## 3 Results
This result is valid for Figure 8, too, and, when replaced in Equation (9), it effortlessly leads to:  
\[\gamma_{1}=-\frac{s_{\rm f}+\widetilde{\rm s}+\widetilde{\rm s}^{\prime}}{s_{\rm f }\cdot\left(s_{\rm c}+\widetilde{\rm s}^{\prime}\right)}\cdot y_{\alpha}^{\prime} \tag{13}\]  
As a consequence, for the right line equation from relation (7), together with the equations from relations (8) and (9), the \(\gamma_{0}=y_{\alpha}^{\prime}\) coefficient stays valid, but the coefficient \(\gamma_{1}=\frac{\gamma_{\rm B}-y_{\alpha}^{\prime}}{s_{\rm f}}\) can be expressed through Equation (13), too. It results the possibility to express the (\(y_{\rm B}\)) tooth's apex movement in relation to the rotation point (center) (\(s_{\rm c}\)) position, which implies an advantage in the computations' interpretation, presented next in the paper. By replacing in Equation (11) the \((\gamma_{0})\) and \((\gamma_{1})\) coefficients from Equations (8) and (13), and considering the equality between Equations (10) and (11), we obtain:  
\[ \begin{array}{l}\mathrm{u}_{0}(t)\cdot s_{\rm c}=\int_{\widetilde{\rm s}}^{ \widetilde{\rm s}+s_{\rm f}}\sigma\cdot\left(s_{0}+\delta_{1}s+\delta_{2}s^{2 }\right)\mathrm{ds}\\ =\sigma\cdot\left[\delta_{0}\cdot\left(\widetilde{\rm s}+s_{\rm f}\right)- \widetilde{\rm s}\right]+\frac{\delta_{2}}{2}\cdot\left[(\widetilde{\rm s}+s _{\rm f})^{2}-\widetilde{\rm s}^{2}\right]+\frac{\delta_{2}}{2}\cdot\left[( \widetilde{\rm s}+s_{\rm f})^{3}-\widetilde{\rm s}^{3}\right)\right]\\ =\sigma\cdot\left[\delta_{0}\cdot s_{\rm f}+\frac{\delta_{1}}{2}\cdot\left(s_ {\rm f}^{2}+2\cdot s_{\rm f}\cdot\widetilde{\rm s}\right)+\frac{\delta_{2}}{3 }\cdot\left(s_{\rm f}^{3}+3\cdot s_{\rm f}^{2}\cdot\widetilde{\rm s}+3\cdot s _{\rm f}\cdot\widetilde{\rm s}^{2}\right)\right]\end{array}  \tag{14}\]  
This transcendent equation in relation to variables (t) and (s) represents the balance (equality) equation between the elastic moment from the left member and the plastic moment, expressed through an integral expression from \((\widetilde{\rm s})\) to \((\widetilde{\rm s}+s_{\rm f})\). After the computation of the integral presented in Equation (11), the \((\delta_{0})\), \((\delta_{1})\), and \((\delta_{2})\) coefficients result, respectively: \(\delta_{0}=\gamma_{0}\cdot s_{\rm c}\); \(\delta_{1}=\gamma_{1}\cdot s_{\rm c}-\gamma_{0}\); and \(\delta_{2}=-\gamma_{1}\). Moreover, we have previously proven that \(\gamma_{0}=y_{\alpha}^{\prime}\) and \(\gamma_{1}=-\frac{s_{\rm c}+\widetilde{\rm s}+\widetilde{\rm s}^{\prime}}{s_ {\rm c}\cdot\left(s_{\rm c}+\widetilde{\rm s}^{\prime}\right)}\cdot y_{\alpha}^ {\prime}\). Of all the coefficients presented in this paragraph, only (\(\sigma\)) and (\(\gamma_{0}\)) are constant. The other coefficients \((\delta_{0})\), \((\delta_{1})\), \((\delta_{2})\), and (\(\gamma_{1}\)) depend on the rotation center (\(s_{\rm c}\)), which is collinear with the tooth's fictional axis (which has the \((\widetilde{\rm s}^{\prime}+\widetilde{\rm s}+s_{\rm f})\) length) presented in Figures 7 and 8. The elastic force (\(\mathrm{u}_{0}(t)\)) highlighted in Equation (14) has the extended form presented in Equation (6), and (\(s_{\rm c}\)) represents the arm of this force (torque's arm). The result of the multiplication between (\(\mathrm{u}_{0}(t)\)) and (\(s_{\rm c}\)) gives the elastic moment (torque) which, consequently, becomes dependent on the (t) and (s) variables. The iterative solving of the transcendental equation presented in Equation (14), through numerical integration, is made for the following constant moments \(\mathrm{t}_{\rm k}=\{0;0.5;1;\ldots 6.5;7\}\) weeks, using the same initial data as in the case of the (\(\mathrm{K}_{y}\)) proportionality coefficient scaling (the same initial data used for obtaining the simulations results presented in Tables 1 and 2).