The corrected values of log d were then plotted against the observed magnitudes. This amounts to shifting the approximately parallel correlation curves for the separate types along the axis of log d until they coincide. Since the mean magnitudes of the various types are nearly constant, the relative shifts will very nearly equal the differences in the mean observed log d, and hence the effect of errors in the first approximation to the values of K will be negligible. [Illustration: Fig. 1.—Frequency distribution of apparent magnitudes among nebulae in Holetschek’s list.] [Illustration:Fig. 2.—Relation between luminosity and diameter among nebulae at the beginning of the sequence of types—E0 and E1 nebulae. The plot is shown in Figure 7, in which the two Magellanic Clouds have been included in order to strengthen the bright end of the curve which would otherwise be unduly influenced by the single object, M 31. The magnitudes +0.5 and +1.5, which were assigned to the Clouds, are estimates based upon published descriptions. [Illustration: Fig. 3.—Relation between luminosity and diameter among nebulae at the middle of the sequence of types—E7, Sa, and SBa nebulae.] [Illustration: Fig. 4.—Relation between luminosity and diameter among nebulae at the end of the sequence of types—Sc and SBc nebulae.] The correlation of the data is very closely represented by the formula $m_{T} = 13.0 - 5 \log d.$ (2) This falls between the two regression curves derived from least-square solutions and could be obtained exactly by assigning appropriate weights to the two methods of grouping. The nature of the data is such that a closer agreement can scarcely be expected. No correction to the assumed value of the slope appears to be required. The material extends over a range of 12 mag., and the few cases which have been investigated indicate that the correlation can be extended another 3 mag., to the limit at which nebulae can be classified with certainty on photographs made with the 100-inch reflector. The relation may therefore be considered to hold throughout the entire range of observations. [Illustration:Fig. 5.—Relation between luminosity and diameter among the irregular nebulae. The Magellanic Clouds are included. N.G.C. 4656 is an exceptional case in that it shows a narrow, greatly elongated image in which absorption effects are very conspicuous; hence the maximum diameter is exceptionally large for its apparent luminosity. The residuals without regard to sign average 0.87 mag., and there appears to be no systematic effect due either to type or luminosity. The scatter, however, is much greater for the spirals, especially in the later types, than for the elliptical nebulae. The limiting cases are explained by peculiar structural features. The nebulae which fall well above the line usually have bright stellar nuclei, and those which fall lowest are spirals seen edge-on in which belts of absorption are conspicuous.