$\cos^2{i} = \frac{1 - b_{1}^2}{1 - b^2},$
where
$b_{1} = 1 - e_{1},\;b = i - e.$
From these equations, the values of i can be determined for all
possible values of e_{1}. The limits for the observed classes E0 to E7
were chosen midway between the consecutive tenths, E0 ranging from
e = 0 to e = 0.05; E1, from e = 0.05 to e = 0.15; E7, from e = 0.65
to e = 0.75. The relative frequencies of the various observed classes
are then proportional to the differences in sin i corresponding to the
two limiting values of e_{1}. These frequencies must be calculated
separately for nebulae of different actual ellipticities.
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.
TABLE X
========+===========================================
Actual | Apparent
+---------+--------+--------+--------+------
| E0 | E1 | E2 | E3 | E4
--------+---------+--------+--------+--------+------
E7 | 0.055 | 0.111 | 0.114 | 0.116 | 0.121
E6 | .059 | .123 | .126 | .133 | .148
E5 | .067 | .140 | .148 | .166 | .216
E4 | .079 | .169 | .190 | .250 | 0.312
E3 | .100 | .225 | .299 | 0.376 |
E2 | .145 | .378 | 0.477 | |
E1 | 0.300 | 0.700 | | |
E0 | 1.000 | | | |
+---------+--------+--------+--------+------
Total | 1.805 | 1.846 | 1.354 | 1.041 | 0.797
| 0.226 | 0.231 | 0.169 | 0.130 | 0.100
--------+---------+--------+--------+--------+------
========+=====================================
Actual | Apparent
--------+---------+--------+--------+---------
| E5 | E6 | E7 | Total
--------+---------+--------+--------+---------
E7 | 0.132 | 0.164 | 0.187 | 1.000
E6 | .187 | 0.224 | |
E5 | 0.263 | | |
+---------+--------+--------+---------
Total | 0.582 | 0.388 | 0.187 | 8.000
| 0.073 | 0.049 | 0.023 | 1.000
--------+---------+--------+--------+---------
From this table and the actual numbers in the observed classes as
read from a smoothed curve, the numbers of each actual ellipticity
mingled in the observed classes can be determined. For instance, the
four nebulae observed as E7 represent 0.187 of the total number of
actual E7. The others are distributed among the observed classes E0
to E6 according to the percentages listed in Table X. Six nebulae are
observed as E6, but 3.6 of these are actually E7. The remaining 2.4
actual E6 nebulae represent 0.224 of the total number of that actual
ellipticity, the others, as before, being scattered among the observed
classes E0 to E5. Table XI gives the complete analysis and is similar
to Table X except that the percentages in the latter are replaced by
the actual numbers indicated by the observations.
Finally, the mean values of C_{7} – C_{e} are calculated from the
numbers of nebulae in the various columns of Table XI together with
the values of C_{7} – C_{e} for the pure classes as derived from
formula (4). The results are listed in the fourth column of Table XII
following those for the pure and the observed classes. In determining
the observed values, N.G.C. 524 and 3998 are included as E0 and
E1, although in Table I they are listed as peculiar, because they are
obviously much flattened nebulae whose minor axes are close to the line
of sight.
TABLE XI
============+=========================================
| Apparent
Actual +--------+--------+--------+--------+-----
| E0 | E1 | E2 | E3 | E4
------------+--------+--------+--------+--------+-----
E7 | 1.2 | 2.4 | 2.5 | 2.5 | 2.6
E6 | .6 | 1.3 | 1.4 | 1.5 | 1.6
E5 | .8 | 1.7 | 1.8 | 2.0 | 2.7
E4 | .8 | 1.7 | 1.9 | 2.5 | 3.1
E3 | 0.9 | 2.1 | 2.8 | 3.5 |
E2 | 1.1 | 2.9 | 3.6 | |
E1 | 1.7 | 3.9 | | |
E0 | 9.9 | | | |
------------+--------+--------+--------+--------+-----
Total* | 17.0 | 16.0 | 14.0 | 12.0 | 10.0
------------+--------+--------+--------+--------+-----
=========+=========================================
| Apparent
Actual +--------+--------+--------+-------
| E5 | E6 | E7 | Total
---------+--------+--------+--------+-------
E7 | 2.9 | 3.6 | 4.0 | 21.7
E6 | 2.0 | 2.4 | | 10.8
E5 | 3.1 | | | 12.1
E4 | | | | 10.0
E3 | | | | 9.3
E2 | | | | 7.6
E1 | | | | 5.6
E0 | | | | 9.9
---------+--------+--------+--------+-------
Total* | 8.0 | 6.0 | 4.0 | 87.0
---------+--------+--------+--------+-------
* The totals represent the numbers in the observed classes as read from
a smooth curve.
The observed values in general fall between those for the pure
classes and those corresponding to random orientation. They are of
the same order as the latter, and the discrepancies are perhaps not
unaccountably large in view of the nature and the limited extent of the
material. There is a systematic difference, however, averaging about
0.2 mag., in the sense that the observed values are too large, and
increasing with decreasing ellipticity. One explanation is that the
observed classes are purer than is expected on the assumption of random
orientation. This view is supported by the relatively small dispersion
in C, as may be seen in Table I and Figure 2, among the nebulae of a
given class, but it is difficult to account for any such selective
effect in the observations. The discrepancies may be largely eliminated
by an arbitrary adjustment of the numbers of nebulae with various
degrees of actual ellipticity; for instance, the values in the last
column of Table XII, calculated on the assumption of equal numbers,
agree very well with the observed values, although the resulting
numbers having the various apparent ellipticities differ slightly from
those observed. The observed values, however, can again be accounted
for by the inclusion of some flatter nebulae among the classes E6 and
E7. Very early Sa or SBa nebulae might easily be mistaken for E nebulae
when oriented edge-on, although they would be readily recognized when
even slightly tilted. If the numerical results fully represented actual
statistical laws, the explanation would be sought in the physical
nature of the nebulae. The change from ellipsoidal to lenticular
figures, noticeable in the later-type nebulae, would affect the results
in the proper direction, as would also a progressive shortening of the
polar axis. The discrepancies, however, are second-order effects, and
since they may be due to accidental variations from random orientation,
a further discussion must await the accumulation of more data.
TABLE XII
Differential Values of C
============+===============+===========+======================
| | | Random Orientation
| | +-----------+----------
Class | Pure Classes | Observed | No. as |
| | | Observed | Equal No.
------------+---------------+------------+-----------+----------
C_{7}–C_{7} | 0.00 | 0.00 | 0.00 | 0.00
C_{6} | 0.63 | 0.35* | 0.25 | 0.35
C_{5} | 1.10 | 0.70* | 0.58 | 0.70
C_{4} | 1.51 | 0.85 | 1.11 | 1.01
C_{3} | 1.84 | 1.42 | 0.87 | 1.28
C_{2} | 2.13 | 1.67 | 1.33 | 1.55
C_{1} | 2.39 | 2.01† | 1.54 | 1.83
C_{0} | 2.62 | 2.17† | 2.15 | 2.25
------------+---------------+------------+-----------+----------
* Read from smooth curve in Fig. 6. The small numbers of observed E5
and E6 nebulae justify this procedure. The other values are the means
actually observed.
† N.G.C. 524 and 3998 are included as E0 and E1, respectively.
Meanwhile, it is evident that, to a first approximation at least,
the polar diameters alone determine the total luminosities of all
elliptical nebulae, and the entire series can be represented by the
various configurations of an originally globular mass expanding
equatorially. A single formula represents the relation, in which the
value of C is that corresponding to the pure type E0. From Table
XII, this is found to be 2.62 mag. less than the value of C_{7} The
latter is observed to be 12.75, hence
$m_{T} + 5 log b = 10.13.$ (5)
If this relation held for the spirals as well, the polar diameters
could be calculated from the measured magnitudes. Unfortunately, it has
not been possible to measure accurately the polar diameters directly,
and hence to test the question, but they have been computed for the
mean magnitudes of the Sa, Sb, and Sc nebulae as given in Table III,
and the ratios of the axes have been derived by a comparison of these
hypothetical values with the means of the measured maximum diameters.
The results, 1 to 4.4, 1 to 5.7, and 1 to 7.3, respectively, although
of the right order, appear to be somewhat too high. An examination of
the photographs indicates values of the order of 1 to 5.5, 1 to 8, and
1 to 10, but the material is meager and may not be representative. The
comparison emphasizes, however, the homogeneity and the progressive
nature of the entire sequence of nebulae and lends some additional
color to the assumption that it represents various aspects of the same
fundamental type of system.
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.