  Let $K/F$ be a finite Galois extension of number fields with Galois group
$G$, let $A$ be an abelian variety defined over $F$, and let ${\cyr W}(A_{^{/
K}})$ and ${\cyr W}(A_{^{/ F}})$ denote, respectively, the Tate-Shafarevich
groups of $A$ over $K$ and of $A$ over $F$. Assuming that these groups are
finite, we derive, under certain restrictions on $A$ and $K/F$, a formula for
the order of the subgroup of ${\cyr W}(A_{^{/ K}})$ of $G$-invariant elements.
As a corollary, we obtain a simple formula relating the orders of ${\cyr
W}(A_{^{/ K}})$, ${\cyr W}(A_{^{/ F}})$ and ${\cyr W}(A_{^{/ F}}^{\chi})$ when
$K/F$ is a quadratic extension and $A^{\chi}$ is the twist of $A$ by the
non-trivial character $\chi$ of $G$.
