## 3 Method of the Investigation
### Statistical Methods
This test was significantly improved by [PERSON] (2017), in which higher Fourier modes are taken into account. In the Fourier test, deviation from isotropy is a slowly varying function of the angle \(\theta\), according to the relationship:  
\[N_{k}=N_{0,k}(1+\Delta_{11}\cos 2\theta_{k}+\Delta_{21}\sin 2\theta_{k}+ \Delta_{12}\cos 4\theta_{k}+\Delta_{22}\sin 4\theta_{k}+\ldots) \tag{10}\]  
In this case, all \(N_{0,k}\) are equal. The Fourier coefficients are given by the following formulas:  
\[\Delta_{1J}=\frac{\sum_{k=1}^{n}(N_{k}-N_{0,k})\cos 2\beta_{k}}{\sum_{k=1}^{n}N _{0,k}\cos^{2}2\beta_{k}} \tag{11}\]  
\[\Delta_{2J}=\frac{\sum_{k=1}^{n}(N_{k}-N_{0,k})\sin 2\beta_{k}}{\sum_{k=1}^{n}N _{0,k}\sin^{2}2\beta_{k}} \tag{12}\]  
with standard deviations given by:  
\[\sigma(\Delta_{1J})=\left(\sum_{k=1}^{n}N_{0,k}\cos^{2}2\beta_{k}\right)^{- \frac{1}{2}} \tag{13}\]  
\[\sigma(\Delta_{2J})=\left(\sum_{k=1}^{n}N_{0,k}\sin^{2}2\beta_{k}\right)^{- \frac{1}{2}} \tag{14}\]  
Analysis using the Fourier test allows us to determine the direction of departure from isotropy. For negative values of the coefficient \(\Delta_{11}\), there is a predominance of galaxies with the position angle parallel to the Local Supercluster plane. For positive values of this coefficient, there is a predominance of galaxies with the position angle being perpendicular to the Local Supercluster plane. When analyzing the angle \(\eta\), negative values of the coefficient \(\Delta_{11}\) indicate a predominance of galaxies with \(\eta\) perpendicular to the 0 meridian coordinate system, while positive values of this coefficient indicate a predominance of galaxies with \(\eta\) pointed parallel to the 0 meridian. However, when investigating the angles, negative values of the coefficient \(\Delta_{11}\) indicate a predominance of galaxies with normals parallel to the main plane of the coordinate system, while positive values of this coefficient indicate a predominance of normals perpendicular to the main plane. The probability that the amplitude  
\[\Delta_{J}=\left(\Delta_{1J}^{2}+\Delta_{2J}^{2}\right)^{-\frac{1}{2}} \tag{15}\]  
described by the two-dimensional Gaussian distribution, is greater than a certain chosen value is given by the formula:  
\[P(>\Delta_{J})=\exp\left(-\frac{1}{2}\left(\frac{\Delta_{1J}^{2}}{\sigma( \Delta_{1J}^{2})}+\frac{\Delta_{2J}^{2}}{\sigma(\Delta_{2J}^{2})}\right)\right). \tag{16}\]  
By using the auxiliary variable \(J=\Sigma_{i}\Sigma_{j}G_{ij}I_{i}I_{j}\), where \(G\) is the inverse matrix to the covariance matrix of \(\Delta_{ij}\), it is possible to write:  
\[P(>\Delta_{1})=\exp\left(-\frac{1}{2}J\right), \tag{17}\]  
where the vector \(\mathbf{I}\) is:  
\[\mathbf{I}=\left( \begin{array}{c}\Delta_{11}\\ \Delta_{21}\end{array} \right). \tag{18}\]  
The probability that the amplitude  
\[\Delta=\left(\Delta_{11}^{2}+\Delta_{21}^{2}+\Delta_{12}^{2}+\Delta_{22}^{2} \right)^{-\frac{1}{2}} \tag{19}\]  
described by the four-dimensional Gaussian distribution, is greater than a certain chosen value is given by the formula:  
\[P(>\Delta)=\left(1+\frac{1}{2}J\right)\exp\left(-\frac{1}{2}J\right), \tag{20}\]  
where vector \(\mathbf{I}\) has the form:  
\[\mathbf{I}=\left( \begin{array}{c}\Delta_{11}\\ \Delta_{21}\\ \Delta_{12}\\ \Delta_{22}\end{array} \right) \tag{21}\]  
(a detailed discussion of the Fourier test is included in Appendix A). It should be mentioned here that, against the use of the Fourier test [26], it has been argued that, from a theoretical point of view, the application of the Fourier transform does not lead to the correct result, because the conditions under which we get exponential probability formulas (Equation (16)) rarely work in practice (see [60], Chapter 5). This problem is well-known and discussed many times in the literature in the context of the utility of power spectrum analysis (PSA) [61; 62; 63]. [PERSON] et al. (1994) [62] pointed out that [PERSON] (1969) showed that the Power Spectrum Analysis version [61] only applies exactly when the uniform distribution function is tested.