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We study the differentiability of the metric and other fields at any of thehorizons of multi center Reissner-Nordstrom black hole solutions in $d \ge 5$and of multi center $M2$ brane solutions. The centers are distributed in aplane in transverse space, hence termed coplanar. We construct the Gaussiannull co-ordinate system for the neighborhood of a horizon by solving thegeodesic equations in expansions of (appropriate powers of) the affineparameter. Organizing the harmonic functions that appear in the solution interms of what can be called generalized Gegenbauer polynomials is key toobtaining the solution to the geodesic equations in a compact and manageableform. We then compute the metric and other fields in the Gaussian nullco-ordinate system and find that the differentiability of the coplanar solutionis \emph{identical to} the differentiability of the collinear solution (centersdistributed on a line in transverse space). The results of this paper thus runcounter to a suggestion in the literature that posits reduction in the degreeof smoothness to accompany reduction in symmetries. We end the paper with aconjecture on the degree of smoothness of the most general multi centersolution, the one with centers distributed arbitrarily and hence possessing notransverse spatial isometries.

On the smoothness of multi center coplanar black hole and membrane horizons