## Appendix A Fourier Test  
Equation (10), taking into account only the first two modes, can be written in the form:  
\[\frac{N_{k}-N_{0,k}}{N_{0,k}}=\Delta_{11}\cos 2\theta_{k}+\Delta_{21}\sin 2 \theta_{k}+\Delta_{12}\cos 4\theta_{k}+\Delta_{22}\sin 4\theta_{k}. \tag{11}\]  
the expressions for matrix \(\mathbf{G}\) and vector \(\mathbf{H}\) will take the form:  
\[\mathbf{G}=\left(\begin{array}{cccc}A&Y&U&V\\ Y&B&X&W\\ U&X&C&Z\\ V&W&Z&D\end{array}\right) \tag{10}\]  
and  
\[\mathbf{H}=\left(\begin{array}{c}K\\ L\\ M\\ N\end{array}\right). \tag{11}\]  
As a consequence, the \(\Delta_{ij}\) coefficients of the vector I described by Equation (21) will be expressed by:  
\[\begin{split}\Delta_{11}&=\frac{S_{1}K+S_{5}M+S_{7}L+S_{8}N}{R} \\ \Delta_{21}&=\frac{S_{2}L+S_{6}N+S_{7}K+S_{9}M}{R}\\ \Delta_{12}&=\frac{S_{3}M+S_{5}K+S_{9}L+S_{10}N}{R}\\ \Delta_{22}&=\frac{S_{4}N+S_{6}L+S_{8}K+S_{10}M}{R}\end{split} \tag{12}\]  
where  
\[\begin{split} S_{1}&=C(BD-W^{2})+X(2 WZ-DX)-Z^{2}B\\ S_{2}&=D(AC-U^{2})+V(2 UZ-CV)-Z^{2}A\\ S_{3}&=A(BD-W^{2})+Y(2 WV-DY)-V^{2}B\\ S_{4}&=B(AC-U^{2})+Y(2 UX-CY)-X^{2}A\\ S_{5}&=-U(BD-W^{2})+Y(DX-WZ)+V(BZ-WX)\\ S_{6}&=-W(AC-U^{2})+Z(AX-UY)+V(CY-UX)\\ S_{7}&=V(CW-XZ)+Y(Z^{2}-CD)+U(DX-WZ)\\ S_{8}&=Y(CW-XZ)+V(X^{2}-BC)+U(BZ-WX)\\ S_{9}&=Z(AW-YY)+X(V^{2}-AD)+U(DY-VW)\\ S_{10}&=X(AW-YY)+Z(Y^{2}-AB)+U(BV-WY)\\ R&=(AC-U^{2})(BD-W^{2})+Y(2W(CV-UZ)+2X(DU-ZV)+\\ &+Y(Z^{2}-CD))+V(2U(BZ-XW)-BCV)+X^{2}(V^{2}-AD)+ZA(2 XW-ZB)\end{split} \tag{13}\]  
The variable \(J\) appearing in the Formula (20) will be expressed by the relation:  
\[\begin{split} J&=A\Delta_{11}^{2}+B\Delta_{21}^{2}+C \Delta_{12}^{2}+D\Delta_{22}^{2}+2Y\Delta_{11}\Delta_{21}+2U\Delta_{11}\Delta_ {12}+\\ &+2V\Delta_{11}\Delta_{22}+2X\Delta_{21}\Delta_{12}+2W\Delta_{21} \Delta_{22}+2 ZZ\Delta_{12}\Delta_{22}\end{split} \tag{14}\]For the case of a symmetric theoretical distribution, i.e., when \(N_{0i}=N_{0n-i}\), the values of the coefficients \(X\), \(Y\), \(V\), \(Z\) are equal to 0 [9] and, consequently, Formulas (A10) are simplified to the form  
\[\begin{split}\Delta_{11}&=\frac{CK-UM}{AC-Ul^{2}}\\ \Delta_{21}&=\frac{DL-WN}{BD-W^{2}}\\ \Delta_{12}&=\frac{AM-UK}{AC-Ul^{2}}\\ \Delta_{22}&=\frac{BN-WL}{BD-W^{2}}\end{split}\] (A13)  
which is similar to the formula for the variable \(J\), reduced to the form  
\[J=A\Delta_{11}^{2}+B\Delta_{21}^{2}+C\Delta_{12}^{2}+D\Delta_{22}^{2}+2U \Delta_{11}\Delta_{12}+2W\Delta_{21}\Delta_{22}\] (A14)  
It should be noted that the applicability limits of Formulas (A13) may be extended in cases for which the Formulas (A10) should be applied, if it is possible to indicate a relatively small asymmetry of theoretical distributions.  
When only the first modes are taken into account in the Fourier test, i.e., when vector \(\mathbf{I}\) is given by relation (18), then vector \(\mathbf{A}\) takes the form  
\[\mathbf{A}=\left(\begin{array}{cc}\cos 2\theta_{1}&\sin 2\theta_{1}\\ \cos 2\theta_{2}&\sin 2\theta_{2}\\ \vdots&\vdots\\ \cos 2\theta_{k}&\sin 2\theta_{k}\end{array}\right)\] (A15)  
following this, matrix \(\mathbf{G}\) and vector \(\mathbf{H}\) are given by:  
\[\mathbf{G}=\left(\begin{array}{cc}A&Y\\ Y&B\end{array}\right)\] (A16)  
and  
\[\mathbf{H}=\left(\begin{array}{c}K\\ L\end{array}\right)\] (A17)  
In the most general form, \(\Delta_{1j}\) coefficients are then expressed by formulas  
\[\begin{split}\Delta_{11}&=\frac{BK-YL}{AB-Y^{2}}\\ \Delta_{21}&=\frac{AL-YK}{AB-Y^{2}}\end{split}\] (A18)  
Additionally, the variable \(J\) in the Formula (17) has the form  
\[J=A\Delta_{11}^{2}+B\Delta_{21}^{2}+2Y\Delta_{11}\Delta_{21}\] (A19)  
In the case of a symmetric theoretical distribution, \(Y\) is equal to 0, so these formulas are simplified to (11), (12) and (16).