Defining a family of geodesics by $\gamma_s$, parametrized by the affine parameter $t$, the coordinates are defined by $x^{\alpha}(s,t)$. The vector tangent to the geodesics is defined as $u^{\alpha} = \frac{d x^{\alpha}}{dt}$ and the family of tangent vectors to the family of geodesics is $\xi^{\alpha} = \frac{d x^{\alpha}}{s}$.
Following Eric Poisson's book (The mathematics of black hole mechanics, equation 1.43), he writes: we wish to derive an expression for its acceleration
$$ \frac{D^2 \xi^{\alpha}}{dt^2} \equiv (\xi^{\alpha}_{; \beta} u^{\beta})_{;\gamma}u^{\gamma} $$
and the rest of the section is showing how this may be written in terms of the Riemann tensor. However, he does not show where this equation comes from. How do we deduce this acceleration equation?