There is in nature a tendency to wide variation, coupled with a coordinate tendency to uniformity in averages, when the number of classes is limited. Thus the land mammals range in size from elephants, say 15 feet long, to mice and shrews of a few inches. If we divide the earth into a good many faunal regions, the average sizes of the mammals in the different provinces may vary considerably; but if we divide the earth into only two halves, the averages will be almost identical.
For the present research, I take Sir John Herschel's "General Catalogue of Nebulæ and Clusters of Stars," which, coming from a single hand, and that the hand of a master, may be considered fairly homogeneous; and excluding the clusters which are known to be associated with the Milky Way, and are therefore comparatively near, I divide the remaining objects into two classes: (1) large nebulæ, or those having a diameter greater than 2′; and (2) small nebulæ, or those which are less than 2′ across; and I shall assume that the small nebulae are on the average farther away than the large nebulæ in the ratio, x:1, leaving the value of the ratio to be determined by considerations to be drawn from the result, and which will appear in the sequel.
A point-source of light diminishes in brightness as the square of its distance increases; but light from a large number of points so close together that they can not be discriminated must be treated as a luminous surface; and since the angular area of a surface also diminishes proportionally to the inverse square of the distance, the intrinsic brightness, or the brightness of the unit of angular area, does not change with the varying distances of the nebulæ. We must therefore inquire: Is the intrinsic brightness of a small, and therefore presumably distant, white nebula equal to, or less than that of a large one? If the average brightness of the unit of angular area is less for the smaller white nebulæ the presumption is that the light of the smaller and more distant objects has been absorbed in passing through space. To apply this test, I further subdivide each class into three groups—(vf) very faint, (f) faint and (b) bright, or, if desired, the last two may be combined into a single group.
Dividing the nebulæ in Herschel's catalogue into groups of four hundred each, and taking the ratios of the small to the large nebulæ in each of the thirteen groups, I find that without exception the faint and small nebulæ are more numerous than the bright and small in a relatively very much larger ratio than occurs in the corresponding divisions of the large nebulæ. With only three exceptions the same relation is obtained by comparing the very faint and the faint nebulæ. Treating the groups separately, and taking the mean of the ratios, I find
Small divided by large: vf, 8.38; f, 6.83; b, 1.48.
The sums for the entire catalogue are
