[Illustration: Fig. 6.—Progressive characteristics in the sequence of types. The upper curve represents the progression in total magnitude with type for nebulae having maximum diameters of one minute of arc. The elliptical nebulae and the normal spirals are included as representing the normal sequence, but the barred spirals and the irregular nebulae are omitted. The figures give the number of objects observed in each type. Among the later elliptical nebulae the numbers are so small that means of adjacent types have been plotted. The lower curve represents the progression in diameter along the normal sequence for nebulae of the tenth magnitude.] TABLE VIII =======+=========+========== Type | C | Δ log d -------+---------+---------- E0 | 10.30 | +0.54 1 | 10.65 | .47 2 | 11.00 | .40 3 | 11.35 | .33 4 | 11.70 | .26 5 | 12.05 | .19 6 | 12.40 | .12 7 | 12.75 | +0.05 | | Sa | 13.31 | –0.06 Sb | 13.90 | .18 Sc | 14.45 | .29 | | SBa | 13.00 | .00 SBb | 13.16 | .03 SBc | 14.41 | .28 | | Irr | 13.68 | –0.14 ------+---------+---------- [Illustration: Fig. 7.—Relation between luminosity and diameter among extra-galactic nebulae. The nebulae have been reduced to a standard type, S0, which, being the mean of E7, Sa, and SBa, represents a hypothetical transition point between elliptical nebulae and spirals. The Magellanic Clouds have been included in order to strengthen the brighter end of the plot. SIGNIFICANCE OF THE LUMINOSITY RELATION The correlations thus far derived are between total luminosities and maximum diameters. In the most general sense, therefore, they express laws of mean surface brightness. The value, K = 5.0, in formula (1) indicates that the surface brightness is constant for each separate type. The variations in C indicate a progressive diminution in the surface brightness from class to class throughout the entire sequence. The consistency of the results amply justifies the sequence as a basis of classification, since a progression in physical dimensions is indicated, which accompanies the progression in structural form. Although the correlations do not necessarily establish any generic relation among the observed classes, they support in a very evident manner the hypothesis that the various stages in the sequence represent different phases of a single fundamental type of astronomical body. Moreover, the quantitative variation in C is consistent with this interpretation, as is apparent from the following considerations.