  Axisymmetric, rigidly rotating polytropes are considered in the framework of
both the original Chandrasekhar (C33) approximation and a different version
(extended C33 approximation). Special effort is devoted to two specific points,
namely (i) a contradiction between the binomial series evaluation and the
vanishing density on the boundary, which affects the self-consistency of the
above mentioned approximations, and (ii) the continuity of selected parameters
as a function of the polytropic index. Concerning (i), it is shown the
Emdem-Chandrasekhar (EC) associated functions are defined at any internal point
even if related EC associated equations hold only for a particular subvolume,
in the framework of both the C33 and the extended C33 approximation. Concerning
(ii), the continuity may safely be established, on the boundary of the domain,
for part of the parameters, while additional data are needed for the remaining
part. Simple fitting curves, valid to a good extent for a wide range of
polytropic index, involve exponential functions and, in a single case, two
straight lines joined by a parabolic segment. The expression of physical
parameters in terms of the polytropic index can be used in building up
sequences of configurations with changing density profile for assigned mass and
angular momentum.
