  Recently in a series of papers it is observed that in many Banach spaces,
which include classical spaces $C(\Omega)$ and $L_p$-spaces, $1 \leq p <
\infty, p \neq 2$, any generalized bi-circular projection $P$ is given by $P =
\frac{I+T}{2}$, where $I$ is the identity operator of the space and $T$ is a
reflection, that is, $T$ is a surjective isometry with $T^2 = I$. For
surjective isometries of order $n \geq 3$, the corresponding notion of
projection is generalized $n$-circular projection as defined in \cite{AD}. In
this paper we show that in a Banach space $X$, if generalized bi-circular
projections are given by $\frac{I+T}{2}$ where $T$ is a reflection, then any
generalized $n$-circular projection $P$, $n \geq 3$, is given by $P =
\frac{I+T+T^2+...+T^{n-1}}{n}$ where $T$ is a surjective isometry and $T^n =
I$. We prove our results for $n=3$ and for $n > 3$, the proof remains same
except for routine modifications.
