Page:Popular Science Monthly Volume 82.djvu/300

296 $$\begin{align} \text{Small:}&& vf=&1765, & f&=897, & b&=241.\\ \text{Large:}&& vf=&\;\;235, & f&=172, & b&=204. \end{align}$$

The division into separate groups with the result that the same general law is given by every one of the groups is of course the more severe test; but taking the ratios for the sums as answering our present purpose, we have for the ratio of

$$ \frac{\text{Small nebulæ}}{\text{Large nebulæ}}\colon \quad vf, 7.51;\quad f, 5.22;\quad b,1.18;$$

or approximately vf:f:b = 6:4:1; that is, the very faint nebulæ are in excess over the bright ones among the small nebulæ in the ratio 6:1, but are of nearly equal frequency among the large nebulæ. In other words, the large nebulæ are intrinsically much brighter than the small ones.

I next performed the same operation with the 744 objects in a "Catalogue of New Nebulæ Discovered on the Negatives" taken with the Crossley reflector at the Lick Observatory, dividing them into two groups: (1) very small, or not over one half minute in diameter, and (2) those which are above this size and which may be called "large." These groups were divided into two classes: (a) very faint, including those which are described as "very faint" and "very very faint," and (b) pretty bright, or those given in the catalogue as "faint" to "bright." The result of this examination is that three fourths of the large nebulæ are pretty bright, and one fourth very faint; while the very small nebulæ have just the opposite distribution of brightness, three fourths of them being very faint, and only one fourth pretty bright.

In comparing the two catalogues, it must be recognized that the photographic method is far more delicate. Most of the objects in the photographic catalogue could not be detected by visual examination. The photograph also includes faint margins and therefore increases the apparent size of such nebulæ as are visually perceptible. Consequently, Herschel's small nebulæ are about equivalent to the "large" nebulas of the photographic catalogue, and we should expect that the photograph would include a much wider range of brightness—all of which is confirmed by a discussion of the observations.

Let us suppose that the average distances of the several classes of nebulæ are given in andromedes, and denoted by the letter a, and that the coefficient of transmission of light through space is $$t^{a}$$; also that the mean distances are inversely proportional to certain assumed apparent diameters which are fairly typical. Each class of nebulæ includes objects having a considerable range of actual diameter, that is, the variation of distance is not as great as that of the apparent diameter. Instead of taking a mean value of to represent the diameter of that class which includes nebulæ less than  in diameter, I take