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The results are given in Table X, where the actual ellipticities, listed in the first column, are followed across the table by the percentages which, on the assumption of random orientation, will be observed as having the various apparent ellipticities. The bottom row will be seen to show the percentages of apparent ellipticities observed in an assembly of nebulae in which the numbers for each actual ellipticity are equal and all are oriented at random.

+ N.G.C. 524 and 3998 are included as E0 and E1, respectively.

From the dynamical point of view, the empirical results are consistent with the general order of events in Jeans's theory. Thus interpreted, the series is one of expansion, and the scale of types becomes the time scale in the evolutionary history of nebulae. In two respects this scale is not entirely arbitrary. Among the elliptical nebulae the successive types differ by equal increments in the ellipticity or the degree of flattening, and among the spirals the intermediate stage is midway between the two end-stages in the structural features as well as in the luminosity relations.

One other feature of the curves may be discussed from the point of view of Jeans's theory before returning to the strictly empirical attitude. The close agreement of the diameters for the stages E7 and Sa suggests that the transition from the lenticular nebula to the normal spiral form is not cataclysmic. If the transition were gradual, however, we should expect to observe occasional objects in the very process, but among the thousand or so nebulae whose images have been inspected, not one clear case of a transition form has been detected. The observations jump suddenly from lenticular nebulae with no trace of structure to spirals in which the arms are fully developed.

If the numerical data could be fully trusted, the SBa forms would fill the gap. Among these nebulae, the transition from the lenticular to the spiral with arms is gradual and complete. It is tempting to suppose that the barred spirals do not form an independent series parallel with that of the normal spirals, but that all or most spirals begin life with the bar, although only a few maintain it conspicuously throughout their history. This would also account for the fact that the relative numbers of the SBa nebulae are intermediate to those of the lenticular and of the Sa. The normal spirals become more numerous as the sequence progresses, while the numbers of barred spirals, on the contrary, actually decrease with advancing type.

RELATION BETWEEN NUCLEAR LUMINOSITIES AND DIAMETERS

where the slope differs by about 1 per cent from that in formula . The list contains 16 elliptical nebulae, 15 normal, and 6 barred spirals. The nebulae are fairly representative, except that few late-type spirals are included. This is an effect of selection due to the fact that nuclei become less and less conspicuous as the sequence progresses.

The parallelism of the two curves representing formulae and indicates that the regular extra-galactic nebulae, when reduced to the standard type, are similar objects. The mean surface brightness is constant, and the luminosity of the nucleus, as measured by Hopmann, is a constant fraction, about one-fourth, of the total luminosity of the nebulae. If there is a considerable range in absolute magnitude and hence in actual dimensions, the smaller nebulae must be faithful miniatures of the larger ones.

ABSOLUTE MAGNITUDES OF EXTRA-GALACTIC NEBULAE

Reliable values of distances, and hence of absolute magnitudes, are restricted to a very few of the brightest nebulae. These are derived from a study of individual stars involved in the nebulae, among which certain types have been identified whose absolute magnitudes in the galactic system are well known. The method assumes that the stars involved in the nebulae are directly comparable with the stars in our own system, and this is supported by the consistency of the results derived from the several different types which have been identified.

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.

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

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.

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

DIMENSIONS OF EXTRA-GALACTIC NEBULAE

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 1.4x10^9 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.

MASSES OF EXTRA-GALACTIC NEBULAE

Spectroscopic rotations are available for the spirals M 31 and N.G.C. 4594, 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 ?pik 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. ?pik computed the ratio of luminosity to mass for our own system in terms of the sun as unity, using Jeans's value for the relative proportion of luminous to non-luminous material. The relation is

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 3.5x10^9 ? ?pik's method 1.6x10^9

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 2.0x10^9 ? ?pik's method 2.6x10^8

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.

?pik'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 x 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 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 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 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 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.

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

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 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 ?pik's method. The new value is then

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, 2.6x10^8 ?. Then, since the mass of the sun in grams is 2x10^ and 1 parsec is 3.1x10^ cm,

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 made the calculations some years ago, but used values for the density, 10^ 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. When R, V, M, and ? represent the radius of curvature, volume, mass, and density, and k and c are the gravitational constant and the velocity of light,

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