  We study Lauricella's hypergeometric function $F_A$ of $m$-variables and the
system $E_A$ of differential equations annihilating $F_A$, by using twisted
(co)homology groups. We construct twisted cycles with respect to an integral
representation of Euler type of $F_A$. These cycles correspond to $2^m$
linearly independent solutions to $E_A$, which are expressed by hypergeometric
series $F_A$. Using intersection forms of twisted (co)homology groups, we
obtain twisted period relations which give quadratic relations for Lauricella's
$F_A$.
