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We study the differentiability of the metric and other fields at any of thehorizons of the \emph{most generic} multi center Reissner-Nordstrom black holesolutions in $d \ge 5$ and of multi center $M2$ brane solutions. Most genericmeans that the centers are generically located in transverse space andconsequently the solutions do not have any transverse spatial isometries. Weconstruct the Gaussian null co-ordinate system for the neighborhood of ahorizon by solving (all) the geodesic equations in expansions of (appropriatepowers of) the affine parameter. Organizing the harmonic functions that appearin the solution in terms of generalized Gegenbauer polynomials, introduced in\cite{Gowdigere:2014aca}, is key to obtaining the solution to the geodesicequations in a compact and manageable form. We then compute the metric andother fields in the Gaussian null co-ordinate system and find that thedifferentiability of the horizon in the most generic solution is\emph{identical to} the differentiability of the horizon in the twocenter/collinear solution (centers distributed on a line in transverse space).We isolate those aspects of the computation that are most relevant to thisresult. We perform these computations in some cases, in several co-ordinatesystems.

On the smoothness of horizons in the most generic multi center black hole and membrane solutions