be perfection itself, as is undoubtedly the case with the 60-inch reflector. The reflector of small angular aperture will give a usable field of larger angular diameter than will a reflector of great angular aperture, while the former will be slower than the latter, on extended objects like nebulae, in the ratio of the squares of the focal ratios. For example, at a distance of 1° from the optical axis a reflector whose focal ratio is 1:10 will show aberrations of 673 and 678, while a reflector of focal ratio 1:3, over eleven times as rapid, will show at a distance of 1° from the optical axis corresponding aberrations of 2070 and 7479. The former will show rather poor, but usable, images on the edges of a field 2" in diameter; the latter will show images on the edge of the same field so broadened and distorted as to be absolutely unusable.



Round, with much brighter center; 013 in diameter; structureless. 10 magn.
star follows 30", but is not connected with the nebula. None of the many
small nebulae on this plate appear to coincide with N. G. C. 21 (Swift).
41 s.n.
A small, fairly bright, 0-type spiral, not noted in the N. G. C.
Somewhat irregular spiral V.5 x O1o in p. a. 160°. Nucleus almost stellar; a 13
magn. star is close to whorl on north, but probably not physically connected.
Planetary. 1 s.n.

Described as eF, eL, Dif. No trace in an exposure of lh 2n>; either exceedingly faint or non-existent. 8 s.n. 68 0 13.2 +29 31 A rather faint, slightly oval spiral about V.5 in diameter, with a fairly bright,

almost stellar nucleus; two 14 magn. stars are involved, but are evidently not physically connected with the nebula. This region is unusually thick with small nebulae and nebulous stars: several of these are small spirals. N. G. C. 67, 69-72, and 75 are all small nebulae or nebulous stars. 98 s.n. A small area of diffuse nebulosity about 3'x2'; very patchy, with a number of

nebulous condensations. Exceedingly faint. 6 s.n. Thirteen nebulae are catalogued in this vicinity in the N. G. C; in reality there are fifty or more small nebulae and nebulous stars. 83 is 15" in diameter, round, bright, structureless. 91 is an interesting two-branched spiral 1' x 0'.3 in p.a. 90°. 55 s.n. Description in Bull. 248 erroneous; not yet photographed.

A fine, symmetrical, rather open spiral about 4' x 2' in p.a. 80°. Whorls are faint; the nucleus large and moderately bright. 157 0 29.7 — 8 57 A fine, bright, slightly oval spiral 215 in length. Almost stellar nucleus; num

erous almost stellar condensations. 17 s.n. A small, bright, very compact oval spiral 0'.6 long; almost stellar nucleus. Relatively bright center; the spiral whorls merge into a very faint oval ring

1'8 long. 48 s.n. A small and very faint oval, 20" long. Almost stellar; structureless; bright nucleus. A very faint spindle is 1'5 s.p.

32 s.n. Very faint, irregular spiral, 018 long; very faint stellar nucleus. Nearly round, 1' in diameter. A very faint, rather regular spiral. Nucleus

almost stellar.
This is a double nebula; the centers are about 28" apart. The northern one is
spindle-shaped, a spiral seen nearly edgewise, with evidence of an absorbing
lane at one side; V.2 long. The southern nebula is a bright oval 013 long;
probably a spiral. See Abs. Eff.
Oval, 017 long; center considerably brighter; probably spiral.
Rather irregular; slightly oval, 3' long; there are two curious rifts near the
nucleus; it appears to be an irregular spiral. The nebular matter is faint
and diffuse. A star of magn. 14 is north of the very faint nucleus. 30 s.n.
Nearly round; 015 in diameter; much stronger center; probably spiral.
Round; O'.'S in diameter; fairly bright; no structure discernible.
Nearly round spiral 1' in diameter; quite faint.

Slightly oval spiral 018 long whorls faint; center moderately bright.
Rather faint; the nucleus is almost stellar; an interesting two-branched spiral
118 i 014. In addition to the nine N. G. C. objects found on this plate, there
are sixty-five other small nebulae. 65 s.n.
Small, faint, considerably elongated spiral 015 long.

Considerably elongated; 017 long; center much brighter; doubtless spiral. Slightly oval; much brighter center; O'.S in diameter; probably spiral. The companion n.p. the nebula in Andromeda. The bright central portion is about 2' in diameter, showing traces of rather irregular spiral structure; the nucleus is almost stellar. There are two small dark patches near the brighter central portion. Very much fainter matter forms the outer portion of the nebula in an oval about 8'x3'; no whorls can be made out in this outer portion; doubtless a spiral of the Andromeda type. 2 s.n. 221 0 37.2 +40~ 19 The well-known companion south of the nebula in Andromeda. Exceedingly

bright. In the long exposures it appears as a "burnt-out" oval 216 x 1'8, with no trace of spiral character in the outer portions. It lies in p.a. 150° ±. The shortest exposures show a nucleus which is nebulous, surrounded by bright nebular matter far brighter than the brightest parts of the Nebula in Andro'eda. The nucleus and inner nebular matter show strongly in lm on S23.

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. DESCRIPTION— (Continued)

0 37 +40 43 The Great Nebula in Andro'eda, Vol. VIII, Plate 1. This wonderful object, the

largest of the spiral nebulae, is too well known to need description. Exposures of lm to 3m on S23 show an almost stellar nucleus, with traces of spiral structure in the surrounding nebular matter. See Abs. Eff.

Planetary. 11 s.n.

A faint, very large, spiral, 18'x 5' in p.a. 175°. Nucleus stellar; many almost stellar condensations. Whorls somewhat irregular, and indistinct. 22 s.n.

Vol. VIII, Plate 2. One of the most beautiful spirals known, 21' x 4' in p.a. 52°. There is perhaps a very faint, almost stellar nucleus and numerous almost stellar condensations. Some evidence of absorption lane effects on the northern side. 23 s.n.

Nearly round; 1' in diameter; sharp nucleus; a rather compact spiral.

A very bright, compact, nearly round spiral, 112 in diameter. The nucleus is well defined and shows in a 5m exposure. The matter in the inner whorls is quite bright. 6 s.n.

A large patch of faint, diffuse, irregular nebulosity 16' across. Shows considerable detail and a clearly defined "hole" with clear-cut edges. A large "bay" cuts into the s.p. side. 1? s.n.

Two enormous, fan-shaped masses of diffuse nebulosity n.f. y Caxsiopeiae. Each one is about 12' across at the base, while the apices point approximately toward the bright star. They show considerable structural detail. The brightest part is a streak at the apex of the southern nebula. See figure 4. 0 s.n.

Described as vF, eeL by, Wolf. There are a number of small nebulae in this region, but no trace of II 1613 in an exposure of lh 44m. 33 s.n.

Rather faint, considerably elongated spiral V.5 in length; nucleus well defined.

Round; 113 in diameter; quite bright. The nucleus is stellar; no spiral structure is apparent, but there is a minute, curved, dark lane n.f. the nucleus. 26 s.n.

A spindle 1' long; moderately bright. Probably spiral; an absorbing lane is suspected.

Slightly oval; 1' long; nucleus almost stellar; no whorls or spiral structure discernible, but is probably a spiral. 46 s.n.

Very small; binuclear. The almost stellar nuclei are 7" apart in p.a. 142°.

This is an extremely irregular mass of nebulosity 4' x 212. It shows an irregular looped structure with numerous stellar condensations; probably a greatly distorted spiral. 47 s.n.

Quite bright, round, 012 in diameter; nucleus almost stellar; no spiral structure.

A rather irregular, somewhat patchy spiral, whose whorls form an oval ring U6 long in p.a. 155°; there is an unsymmetrical fainter extension at the south. Nucleus 13 magn.; almost stellar. 14.9 +2 53 Round; 015 in diameter. Center very bright, with much fainter matter outside. No spiral structure discernible. 28 s.n.

Very small and faint; nearly round; probably spiral.

Nearly round; 3' in diameter; very bright, almost stellar nucleus. An unusually symmetrical, rather faint spiral. The whorls are very delicate, and so close together that at first sight they appear to be rings around the nucleus instead of spiral whorls. 31 s.n.

Nebulous star.

There are two nebulae at this point, about 1' apart. The preceding one is very faint, nearly round, 0'A in diameter, structureless. The following one is probably 492; 015 in diameter; stellar nucleus; a small spiral.

Very small; almost stellar; perhaps spiral.

A faint, nearly round spiral; 2' in diameter; quite regular and symmetrical. Nucleus almost stellar; a number of almost stellar condensations. 19.6 +9 1 Round; bright; 018 in diameter. Nucleus almost stellar. Very indistinct traces of spiral character. A number of small spindle-shaped nebulae in this region. 23 s.n. 26.3 — 7 23 Quite bright, with strong central portion and a sharp nucleus; 2' long; considerably elongated; no whorls visible. 28.2 +30 9 Vol. VIII, Plate 3. A close rival to the Nebula of Andromeda as the most beautiful spiral known. With its faintest extensions it covers an area at least 55' x 40'. Messier 33 Trianguli. 5S8, 592, 595, and 603 are simply brighter portions of 598. It is uncertain whether there is an actual stellar nucleus. A multitude of stellar condensations in the whorls; the spiral which furnishes the best known example of "resolution" into stars. 7 s.n.

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