\begin{array}{l}
\displaystyle
P_1(t)=\int_0^tdx_1,\\
\displaystyle
P_2(t)=\int_0^tdx_1\int_0^{t-x_1}dx_2,\\
\vdots\\
\displaystyle
P_n(t)=\int_0^tdx_1\int_0^{t-x_1}dx_2\cdots\int_0^{t-(x_1+\cdots+x_{n-1})}dx_n.
\end{array}