Abstract: Filament Winding is one of the widely used methods for fabrication of axisymmetric composite structures. Geodesic curve on the surface is the ideal path for placement of fibers since these curves are stable path and there is no possibility of slippage of fibers during winding. Pressure vessel with integral end domes having unequal opening at two ends is a typical example. All such examples require deviation from geodesic path during winding. Non-geodesic winding is inherently unstable and prone to slippage. Tendency of slippage can be resisted only if adequate friction is available. Thus, non-geodesic winding requires investigations of stability of fiber path. This study deals with theoretical investigations related to stability of non-geodesic winding on axi-symmetric surface. The relation between the slippage coefficient and the winding angle is obtained to meet stable winding requirements. Limits of deviation from geodesic path and stability of a predefined path is established through application of differential geometry. Simple design nomograms for cylindrical sections are established.