\begin{table}%t1b \caption{\label{tabla 1b}Coefficients $a_{z} {(i,j)}$ and $ b_{z} {(i,j)}$ of the power series for $z_{\star}$, corresponding to a stratified medium with an arbritrary density distribution.} \par %\centering \par \small\begin{tabular}{c c l l} \hline \hline $i$ & $j$ & $a_{z} {(i,j)}$ & $ b_{z}{(i,j)}$ \\ \hline 1 & 1 & 1 & 0 \\ & 2 & --- & -$b_{r}(1,1)$ \\ \hline 2 & 1 & $a_{r}(2,1) $ & $ {f}_{1}^{3}-4 {f}_{1} {f}_{2} + 6 {f}_{3}$ \\ & 2 & $a_{r}(2,2) $ & $-b_{r}(2,1) $ \\ & 3 & --- & $-b_{r}(2,2) $ \\ \hline 3 & 1 & $2 {f}_{1}^{4}-16 {f}_{2}^{2}-12 {f}_{1} {f}_{3}+ 96 {f}_{4}$ & $-4 {f}_{1}^{5}+16 {f}_{1}^{3} {f}_{2} -120 {f}_{2} {f}_{3}+48 {f}_{1} {f}_{4}+480 {f}_{5}$ \\ & 2 & $a_{r}(3,2) $ & $-7 {f}_{1}^{5}-36 {f}_{1}^{3} {f}_{2} +120 {f}_{1} {f}_{2}^{2}+120 {f}_{1}^{2} {f}_{3}-60 {f}_{2} {f}_{3}-312 {f}_{1} {f}_{4}-840 {f}_{5}$ \\ & 3 & $a_{r}(3,3) $ & $-b_{r}(3,2) $ \\ & 4 & --- & $-b_{r}(3,3) $ \\ \hline 4& 1& $ -5 {f}_{1}^{6}- 114 {f}_{1}^{4} {f}_{2}+ 528 {f}_{1}^{2} {f}_{2}^{2} -240 {f}_{2}^{3} + 324 {f}_{1}^{3} {f}_{3} $ & $ 15 {f}_{1}^{7}+168 {f}_{1}^{5} {f}_{2}-1440 {f}_{1}^{3} {f}_{2}^{2}+2112 {f}_{1} {f}_{2}^{3} -666 {f}_{1}^{4} {f}_{3} $ \\ & & $-1440 {f}_{1} {f}_{2} {f}_{3}-540 {f}_{3}^{2}-384 {f}_{1}^{2} {f}_{4} + 96 {f}_{2} {f}_{4} $ & $ +5112 {f}_{1}^{2} {f}_{2} {f}_{3} -5040 {f}_{2} {f}_{3} -4860 {f}_{1} {f}_{3}^{2}+1296 {f}_{1}^{3} {f}_{4} -7200 {f}_{1} {f}_{2} {f}_{4} $ \\ & & $+4800 {f}_{1} {f}_{5}+ 10800 {f}_{6} $ & $ - 4032 {f}_{3} {f}_{4}+2880 {f}_{1}^{2} {f}_{5}+ 10080 {f}_{2} {f}_{5}+50400 {f}_{1} {f}_{6} + 75600 {f}_{7} $ \\ & 2& $-9 {f}_{1}^{6}-342 {f}_{1}^{4} {f}_{2} + 528 {f}_{1}^{2} {f}_{2}^{2}+240 {f}_{2}^{3} + 324 {f}_{1}^{3} {f}_{3} $ & $26 {f}_{1}^{7}+864 {f}_{1}^{5} {f}_{2}-2880 {f}_{1}^{3} {f}_{2}^{2}-1332 {f}_{1}^{4} {f}_{3} +10080 {f}_{2}^{2} {f}_{3} +9720 {f}_{1} {f}_{3}^{2} $ \\ & & $ +1440 {f}_{1} {f}_{2} {f}_{3} -1260 {f}_{3}^{2}+384 {f}_{1}^{2} {f}_{4} $ & $+14400 {f}_{1} {f}_{2} {f}_{4} -16128 {f}_{3} {f}_{4} -5760 {f}_{1}^{2} {f}_{5}-40320 {f}_{2} {f}_{5}-109440 {f}_{1} {f}_{6}-131040 {f}_{7} $ \\ & & $-3360 {f}_{2} {f}_{4}-6720 {f}_{1} {f}_{5}-7920 {f}_{6}$ & \\ & 3& $ a_{r}(4,3) $ & $16 {f}_{1}^{7}+1416 {f}_{1}^{5} {f}_{2}+1632 {f}_{1}^{3} {f}_{2}^{2}-6016 {f}_{1} {f}_{2}^{3} +1872 {f}_{1}^{4} {f}_{3} $ \\ & & $ $ & $ -21312 {f}_{1}^{2} {f}_{2} {f}_{3} +2016 {f}_{2} {f}_{3} +4752 {f}_{1} {f}_{3}^{2} -8640 {f}_{1}^{3} {f}_{4} $ \\ & & $ $ & $ +15552 {f}_{1} {f}_{2} {f}_{4}+44352 {f}_{3} {f}_{4}+21600 {f}_{1}^{2} {f}_{5}+ 60480 {f}_{2} {f}_{5} $ \\ &&&$+106560 {f}_{1} {f}_{6} + 80640 {f}_{7}$\\ & 4& $ a_{r}(4,4)$ & $-b_{r}(4,3) $ \\ & 5& --- & $-b_{r}(4,4) $ \\ \hline \end{tabular} \end{table}