\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}