An analysis of the distribution provides strong evidence for the existence of second-order clusters, that is, clusters of galaxies. Counts of individual galaxies are useful in the study of the large-scale distribution of cosmic matter. Galaxy clusters, on the other hand, provide an independent approach to the problem of the overall distribution of matter.
How galactic absorption actually affects the results of the investigation will be discussed in section M. Quantitative estimates of the consistency of the sizes can be made, and are described in section G. Sizes of a number of galaxies therefore had to be determined before calibration of the images on the films was possible.
The validity of (ii) depends on the shape of the bright end of the cluster brightness function. The bright ends of the apparent brightness functions for these clusters were approximately determined using the. Sandage (1) also measured the magnitudes of the tenth brightest members of the redshift clusters.
Extinction: The magnitudes of the tenth-brightest members of all clusters were corrected for the effect of atmospheric extinction.
PART II
GROUP 6
GROUP 5
The maximum is particularly well developed for group 6, and also for the combined groups 1 to 4, although due to the small sample size, the non-randomness in the distribution of the nearest groups is only marginally significant (around the 5 percent level ). In any case, for each group there appears to be an average linear dimension corresponding to the degree of clustering. The result is that expected if it is assumed that cluster clusters tend to be less the same size everywhere in space.
The observed non-random distribution of clusters cannot be accounted for by the assumption of galactic or intergalactic obscuration. The possibility was not one of the subjects of investigation in this study and nothing can be said about it here. In three of the cases Shane-Wirtanen clouds (numbers 4, 5 and 6) correspond to visible clusters of two or more clusters in
Clusters in each distance group were counted over the entire area of the sample and within only 30° of the galactic poles. The assumption that the average surface density of clusters is the same in both hemispheres was then checked with a. There is therefore no reason to assume that there are more clusters on one side of the galactic plane than on the other.
Probability that the average areal density of cluster centers is the same in the Northern and Southern Galactic Hemispheres. The square of the lumpiness index, K (s), corresponding to a right-angled solid angle 2ct1 x 2sa2, is a non-decreasing function of s ••• 11. The authors also show that as both dimensions of the solid angle increase, K2 will increase also grow.
These theorems, which are quite general, imply that if all galaxies are clustered, and if there is no obscuring interstellar or intergalactic matter, then k2. z,n) will statistically be a non-decreasing function of the area of the cells in which the galaxies are counted. Thus, if one considers the hypothesis that all galaxy clusters are members of second-order clusters and that second-order clusters are distributed according to Poisson's law, the square of the clustering index, defined analogously to equation 19, will be a non-decreasing function of the area of the cells in which the clusters are counted. The statistic k(z,n) defined analogously to equation 18, with the variance of the Poisson distribution sg [equal to the mean of the Poisson distribution (40)] estimated as the sample mean, was calculated for distance groups 5 and 6 , as for the entire area of.
GROUPS 1-4
GROUP 5
Although there is considerable scatter around a smooth curve, which is expected for a sample of this size, there is no evidence of a maximum of k2(z,n) in either case. An attempt was made to determine the correlation coefficient of scores in different directions for distance groups 5 and 6. It was desired to find the correlation coefficient, f'Ce). between counts in cells e degrees apart, defined by. i ie degrees apart, and m is the average number of clusters per cell for the distance group under consideration.
If no second-order clustering exists, and if there is no interstellar or intergalactic eclipse, one would expect no correlation between counts in different cells, and . rce) = o(e) ' (21). However, as a second-order grouping r(e) exists. will have the value unity when 8 = 0 and will gradually decrease with increasing e, the rate of decrease being determined by the angular dimensions of the second-order clusters. An estimate of the correlation coefficient was obtained as follows: Values of n.-m for all the cells in each latitude.
All the products from all the latitude zones were then averaged for each value of e and divided by n~ to obtain r (e). The resulting estimates of the correlation coefficient are given in Table 19 and are plotted in Figure 27. The correlation coefficient for Group 6 appears to approach a lower limit as e increases, at least over the range of e considered.
Otherwise, the correlation coefficients decline more or less gradually, consistent with the second-order clustering assumption. Unfortunately, the size of the counted cells was too large to significantly determine the shape of j1(e). Unfortunately, the calculations involved are too laborious to complete in the time available for this investigation;. the project will probably only be feasible with an electronic computer.
In particular, around galactic longitude J00° and extending in the northern galactic hemisphere to at least latitude +60°, there exists galactic absorption of the order of several tenths of a magnitude (photored) greater than at corresponding latitudes around longitude 100°. The data strongly suggest the existence of second-order clusters, or clusters of galaxies. Unfortunately, the cell division used in the calculation of correlation coefficients was not sufficiently fine to give results of high statistical significance. 1936) The Realm of the Nebulae, (Yale University Press, New Haven), ch.
