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In this paper we calculate the potential for a prolate spheroidaldistribution as in a dark matter halo with a radially varying eccentricity.This is obtained by summing up the shell-by-shell contributions of isodensitysurfaces, which are taken to be concentric and with a common polar axis andwith an axis ratio that varies with radius. Interestingly, the constancy ofpotential inside a shell is shown to be a good approximation even when theisodensity contours are dissimilar spheroids, as long as the radial variationin eccentricity is small as seen in realistic systems. We consider three caseswhere the isodensity contours are more prolate at large radii, or are lessprolate, or have a constant eccentricity. Other relevant physical quantitieslike the rotation velocity, the net orbital and vertical frequency due to thehalo and an exponential disc of finite thickness embedded in it are obtained.We apply this to the kinematical origin of Galactic warp, and show that aprolate shaped halo is not conducive to making long-lived warps - contrary towhat has been proposed in the literature. The results for a prolate massdistribution with a variable axis ratio obtained are general, and can beapplied to other astrophysical systems such as prolate bars, for a morerealistic treatment.

Dynamics of prolate spheroidal mass distributions with varying eccentricity