the upper limit of 1′2 as representing the class of very small nebulæ. For the intermediate class which includes those objects called "small" by Herschel and "large" in the Lick catalogue, and which may be designated as medium, I take a diameter five times as great, or 21′2; and for Herschel's "large" nebulæ, I take a diameter of 5′. The reason for these selections shall now be given.
I take for the diameter of the Andromeda nebula, 110′. This subtends the longer axis of the oval figure of the more condensed spiral arms. The fainter extensions are omitted because these are seldom included in the more distant nebulæ. Taking a suitable value for the coefficient of transmission, the curves giving the relation between brightness and distance become congruous for the two catalogues, if we take x, the unknown ratio of distance for large and small nebulæ, equal to 2 for Herschel's catalogue, and x = 5 for the Lick catalogue. This gives the following sequence:
| Nebular Class | Diameter | Distance | Transmission | |||||
| Andromeda | 110′.0, | a1 | = | 1.0 | andromede, | t | = | 0.996 |
| Large nebulæ | 5′.0, | a2 | = | 62.5 | andromede, | ta | = | 0.778 |
| Medium nebulæ | 2′.5, | a3 | = | 125.0 | andromede, | ta | 0.606 | |
| Very small nebulæ | 0′.5, | a4 | = | 625.0 | andromede, | ta | = | 0.082 |
The statement which was made for the ratio of brightness among the groups in the Lick catalogue (vf and f + b for large and small nebulæ) can be repeated in identical language for Herschel's catalogue by merely substituting the fraction 23 instead of 34; that is to say, Herschel's nebulæ are not only nearer than the Lick nebulæ, but are more nearly at a common distance; and the fraction expressing the ratio of brightness for the two groups of near and distant objects among the Herschelian nebulæ approaches nearer to the value of equality which it would have if all of the nebulæ were at the same distance, for then there would be equal absorption, and large and small objects should be equally grouped about a mean value.
| Ratio of brightness for large and for small nebulæ | If equidistant, | 1 : 1 | |||
| Herschel, | 2 : 1 | ||||
| Lick Obs'y | 3 : 1[1] |
The absorption exerted by the medium between us and the nebula? is in the main a non-selective one. If it were not so, but resembled the ordinary selective absorption of the planetary atmospheres, the most distant nebulæ should be deep red instead of white. Some selective absorption may, however, be exercised by the misty quasi-atmospheric envelopes which we have reason to believe are associated with some or
- ↑ For the details of this investigation reference may be made to my paper, "Are the White Nebulæ Galaxies?" Astronomische Nachrichten, No. 4536, Bd. 189, 441–454, November, 1911.
