Let G be a cyclic group of order 2 and let $S^1$ denote the unit circle in $R^2$ with the standard metric. We consider smooth G-vector bundles over $S^1$ when G acts on $S^1$ by reflection. Then the fixed point set of G on $S^1$ is two points ${z_0, z_1}$. Let $E$mid$_{z_0} and E$mid$_{z_1}$</TEX> be the fiber G-representation spaces at $z_0$ and $z_1$ respectively. We associate an orthogonal G-representation $ ho_i : G o O(n)$ to $E$mid$_{z_i}, i = 0, 1$</TEX>. Let det $p ho_i(g), g eq 1$, be denoted by det $E$mid$_{z_i}$</TEX> since det $ ho_i(g)$ is independent of choice of $ ho_i$.