## 3 Method of the Investigation  
The proposed new method for investigating the orientation of galaxies in clusters is an extension of the approach presented in [9; 22; 27; 31; 48; 49]. According to the method developed by [PERSON] and [PERSON] [31], and by [PERSON] [9; 22], one calculates the \"polar\"angle \(\delta_{D}\), which is the angle between the normal to the galaxy plane and the main plane of the coordinate system, and the \"azimuth\" angle \(\eta\), describing the direction between the projection of this normal onto the main plane and the direction towards the zero initial meridian. Using a supergalactic coordinate system (in this case, the main plane is supergalactic equator), these angles can be expressed as the relationship between the latitude and longitude angles, \(B\) and \(L\), the position angle \(P\), and the inclination angle \(i\) (Figure 1). Since galaxies are oblate spheroids [32], determining a galaxy's angular return using only its image is problematic, all the more so as the galaxy can be aligned in relation to the observer in four ways, giving the same number of possible rotation axes, as it is easily seen in Figure 1. If (as in our case) we do not have data on the rotational direction of the galaxies, it is sufficient to study only the line segment directed normal to the galaxy planes inside the sphere. This implies the need to consider only two possible orientations of galaxies. As a result, the formulas for calculating the angles \(\delta_{D}\) and \(\eta\) take the form:  
\[\sin\delta_{D}=-\cos i\sin B\pm\sin i\cos r\cos B, \tag{1}\]  
\[\sin\eta=(\cos\delta_{D})^{-1}[-\cos i\cos B\sin L+\sin i(\mp\cos r\sin B\sin L \pm\sin r\cos L)], \tag{2}\]  
\[\cos\eta=(\cos\delta_{D})^{-1}[-\cos i\cos B\cos L+\sin i(\mp\cos r\sin B\cos L \mp\sin r\sin L)], \tag{3}\]  
\[S=-\cos i\cos B\mp\sin i\cos r\sin B, \tag{4}\]  
\[r=P-\frac{\pi}{2}. \tag{5}\]  
Obtaining a random distribution obliges us to change the sign of the \(\delta_{D}\) angle to the opposite one for any value of parameter \(S\) that is smaller than zero [22; 31]. Please note, that for face-on galaxies, the inclination angle \(i\) is close to 0. As a result, \(\sin i\) is also close to zero, and the position angle has a minimal effect on the spatial orientations of the face-on galaxies. Figure 1: A schematic illustration of angles \(\delta_{D}\) (the polar angle between the normal to the galaxy plane and the main plane of the coordinate system) and \(\eta\) (the azimuth angle between the projection of this normal onto the main plane and the direction towards the zero initial meridian). The angle of inclination \(i\) is the angle between the normal to the plane of the galaxy observer’s line of sight, while \(P\) is the angle of the position in the frame of reference. \(N_{1}\) and \(N_{2}\) are possible positions of the normal to the galaxy plane-directed line segment inside the sphere. If we were also interested in the rotation  
direction of the galaxy, we would have two more normal vectors, \(N_{1}^{\prime}\) and \(N_{2}^{\prime}\), which are different from vectors \(N_{1}\) and \(N_{2}\) in that they would be comprise a directed line segment outside the sphere. \(L\) and \(B\) are the longitude and latitude of the reference coordinate system (for more details see [PERSON] [9], [PERSON] [31], [PERSON] et al. [50]). Any statistical study involving orientations must take into account the fact that directional variables are cyclic [51], and moreover, in theory, the \"random\" distribution of analyzed angles can be not uniform. As a result, tests statistics could be significantly modified and must be obtained from simulations (see [9] for detailed discussion) or from tests specially dedicated to this problem, such as the Cramer-von Mises test [52; 53] and Watson test [54; 55].