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We present some new results on the relations between the rotation index of bar billiards of two nested circles CR and Cr, of radii R and r and with distance d between their centers, satisfying Poncelet's porism property. The rational indices correspond to closed Poncelet transverses, without or with self-intersections. We derive an interesting series arising from the theory of special functions. This relates the rotation number ⅓, of a triangle of Poncelet transverses, to a double series involving R, r, and d. We also provide a Steiner-type formula which gives a necessary condition for a bar billiard to be a pentagon with self-intersections and rotation index ⅖. Finally we show that, close to a pair of circles having Poncelet's porism property for index ⅓, there exist always circle pairs having indices ... they and ⅙; in the case ... they are even unique. [ABSTRACT FROM AUTHOR]