= Solution
The fields $X$ and $S$ arise by pushing forward the two coordinate vectors on the two-parameter family of geodesics, so $[X,S]=0$. Part (i), or equivalently the torsion-free property, gives
$$
\nabla_XS=\nabla_SX.
$$
In components,
$$
X^\mu\nabla_\mu S^\alpha
=S^\mu\nabla_\mu X^\alpha
=S^\mu B^\alpha{}_\mu.
$$
Therefore
$$
\boxed{X^\mu\nabla_\mu S^\alpha=B^\alpha{}_\mu S^\mu.}
$$
This says that the <deformation tensor of a timelike geodesic congruence> gives the relative velocity of neighboring geodesics represented by the <geodesic deviation> vector.
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