The Hurwitz variety in $R^{n}$ is defined as the set $H$, of those points $(\xi_{1},\dots,\xi_{n})$ for which the polynomial $z^{n} + \sum_{k=1}^{n} \, \xi_{k} z^{n-k}$ has all roots with negative real part. k=\ As is shown here, $H_{n}$ coincides with the union of the sets of an $n$-parameter family of $n$-dimensional ellipsoids. The same holds true for $H$ with respect to another $n$-parameter family of $n$-dimensional domains.