  Supposing the spin-triplet superconducting state of Sr$_2$RuO$_4$, the
spin-orbit (SO) coupling associated with relative motion in Cooper pairs is
calculated by extending the method for the dipole-dipole coupling given by
Leggett in the superfluid $^{3}$He. It is shown that the SO coupling works only
in the equal-spin pairing (ESP) state to make the pair angular momentum
$\hbar{\vec L}$ and the pair spin angular momentum ${\rm i}{\vec d}\times{\vec
d}^{*}$ parallel with each other. The SO coupling gives rise to the internal
Josephson effect in a chiral ESP state as in superfluid A-phase of $^3$He with
a help of an additional anisotropy arising from SO coupling of atomic origin
which works to direct the {\bf d}-vector into $ab$-plane. This resolves the
problem of the anomalous relaxation of $^{17}$O-NQR and the structure of {\bf
d}-vector in Sr$_2$RuO$_4$.
