  It has recently been demonstrated that the dynamics of black holes at large
$D$ can be recast as a set of non gravitational membrane equations. These
membrane equations admit a simple static solution with shape $S^{D-p-2} \times
R^{p,1}$. In this note we study the equations for small fluctuations about this
solution in a limit in which amplitude and length scale of the fluctuations are
simultaneously scaled to zero as $D$ is taken to infinity. We demonstrate that
the resultant nonlinear equations, which capture the Gregory- Laflamme
instability and its end point, exactly agree with the effective dynamical
`black brane' equations of Emparan Suzuki and Tanabe. Our results thus identify
the `black brane' equations as a special limit of the membrane equations and so
unify these approaches to large $D$ black hole dynamics.
