Chapter IV: Part II: Statistical Study of Extra-Galactic Nebulae (2)

The range in the stars involved is about 2.5 mag., and in the total luminosities of the nebulae, about 3.8 mag. This latter is consistent with the scatter in the diagram exhibiting the relation between total luminosities and diameters. The associated objects, M 31 and 32, represent the extreme limits among the known systems, and the mean of these two is very close to the mean of them all.

LUMINOSITY OF STARS INVOLVED IN NEBULAE

The number of nebulae of known distance is too small to serve as a basis for estimates of the range in absolute magnitude among nebulae in general. Further information, however, can be derived from a comparison of total apparent magnitudes with apparent magnitudes of the brightest stars involved, on the reasonable assumption, supported by such evidence as is available, that the brightest stars in isolated systems are of about the same intrinsic luminosity.

The most convenient procedure is to test the constancy of the differences in apparent magnitude between the brightest stars involved and the nebulae themselves, over as wide a range as possible in the latter quantities.

An examination of the photographs in the Mount Wilson collection has revealed no stars in the very faint objects or in the bright elliptical nebulae and early-type spirals. This was to be expected from the conclusions previously derived. Observations were therefore confined to intermediate- and late-type spirals and the irregular nebulae to the limiting visual magnitude 10.5. The Magellanic Clouds and N.G.C. 6822 were added to the nebulae in Holetschek’s list. Altogether, data were available for 32 objects, or about 60 per cent of the total number in the sky to the adopted limit. For this reason it is believed that the results are thoroughly representative.

TABLE XVI
Difference in Magnitude between Nebulae and
Their Brightest Stars
==========+=============+============+===============
N.G.C. | m_{s} | m_{T} | m_{s} – m_{T}
----------+-------------+------------+---------------

Sb
----------+-------------+------------+---------------
224 | 15.5 | 5.0 | 10.5
1068 | 17.5 | 9.1 | 8.4
2841 | >19.5 | 9.4 | >10.1
3031 | 18.5 | 8.3 | 10.2
3310 | >19.0 | 10.4 | > 8.6
3623 | >20.0 | 9.9 | >10.1
3627 | 18.5 | 9.1 | 9.4
4438 | >19.0 | 10.3 | > 8.7
4450 | 19.5 | 10.0 | 9.5
4736 | 17.3 | 8.4 | 8.9
4826 | >19.5 | 9.2 | >10.3
5055 | >19.0 | 9.6 | > 9.4
5746 | >19.5 | 10.4 | > 9.1
7331 | 19.0 | 10.4 | 8.6
----------+-------------+------------+---------------

SBb
----------+-------------+------------+---------------
4699 | >19.5 | 10.0 | > 9.5
----------+-------------+------------+---------------

Sc
----------+-------------+------------+---------------
253 | 18.3 | 9.3 | 9.0
598 | 15.6 | 7.0 | 8.6
2403 | 17.3 | 8.7 | 8.6
2683 | >20.0 | 9.9 | >10.1
2903 | 19.0 | 9.1 | 9.9
4254 | 18.5 | 10.4 | 8.1
4321 | 18.8 | 10.5 | 8.3
4414 | >19.5 | 10.1 | > 9.4
4490 | 18.8 | 10.2 | 8.6
5194 | 17.3 | 7.4 | 9.9
5236 | 18.6 | 10.4 | 8.2
5457 | 17.0 | 9.9 | 7.1
----------+-------------+------------+---------------

Irr.
----------+-------------+------------+---------------
LMC | 9.5 | 0.5 | 9.0
SMC | 12.0 | 1.5 | 10.5
3034 | >19.5 | 9.0 | >10.5
4449 | 17.8 | 9.5 | 8.3
6822 | 15.8 | 8.5 | 7.3
----------+-------------+------------+---------------

The data are listed in Table XVI and are shown graphically in Figure 10. The luminosities of the brightest stars are given in photographic magnitudes. For the Magellanic Clouds, M 33, and N.G.C. 6822, these were obtained from published star counts. For M 31, 51, 63, 81, 94, and N.G.C. 2403, they depend upon unpublished counts, for which the magnitudes were determined by comparisons with Selected Areas. For the remaining nebulae, the magnitudes of stars were estimated with varying degrees of precision, but are probably less than 0.5 mag. in error.

The sloping line to the right in Figure 10 represents the limits of the observations, for, from a study of the plates themselves, it appeared improbable that stars fainter than about 19.5 could be detected with certainty on a nebulous background. Points representing nebulae in which individual stars could not be found should lie in this excluded region above the line, and their scatter is presumably comparable with that of the points actually determined below the line. When allowance is made for this inaccessible region, the data can be interpreted as showing a moderate dispersion around the mean ordinate

$m_{s} - m_{T} = 9.0.$ (7)

The range in total magnitudes is sufficiently large in comparison with the dispersion to lend considerable confidence to the conclusion. The total range of four, and the average dispersion of less than 1 mag., are comparable with those in Table XV and in Figure 7, and agree with the former in indicating a constant order of absolute magnitude.

The mean absolute magnitude of the brightest stars in the nebulae listed in Table XV, combined with the mean difference between nebulae and their brightest stars, furnishes a mean absolute magnitude of –15.3 for the nebulae listed in Table XVI. This differs by only 0.2 mag. from the average of the nebulae in Table XV, and the mean of the two, –15.2, can be used as the absolute magnitude of intermediate- and late-type spirals and irregular nebulae whose apparent magnitudes are brighter than 10.5. The dispersion is small and can safely be neglected in statistical investigations.

This is as far as the positive evidence can be followed. For reasons already given, however, it is presumed that the earlier nebulae, the elliptical and the early-type spirals, are of the same order of absolute magnitude as the later. The one elliptical nebula whose distance is known, M 32, is consistent with this hypothesis.

Conclusions concerning the intrinsic luminosities of the apparently fainter nebulae are in the nature of extrapolations of the results found for the brighter objects. When the nebulae are reduced to a standard type, they are found to be constructed on a single model, with the total luminosities varying directly as the square of the diameters. The most general interpretation of this relation is that the mean surface brightness is constant, but the small range in absolute magnitudes among the brighter nebulae indicates that, among these objects at least, the relation merely expresses the operation of the inverse-square law on comparable objects distributed at different distances. The actual observed range covered by this restricted interpretation is from apparent magnitude 0.5 to 10.5. The homogeneity of the correlation diagrams and the complete absence of evidence to the contrary justify the extrapolation of the restricted interpretation to cover the 2 or 3 mag. beyond the limits of actual observation.

These considerations lead to the hypothesis that the nebulae treated in the present discussion are all of the same order of absolute magnitude; in fact, they lend considerable color to the assumption that extra-galactic nebulae in general are of the same order of absolute magnitude and, within each class, of the same order of actual dimensions. Some support to this assumption is found in the observed absence of individual stars in the apparently fainter late-type nebulae. If the luminosity of the brightest stars involved is independent of the total luminosity of a nebula, as is certainly the case among the brighter objects, then, when no stars brighter than 19.5 are found, the nebulae must in general be brighter than absolute magnitude m_{T} – 25.8 where m_{T} is the total apparent magnitude. On this assumption, the faintest of the Holetschek nebulae are brighter than –12.5 and hence of the same general order as the brighter nebulae.

Once the assumption of a uniform order of luminosity is accepted as a working hypothesis, the apparent magnitudes become, for statistical purposes, a measure of the distances. For a mean absolute magnitude of –15.2, the distance in parsecs is

$\log D = 0.2 m_{T} + 4.04.$ (8)

DIMENSIONS OF EXTRA-GALACTIC NEBULAE

When the distances are known, it is possible to derive actual dimensions and hence to calibrate the curve in Figure 6, which exhibits the apparent diameters as a function of type, or stage in the nebular sequence, for nebulae of a given apparent magnitude. The mean maximum diameters in parsecs corresponding to the different mean types are given in Table XVII. For the elliptical nebulae, values are given both for the statistical mean observed diameters and for the diameter as calculated for the pure types.

Spirals at the last stage in the observed sequence have diameters of the order of 3000 parsecs. Assuming 1:10 as the ratio of the two axes, the corresponding volume is of the order of 1.4×10^9 cubic parsecs, and the mean luminosity density is of the order of 7.7 absolute magnitudes per cubic parsec as compared with 8.15 for the galactic system in the vicinity of the sun. These results agree with those of Seares who, from a study of surface brightness, concluded that the galactic system must be placed at the end of, if not actually outside, the series of known spirals when arranged according to density.[19]

TABLE XVII
========+=======================+=======+=====================
Type | Diameter in Parsecs | Type | Diameter in Parsecs
+-----------+-----------+ |
| Obs. | Cal. | |
--------+-----------+-----------+-------+---------------------
E0 | 360 | 340 | Sa | 1450
E1 | 430 | 380 | Sb | 1900
E2 | 500 | 430 | Sc | 2500
E3 | 590 | 490 | |
E4 | 700 | 570 | SBa | 1280
E5 | 810 | 680 | SBb | 1320
E6 | 960 | 850 | SBc | 2250
E7 | 1130 | 1130 | Irr | 1500
--------+-----------+-----------+-------+---------------------

MASSES OF EXTRA-GALACTIC NEBULAE

Spectroscopic rotations are available for the spirals M 31[20] and N.G.C. 4594,[21] and from these it is possible to estimate the masses on the assumption of orbital rotation around the nucleus. The distances of the nebulae are involved, however, and this is known accurately only for M 31; for N.G.C. 4594 it must be estimated from the apparent luminosity.

Another method of estimating masses is that used by Öpik[22] in deriving his estimate of the distance of M 31. It is based on the assumption that luminous material in the spirals has about the same coefficient of emission as the material in the galactic system. Öpik computed the ratio of luminosity to mass for our own system in terms of the sun as unity, using Jeans’s value[23] for the relative proportion of luminous to non-luminous material. The relation is

Mass = 2.6L. (9)

The application of this method of determining orders of masses seems to be justified, at least in the case of the later-type spirals and irregular nebulae, by the many analogies with the galactic system itself. Moreover, when applied to M 31, where the distance is fairly well known, it leads to a mass of the same order as that derived from the spectrographic rotation:

MASS OF M 31

Spectrographic rotation 3.5×10^9 ☉
Öpik’s method 1.6×10^9

The distance of N.G.C. 4594 is unknown, but the assumption that it is a normal nebula with an absolute magnitude of –15.2 places it at 700,000 parsecs. The orders of the mass by the two methods are then

MASS OF N.G.C. 4594

Spectrographic rotation 2.0×10^9 ☉
Öpik’s method 2.6×10^8

Here again the resulting masses are of the same order. They can be made to agree as well as those for M 31 by the not unreasonable assumption that the absolute luminosity of the nebula is 2 mag. or so brighter than normal.

Öpik’s method leads to values that are reasonable and fairly consistent with those obtained by the independent spectrographic method. Therefore, in the absence of other resources, its use for deriving the mass of the normal nebula appears to be permissible. The result, 2.6 × 10^8 ☉, corresponding to an absolute magnitude of –15.2, is probably of the right order. The two test cases suggest that this value may be slightly low, but the data are not sufficient to warrant any empirical corrections.

NUMBERS OF NEBULAE TO DIFFERENT LIMITING MAGNITUDES

The numbers of nebulae to different limiting magnitudes can be used to test the constancy of the density function, or, on the hypothesis of uniform luminosities, to determine the distribution in space. The nebulae brighter than about the tenth magnitude are known individually. Those not included in Holetschek’s list are: the Magellanic Clouds, the two nebulae N.G.C. 55 and 1097, between 9.0 and 9.5 mag., and the seven nebulae N.G.C. 134, 289, 1365, 1533, 1559, 1792, and 3726, all between 9.5 and 10.0 mag.

A fair estimate of the number between 10.0 and 11.0 mag. can be derived from a comparison of Holetschek’s list with that of Hardcastle, an inspection of images on the Franklin-Adams charts and other photographs, and a correlation between known total magnitudes and the descriptions of size and brightness in Dreyer’s catalogues. It appears that very few of these objects were missed by Holetschek in the northern sky—not more than six of Hardcastle’s nebulae. For the southern sky, beyond the region observed by Holetschek, the results are very uncertain, but probable upper and lower limits were determined as 50 and 20, respectively. The brighter nebulae are known to be scarce in those regions. A mean value of 35 leads to a total 295 for the entire sky, and this is at least of the proper order.

The number of nebulae between 11.0 and 12.0 mag. can be estimated on the assumption that the two lists, Holetschek’s and Hardcastle’s, are about equally complete within this range. They are known to be comparable for the brighter nebulae, and, moreover, the total numbers included in the two lists for the same area of the sky, that north of declination –10°, are very nearly equal—400 as compared with 408. The percentages of Holetschek’s nebulae included by Hardcastle were first determined as a function of magnitude. Within the half-magnitude interval 11.0 to 11.5, for instance, 60 per cent are in Hardcastle’s list. If the two lists are equally complete and, taken together, are exhaustive, the total number in the interval will be 1.4 times the number of Holetschek’s nebulae. The latter is found to be 50 from smoothed frequency curves of the magnitudes listed in Tables I–IV. The total number north of –10° is therefore 70. This can be corrected to represent the entire sky by applying the factor 1.75, which is the ratio of the total number of Hardcastle’s nebulae, 700, to the number north of –10°, 400. In this manner a reasonable estimate of 123 is obtained for the number of nebulae in the entire sky between 11.0 and 11.5 mag. Similarly, between 11.5 and 12.0, where 50 per cent of Holetschek’s nebulae are included in Hardcastle’s list, the total number for the entire sky is found to be 236.

The greatest uncertainty in these figures arises from the assumption that the two lists together are complete to the twelfth magnitude. The figures are probably too small, but no standards are available by which they can be corrected. It is believed, however, that the errors are certainly less than 50 per cent and probably not more than 25 per cent. This will not be excessive in view of the possible deviations from uniform distribution where so limited a number of objects is considered.

Beyond 12.0 mag. the lists quickly lose their aspect of completeness and cannot be used for the present purpose. There are available, however, the counts by Fath[24] of nebulae found on plates of Selected Areas made with the 60-inch reflector at Mount Wilson. The exposures were uniformly 60 minutes on fast plates and cover the Areas in the northern sky down to and including the –15° zone. The limiting photographic magnitudes for stars average about 18.5. The counts have been carefully revised by Seares[25] and are the basis for his estimate of 300,000 nebulae in the entire sky down to this limit.

Approximate limiting total magnitudes for the nebulae in two of the richest fields, S.A. 56 and 80, have been determined from extra-focal exposures with the 100-inch reflector. The results are 17.7 in each case, and this, corrected by the normal color-index of such objects, gives a limiting visual magnitude of about 16.7, which can be used for comparison with the counts of the brighter nebulae.

The various data are collected in Table XVIII, where the observed numbers of extra-galactic nebulae to different limits of visual magnitude are compared with those computed on the assumption of uniform distribution of objects having a constant absolute luminosity. The formula used for the computation is

$\log N = 0.6 m_{T} + \textrm{Constant},$

where the constant is the value of log N for m_{T} = 0. The value —4.45 is found to fit the observational data fairly well.

The agreement between the observed and computed log N over a range of more than 8 mag. is consistent with the double assumption of uniform luminosity and uniform distribution or, more generally, indicates that the density function is independent of the distance.

The systematic decrease in the residuals O – C with decreasing luminosity is probably within the observational errors, but it may also be explained as due to a clustering of nebulae in the vicinity of the galactic system. The cluster in Virgo alone accounts for an appreciable part. This is a second-order effect in the distribution, however, and will be discussed at length in a later paper.

TABLE XVIII
Numbers of Nebulae to Various Limits
========+===================+===========+========
| LOG N* | |
m_{T} |---------+---------| O – C | LOG D†
| O | C | |
--------+---------+---------+-----------+--------
8.5 | 0.85 | 0.65 | +0.20 | 5.74
9.0 | 1.08 | 0.95 | .13 | 5.84
9.5 | 1.45 | 1.25 | .20 | 5.94
10.0 | 1.73 | 1.55 | .18 | 6.04
10.5 | 1.95 | 1.85 | .10 | 6.14
11.0 | 2.17 | 2.15 | + .02 | 6.24
11.5 | 2.43 | 2.45 | – .02 | 6.34
12.0 | 2.70 | 2.75 | .05 | 6.44
16.7 | 5.48 | 5.57 | –0.09 | 7.38
(18.0) | | (6.35) | | (7.64)
--------+---------+---------+-----------+--------

* Log N = 0.6 m_{T} – 4.45.
† Log D = 0.2 m_{T} + 4.04.

Distances corresponding to the different limiting magnitudes, as derived from formula (8), are given in the last column of Table XVIII. The 300,000 nebulae estimated to the limits represented by an hour’s exposure on fast plates with the 60-inch reflector appear to be the inhabitants of space out to a distance of the order of 2.4 × 10^7 parsecs. The 100-inch reflector, with long exposures under good conditions, will probably reach the total visual magnitude 18.0, and this, by a slight extrapolation, is estimated to represent a distance of the order of 4.4×10^7 parsecs or 1.4×10^8 light-years, within which it is expected that about two million nebulae should be found. This seems to represent the present boundaries of the observable region 3 of space.

DENSITY OF SPACE

The data are now available for deriving a value for the order of the density of space. This is accomplished by means of the formulae for the numbers of nebulae to a given limiting magnitude and for the distance in terms of the magnitude. In nebulae per cubic parsec, the density is

$\begin{align}
\log \rho & = \log N - \log V\\
& = (0.6 m_{T} - 4.45) -\ log\frac {4π}{3} - 3(4.04 + 0.2 m_{T})\\
& = -17.19
\end{align}$ (11)

This is a lower limit, for the absence of nebulae in the plane of the Milky Way has been ignored. The current explanation of this phenomenon in terms of obscuration by dark clouds which encircle the Milky Way is supported by the extra-galactic nature of the nebulae, their general similarity to the galactic system, and the frequency with which peripheral belts of obscuring material are encountered among the spirals. The known clouds of dark nebulosity are interior features of our system, and they do not form a continuous belt. In the regions where they are least conspicuous, however, the extra-galactic nebulae approach nearest to the plane of the Milky Way, many being found within 10°. This is consistent with the hypothesis of a peripheral belt of absorption.

The only positive objection which has been urged to this explanation has been to the effect that the nebular density is a direct function of galactic latitude. Accumulating evidence[26] has failed to confirm this view and indicates that it is largely due to the influence of the great cluster in Virgo, some 15° from the north galactic pole. There is no corresponding concentration in the neighborhood of the south pole.

If an outer belt of absorption is assumed, which, combined with the known inner clouds, obscures extra-galactic nebulae to a mean distance of 15° from the galactic plane, the value derived for the density of space must be increased by nearly 40 per cent. This will not change the order of the value previously determined and is within the uncertainty of the masses as derived by Öpik’s method. The new value is then

ρ = 9×10^{–18} nebulae per cubic parsec. (12)

The corresponding mean distance between nebulae is of the order of 570,000 parsecs, although in several of the clusters the distances between members appear to be a tenth of this amount or less.

The density can be reduced to absolute units by substituting the value for the mean mass of a nebula, 2.6×10^8 ☉. Then, since the mass of the sun in grams is 2×10^{33} and 1 parsec is 3.1×10^{18} cm,

ρ = 1.5×10^{–31} grams per cubic centimeter. (13)

This must be considered as a lower limit, for loose material scattered between the systems is entirely ignored. There are no means of estimating the order of the necessary correction. No positive evidence of absorption by inter-nebular material, either selective or general, has been found, nor should we expect to find it unless the amount of this material is many times that which is concentrated in the systems.

THE FINITE UNIVERSE OF GENERAL RELATIVITY

The mean density of space can be used to determine the dimensions of the finite but boundless universe of general relativity. De Sitter[27] made the calculations some years ago, but used values for the density, 10^{–26} and greater, which are of an entirely different order from that indicated by the present investigations. As a consequence, the various dimensions, both for spherical and for elliptical space, were small as compared with the range of existing instruments.

For the present purpose, the simplified equations which Einstein has derived for a spherically curved space can be used.[28] When R, V, M, and ρ represent the radius of curvature, volume, mass, and density, and k and c are the gravitational constant and the velocity of light,

$R = \frac{c}{\sqrt{4πk}} \cdot \frac{1}{\sqrt{\rho}},$ (14)

$V = 2π^2R^3,$ (15)

$M = πc^2/2k · R.$ (16)

Substituting the value found for ρ, 1.5×10^{–31}, the dimensions become

R = 8.5×10^{28} cm = 2.7×10^{10} parsecs, (17)

V = 1.1×10^{88} cm = 3.5×10^{32} cubic parsecs, (18)

M = 1.8×10^{57} grams = 9×10^{22} ☉. (19)

The mass corresponds to 3.5×10^{15} normal nebulae.

The distance to which the 100-inch reflector should detect the normal nebula was found to be of the order of 4.4×10^{75} parsecs, or about 1⁄600 the radius of curvature. Unusually bright nebulae, such as M 31, could be photographed at several times this distance, and with reasonable increases in the speed of plates and size of telescopes it may become possible to observe an appreciable fraction of the Einstein universe.

Mount Wilson Observatory
September 1926

[1] _Contributions from the Mount Wilson Observatory_, No. 324.

[2] These are the two Magellanic Clouds, M 31, and M 33.

[3] Bailey, _Harvard Annals_, ❰60❱, 1908.

[4] Hardcastle, _Monthly Notices_, ❰74❱, 699, 1914.

[5] This estimate by Seares is based on a revision of Fath’s counts of nebulae in Selected Areas (_Mt. Wilson Contr._, No. 297; _Astrophysical Journal_, ❰62❱, 168, 1925).

[6] “A General Study of Diffuse Galactic Nebulae,” _Mt. Wilson Contr._, No. 241; _Astrophysical Journal_, ❰56❱, 162, 1922.

[7] The classification was presented in the form of a memorandum to the Commission on Nebulae of the International Astronomical Union in 1923. Copies of the memorandum were distributed by the chairman to all members of the Commission. The classification was discussed at the Cambridge meeting in 1925, and has been published in an account of the meeting by Mrs. Roberts in _L’Astronomie_, ❰40❱, 169, 1926. Further consideration of the matter was left to a subcommittee, with a resolution that the adopted system should be as purely descriptive as possible, and free from any terms suggesting order of physical development (_Transactions of the I.A.U._, 2, 1925). Mrs. Roberts’ report also indicates the preference of the Commission for the term “extra-galactic” in place of the original, and then necessarily non-committal, “non-galactic.”

Meanwhile K. Lundmark, who was present at the Cambridge meeting and has since been appointed a member of the Commission, has recently published (_Arkiv für Matematik, Astronomi och Fysik, Band 19B_, No. 8, 1926) a classification, which, except for nomenclature, is practically identical with that submitted by me. Dr. Lundmark makes no acknowledgments or references to the discussions of the Commission other than those for the use of the term “galactic.”

[8] _Problems of Cosmogony and Stellar Dynamics_, 1919.

[9] N.G.C. 4486 (M 87) may be an exception. On the best photographs made with the 100-inch reflector, numerous exceedingly faint images, apparently of stars, are found around the periphery. It was among these that Belanowsky’s nova of 1919 appeared. The observations are described in _Publications of the Astronomical Society of the Pacific_, ❰35❱, 261, 1923.

[10] “Early” and “late,” in spite of their temporal connotations, appear to be the most convenient adjectives available for describing relative positions in the sequence. This sequence of structural forms is an observed phenomenon. As will be shown later in the discussion, it exhibits a smooth progression in nuclear luminosity, surface brightness, degree of flattening, major diameters, resolution, and complexity. An antithetical pair of adjectives denoting relative positions in the sequence is desirable for many reasons, but none of the progressive characteristics are well adapted for the purpose. Terms which apply to series in general are available, however, and of these “early” and “late” are the most suitable. They can be assumed to express a progression from simple to complex forms.

An accepted precedent for this usage is found in the series of stellar spectral types. There also the progression is assumed to be from the simple to the complex, and in view of the great convenience of the terms “early” and “late,” the temporal connotations, after a full consideration of their possible consequences, have been deliberately disregarded.

[11] _Publications of the Lick Observatory_, ❰13❱, 12, 1918.

[12] _H_{β}_ is brighter than N_{2}. Patches with similar spectra are often found in the arms of late-type spirals—N.G.C. 253, M 33, M 101. The typical planetary spectrum, where _H_{β}_ is fainter than N_{2}, is found in the rare cases of apparently stellar nuclei of spirals; for instance, in N.G.C. 1068, 4051, and 4151. Here also the emission spectra are localized and do not extend over the nebulae.

[13] _Monthly Notices_. ❰74❱, 699, 1914.

[14] _Annalen der Wiener Sternwarte_, ❰20❱, 1907.

[15] _Astronomische Nachrichten_, ❰214❱, 425, 1921.

[16] _Publications of the Lick Observatory_, ❰13❱, 1918.

[17] Since C is constant for all nebulae in a given class, the linear relation between Δ log d and C for the different classes is something more than a mere geometrical relation arising from the observed equality of the mean m_{T} in the various classes.

[18] This is apparent even among the observed classes. Referring to formula (3), m_{T} + 5 log b will be constant in so far as C_{e} + 5 log (1 – e) is constant. The following table indicates that the latter term is approximately constant throughout the sequence of elliptical nebulae. The values of C_{e} were read from the smooth curve in Fig. 6.

======+=========+================+==========+===========
e | C_{e} | 5 log (1 – e) | C_{0} | Res.
------+---------+----------------+----------+-----------
0 | 10.30 | 0.0 | 10.30 | –0.14
1 | 10.65 | – .23 | 10.42 | – .02
2 | 11.00 | .48 | 10.52 | + .10
3 | 11.35 | 0.78 | 10.57 | + .13
4 | 11.70 | 1.11 | 10.59 | + .15
5 | 12.05 | 1.50 | 10.55 | + .11
6 | 12.40 | 1.99 | 10.41 | – .03
7 | 12.75 | –2.62 | 10.13 | –0.31
+---------+----------------+----------+-----------
Mean | | | 10.44 | 0.12
------+---------+----------------+----------+-----------

[19] _Mt. Wilson Contr._, No. 191; _Astrophysical Journal_, ❰52❱, 162, 1920.

[20] Pease, _Mt. Wilson Comm._, No. 51; _Proceedings of the National Academy of Sciences_, ❰4❱, 21, 1918.

[21] Pease, _Mt. Wilson Comm._, No. 32: _ibid._, ❰2❱, 517, 1916.

[22] _Astrophysical Journal_, ❰55❱, 406, 1922.

[23] _Monthly Notices_, ❰82❱, 133, 1922.

[24] _Astronomical Journal_, ❰28❱, 75, 1914.

[25] _Mt. Wilson Contr._, No. 297; _Astrophysical Journal_, ❰62❱, 168, 1925.

[26] The latest and most reliable results bearing on the distribution of faint (hence apparently distant) nebulae are found in Seares’s revision and discussion of the counts made by Fath on plates of the Selected Areas with the 60-inch reflector. When the influence of the cluster in Virgo is eliminated the density appears to be roughly uniform for all latitudes greater than about 25°.

[27] _Monthly Notices_, ❰78❱, 3, 1917.

[28] Haas, _Introduction to Theoretical Physics_, ❰2❱, 373, 1925.

Transcriber’s Notes

1. Footnotes have been renumbered and moved to the end of the paper.

2. Tables have been moved to avoid splitting paragraphs. In some
cases the tables have been reconfigured or split in order to work
better with e-readers.

3. Except as mentioned above every effort has been made to replicate
this text as faithfully as possible, including non-standard
punctuation, inconsistently hyphenated words, etc.

4. The original printed text is available from HathiTrust at:
https://hdl.handle.net/2027/uc1.b3805680 (see pg 353). The outline
“CLASSIFICATION OF NEBULAE” starting on page 3 lists N.G.C. 2117
as a type E7 nebula. The version of this paper printed in
_Contributions from the Mount Wilson Observatory_ No. 324 lists
N.G.C 3115. Otherwise the papers are essentially identical.

Special notation for the plain text version:

1. Text enclosed by underscores “_” is italicised.

2. Text enclosed by macrons “¯” has a line draw over it.

3. Text enclosed in braces following an underscore “_{ }” is
subscript.

4. Text enclosed in braces following a circumflex accent “^{ }”is
superscript. If only a single character, the braces may be
omitted.

5. Text enclosed in heavy angle brackets “❰ ❱” is bold.

6. Text enclosed in dollar signs "$" is in TeX notation.