Let R be a ring with centre Z(R). A bi-additive symmetric mapping
D$\left(\cdot,\cdot\right)$ : $R\times R\longrightarrow R$ is called
symmetric bi-derivation if for any fìxed $y\,\epsilon\, R,\, x\mapsto D\left(x,y\right)$is
a derivation. The main result of the present paper states that if
R is a semiprime ring of characteristic different from two and three
which admits a symmetric bi-derivation D such that $\left[\left[D\left(x,x\right)x\right],x\right]\epsilon Z\left(R\right)$
holds for all $y\,\epsilon\, R$, then $\left[D\left(x,x\right)x\right]$=0,
for all $y\,\epsilon\, R$. Further, some commutativity results are
also obtained.
