14. $A B C D$ is a rectangle, $A B=12 \sqrt{3}, B C=$ $13 \sqrt{3}$, diagonals $A C, B D$ intersect at $P$. If the triangle $A B P$ is cut off and $A P, B P$ are joined, find its volume.


14. As shown in the figure, $N$ is the intersection of the altitude from $P$ to $BCD$.

It is easy to prove that $N$ is the circumcenter of $\angle BCD$, and
\[
\begin{array}{l}
r=\frac{n^{2}}{\sqrt{4 n^{2}-m^{2}}} . \\
\because P B=\frac{1}{2} \sqrt{m^{2}+n^{2}}, \text { by the Pythagorean theorem, we have } \\
P N=\frac{m}{2} \sqrt{\frac{3 n^{2}-m^{2}}{4 n^{2}-m^{2}}}, \\
\therefore V_{\mathrm{PBCD}}=\frac{1}{3} P N \cdot S \triangle \mathrm{BCD} \\
\quad=\frac{m^{2}}{24} \sqrt{3 n^{2}-m^{2} .} \\
\because \quad m^{2}=432, n^{2}=507, \\
\therefore \quad V=594 .
\end{array}
\]