The various data are collected in Table XVIII, where the observed numbers of extra-galactic nebulae to different limits of visual magnitude are compared with those computed on the assumption of uniform distribution of objects having a constant absolute luminosity. The formula used for the computation is $\log N = 0.6 m_{T} + \textrm{Constant},$ where the constant is the value of log N for m_{T} = 0. The value —4.45 is found to fit the observational data fairly well. The agreement between the observed and computed log N over a range of more than 8 mag. is consistent with the double assumption of uniform luminosity and uniform distribution or, more generally, indicates that the density function is independent of the distance. The systematic decrease in the residuals O – C with decreasing luminosity is probably within the observational errors, but it may also be explained as due to a clustering of nebulae in the vicinity of the galactic system. The cluster in Virgo alone accounts for an appreciable part. This is a second-order effect in the distribution, however, and will be discussed at length in a later paper. TABLE XVIII Numbers of Nebulae to Various Limits ========+===================+===========+======== | LOG N* | | m_{T} |---------+---------| O – C | LOG D† | O | C | | --------+---------+---------+-----------+-------- 8.5 | 0.85 | 0.65 | +0.20 | 5.74 9.0 | 1.08 | 0.95 | .13 | 5.84 9.5 | 1.45 | 1.25 | .20 | 5.94 10.0 | 1.73 | 1.55 | .18 | 6.04 10.5 | 1.95 | 1.85 | .10 | 6.14 11.0 | 2.17 | 2.15 | + .02 | 6.24 11.5 | 2.43 | 2.45 | – .02 | 6.34 12.0 | 2.70 | 2.75 | .05 | 6.44 16.7 | 5.48 | 5.57 | –0.09 | 7.38 (18.0) | | (6.35) | | (7.64) --------+---------+---------+-----------+-------- * Log N = 0.6 m_{T} – 4.45. † Log D = 0.2 m_{T} + 4.04. Distances corresponding to the different limiting magnitudes, as derived from formula (8), are given in the last column of Table XVIII. The 300,000 nebulae estimated to the limits represented by an hour’s exposure on fast plates with the 60-inch reflector appear to be the inhabitants of space out to a distance of the order of 2.4 × 10^7 parsecs. The 100-inch reflector, with long exposures under good conditions, will probably reach the total visual magnitude 18.0, and this, by a slight extrapolation, is estimated to represent a distance of the order of 4.4×10^7 parsecs or 1.4×10^8 light-years, within which it is expected that about two million nebulae should be found. This seems to represent the present boundaries of the observable region 3 of space. DENSITY OF SPACE