## 3 Method of the Investigation
### Statistical Methods  
Because in the investigation we have an uniform distribution of angles the sphere, numerous ranges of the polar \(\delta_{D}\) and azimuthal \(\eta\) angles should be characterized by successively cosine and uniform distributions. The analysis of these distributions is possible with the use of statistical methods, proposed for the first time in the paper of [PERSON] and [PERSON]: the \(\chi^{2}\) test, the First Autocorrelation test and the Fourier test, and additionally the Kolmogorov-Smirnov test, Cramer-von Mises test and Watson test. Within them, the tested range of the test angle \(\theta\) is divided into \(n\), equal to the width of bins. In the conducted research, the range of the test angle was divided into 18 bins. The test angle is equal, in this case, to \(\delta_{D}+\frac{\eta}{2}\) or \(\eta\). The obtained real numbers of galaxies in the \(k\)-th bin \(N_{k}\) are confronted with the theoretical numbers of galaxies \(N_{0,k}\) in these bins. The \(\chi^{2}\) test uses the statistic:  
\[\chi^{2}=\sum_{k=1}^{n}\frac{\left(N_{k}-Np_{k}\right)^{2}}{Np_{k}}=\sum_{k=1} ^{n}\frac{\left(N_{k}-N_{0,k}\right)^{2}}{N_{0,k}} \tag{8}\]  
where \(N\) is the total number of galaxies in the cluster under consideration, and \(p_{k}\) is the probability that the selected galaxy will fall into the \(k\)-th range. For \(n\) intervals, the number of degrees of freedom of the \(\chi^{2}\) test is \(n-1\), therefore, the expected value is \(E(\chi^{2})=n-1\), and the variance is expected to be \(\sigma^{2}(\chi^{2})=2(n-1)\). For 18 bins, the expected value is \(E(\chi^{2})=17\) with variance \(\sigma(\chi^{2})=5.831\). For the assumed confidence level of \(\alpha=0.05\), the critical value of the test is \(\chi^{2}_{c\tau}=27.587\). It should be noted, however, that the statistic described by the above equation has only limited \(\chi^{2}\) distribution. As a result, in the case of real analysis, the distribution of our statistic given by Equation (8) may be different than that described by the theoretical distribution \(\chi^{2}\)[57; 58; 59], and should be obtained from numerical simulations. The correct ness of the approximation of the distribution of our statistics by \(\chi^{2}\) distributions in galaxy orientation studies was analyzed in [PERSON] et al., 2019 [PERSON] et al. (2019). A different approach is presented by the First Autocorrelation test ([PERSON] and [PERSON], 2019). It should be noted here that the standard and probably the best test of autocorrelation is the von-Neumann-Durbin-Watson test. However, in our article, we do not use full autocorrelation, and although the test limited to the First Autocorrelation test may not perform as well as the von-Neumann-Durbin-Watson test, the idea presented there is also widely used (see, for example, [PERSON] (2010)--especially chapter 6 of the book). So, it seems that the use of this test works well enough. The usefulness and correctness of using the First Autocorrelation test in research in terms of ordering the orientation of galaxies has been shown in the work of [PERSON] et al., 2019 [PERSON] et al. (2019). The First Autocorrelation test determines the correlations between galaxy numbers in neighboring angle bins, which is given by the statistic:  
\[C=\sum_{k=1}^{n}\frac{(N_{k}-N_{0,k})(N_{k+1}-N_{0,k+1})}{[N_{0,k}N_{0,k+1}]^ {\frac{1}{2}}} \tag{9}\]  
where \(N_{n+1}=N_{1}\). In the paper of [PERSON] and [PERSON] (1999), it was calculated that the expected value \(E(C)=0\), while the standard deviation \(\sigma(C)=\sqrt{n}\). According to the paper of [PERSON] (2017), it was shown that this result is only an approximation for the case in which all \(N_{0,k}\) are equal, with the expected value \(E(C)=-1\). The correct value of the variance of the \(C\) statistic is only approximately equal to \(n\), and must be obtained from numerical simulations. The Fourier test originally appeared in the paper of [PERSON] and [PERSON] (1999). However, only the first two modes were analyzed at that time.