Example $15\mathrm{.}4$: The top $(Q)$ of a chimney was sighted from two stations $P$ and $R$ at very different levels, the stations $P$ and $R$ being in line with the top of the chimney. The angle of elevation from $P$ to the top of the chimney was $38{21}^{\text{'}}$ and that from $R$ to the top of the chimney was $21{18}^{\text{'}}.$ The angle of the elevation from $R$ to a vane $2m$ above the foot of the staff held at $P$ was $1511.$ The heights of instrument at $P$ and $R$ were $1\mathrm{.}87m$ and $1\mathrm{.}64m$ respectively. The horizontal distance between $P$ and $R$ was $127m$ and the reduced level of $R$ was $112\mathrm{.}78m.$ Find the R$.$L$.$ of the top of the chimney and the horizontal distance from $P$ to the chimney.