## Appendix A  
We shall closely investigate the validity of the assumption that the calibration-curve shape of the dependence of the measured (not emitted by impure atoms in the light source) analytical-line intensity of the impure atoms on their concentration in an analyzed sample, under the parameters changes of an in-service spectrometer, is constant. Practice shows that in most cases this assumption is valid. However, in a number of cases, calibration-line recalibration carried out according to the algorithm based on this assumption does not result in spectrometer recovery. As a result, we need to determine the conditions under which this assumption is valid. As has been noted, the relation of the analytical-line-radiation intensity \(I_{0}\) at the spectrometer's entrance slit depends on its impurity concentration and the parameters \(\overset{\rightarrow}{Y}\) (see Equation (15)):  
\[I_{0}=I_{0}(C,\overset{\rightarrow}{Y})\to C={I_{0}}^{-1}(I_{0}, \overset{\rightarrow}{Y}), \tag{17}\]  
where \({I_{0}}^{-1}(w,\overset{\rightarrow}{Y})\) is the function inverse of the function \(w=I_{0}(C,\overset{\rightarrow}{Y})\) (\(w\) = some scalar) i.e., for which at any \(w\) the identity is satisfied:  
\[w\equiv I_{0}[{I_{0}}^{-1}(w,\overset{\rightarrow}{Y}),\overset{ \rightarrow}{Y}].\]Considering Equations (17) and (18) as equations to determine concentration \(C\), taking into account Equation (20), we can obtain:  
\[ \begin{array}{c}C={I_{0}}^{-1}\left[\frac{{I-I_{F}}}{{\widetilde{K}}(Z,\overrightarrow {Y})K(\overrightarrow{X})},\overrightarrow{Y}\right]={I_{0}}^{-1}\left[\frac{ {I^{\prime}-I^{\prime}_{F}}}{{\widetilde{K}}(Z,\overrightarrow{Y^{\prime}})K (\overrightarrow{X^{\prime}})},\overrightarrow{Y^{\prime}}\right];\\ \qquad\qquad I_{F}=\widetilde{K}(\overrightarrow{Z},\overrightarrow{Y})[K( \overrightarrow{X})I_{pl}(\overrightarrow{Y})+I_{n}(\overrightarrow{X})];\\ \qquad\qquad I^{\prime}_{F}=\widetilde{K}(\overrightarrow{Z^{\prime}}, \overrightarrow{Y^{\prime}})[\overrightarrow{K(\overrightarrow{X})}I_{pl}( \overrightarrow{Y^{\prime}})+I_{n}(\overrightarrow{X^{\prime}})].\end{array}  \tag{20}\]  
In Equation (20) the type of the inverse function \({I_{0}}^{-1}(w,\overrightarrow{Y})\) is clearly determined by the dependence of the function \({I_{0}}(C,\overrightarrow{Y})\) on the argument \(C\) and parameters \(\overrightarrow{Y}\). It implies that, strictly speaking, the equality  
\[{I_{0}}^{-1}(w,\overrightarrow{Y})={I_{0}}^{-1}(w^{\prime},\overrightarrow{ Y^{\prime}}) \tag{21}\]  
for the arbitrary function \({I_{0}}={I_{0}}(C,\overrightarrow{Y})\) is not true. It is possible if \(\overrightarrow{Y}=\overrightarrow{Y^{\prime}}\). That is, it is correct to apply the conventional recalibration algorithm only if there are not any in-service changes to the spectrometer's unit parameters that are responsible for analytical-line intensity before entering the slit-light entrance system: generator parameters, inter-electrode distance, etc. In this case, the correlations (Formula (20)) provide:  
\[I-I_{F}=\frac{\widetilde{K}(\overrightarrow{Z},\overrightarrow{Y})K( \overrightarrow{X})}{\widetilde{K}(\overrightarrow{Z^{\prime}}, \overrightarrow{Y^{\prime}})K(\overrightarrow{X^{\prime}})}({I^{\prime}}-{I ^{\prime}}_{F})=\varphi(\overrightarrow{X},\overrightarrow{Y}, \overrightarrow{Z},\overrightarrow{X^{\prime}},\overrightarrow{Y^{\prime}}, \overrightarrow{Z^{\prime}})({I^{\prime}}-{I^{\prime}}_{F}) \tag{22}\]  
Thus, we obtained the already-known result. In the case when the transmission and conversion coefficients are independent of the intensity \({I_{0}}\), the relation between values \(I\) and \({I^{\prime}}\) is linear. However, in contrast to previously obtained results, where the conversion coefficients are expressed only through the spectrometer characteristics after their change, in (Formula 4) there are characteristics both before and after the change.