  We consider the system $\mathcal{F}_4(a,b,c)$ of differential equations
annihilating Appell's hypergeometric series $F_4(a,b,c;x)$. We find the
integral representations for four linearly independent solutions expressed by
the hypergeometric series $F_4$. By using the intersection forms of twisted
(co)homology groups associated with them, we provide the monodromy
representation of $\mathcal{F}_4(a,b,c)$ and the twisted period relations for
the fundamental systems of solutions of $\mathcal{F}_4$.
