Herstein proved [1, Theorem 3.3] that any Jordan derivation of a prime ring of characteristic not 2 is a derivation of
R
R
. Our purpose is to extend this result on Lie ideals. We prove the following Theorem. Let
R
R
be any prime ring such that char
R
≠
2
R \ne 2
ana let
U
U
be a Lie ideal of
R
R
such that
u
2
∈
U
{u^2} \in U
for all
u
∈
U
u \in U
. If ,’, is an additive mapping of
R
R
into itself satisfying
(
u
2
)
′
=
u
′
u
+
u
u
′
({u^2})’ = u’u + uu’
for all
u
∈
U
u \in U
, then
(
u
υ
)
′
=
u
′
υ
+
u
υ
′
(u\upsilon )’ = u’\upsilon + u\upsilon ’
for all
u
,
υ
∈
U
u,\upsilon \in U
.