  The Hobby-Rice Theorem states that, given $n$ functions $f_j$ on
${\mathbb{R}}^N$, there exists a multiplier $h$ such that the integrals of
$f_jh$ are all simultaneously zero. This multiplier takes values~$\pm1$ and is
discontinuous. We show how to find a multiplier $h=e^{ig}$ that is infinitely
differentiable, takes values on the unit circle, and is such that the integrals
of $f_jh$ are all zero. We also show the existence of $n$ infinitely
differentiable, real functions $g_j$ such that the $n$ functions $f_j e^{ig_j}$
are pairwise orthogonal.
