From these equations, the values of i can be determined for all possible values of e1. 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 e1. 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 | E5 | E6 | E7 | Total | |
| E7 | 0.055 | 0.111 | 0.114 | 0.116 | 0.121 | 0.132 | 0.164 | 0.187 | 1.000 |
| 6 | .059 | .123 | .126 | .133 | .148 | .187 | 0.224 | ||
| 5 | .067 | .140 | .148 | .166 | .216 | 0.263 | |||
| 4 | .079 | .169 | .190 | .250 | 0.312 | ||||
| 3 | .100 | .225 | .299 | 0.376 | |||||
| 2 | .145 | .378 | 0.477 | ||||||
| 1 | 0.300 | 0.700 | |||||||
| 0 | 1.000 | ||||||||
| Total | 1.805 | 1.846 | 1.354 | 1.041 | 0.797 | 0.582 | 0.388 | 0.187 | 8.000 |
| 0.226 | 0.231 | 0.169 | 0.130 | 0.100 | 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 C7 – Ce are calculated from the numbers of nebulae in the various columns of [Table XI] together with the values of C7 – Ce 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
| Actual | Apparent | ||||||||
|---|---|---|---|---|---|---|---|---|---|
| E0 | E1 | E2 | E3 | E4 | E5 | E6 | E7 | Total | |
| E7 | 1.2 | 2.4 | 2.5 | 2.5 | 2.6 | 2.9 | 3.6 | 4.0 | 21.7 |
| 6 | 0.6 | 1.3 | 1.4 | 1.5 | 1.6 | 2.0 | 2.4 | 10.8 | |
| 5 | .8 | 1.7 | 1.8 | 2.0 | 2.7 | 3.1 | 12.1 | ||
| 4 | .8 | 1.7 | 1.9 | 2.5 | 3.1 | 10.0 | |||
| 3 | .9 | 2.1 | 2.8 | 3.5 | 9.3 | ||||
| 2 | 1.1 | 2.9 | 3.6 | 7.6 | |||||
| 1 | 1.7 | 3.9 | 5.6 | ||||||
| 0 | 9.9 | 9.9 | |||||||
| Total* | 17.0 | 16.0 | 14.0 | 12.0 | 10.0 | 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.