  Cosmic string loops are defined by a pair of periodic functions ${\bf a}$ and
${\bf b}$, which trace out unit-length closed curves in three-dimensional
space. We consider a particular class of loops, for which ${\bf a}$ lies along
a line and ${\bf b}$ lies in the plane orthogonal to that line. For this class
of cosmic string loops one may give a simple analytic expression for the power
$\gamma$ radiated in gravitational waves. We evaluate $\gamma$ exactly in
closed form for several special cases: (1) ${\bf b}$ a circle traversed $M$
times; (2) ${\bf b}$ a regular polygon with $N$ sides and interior vertex angle
$\pi-2\pi M/N$; (3) ${\bf b}$ an isosceles triangle with semi-angle $\theta$.
We prove that case (1) with $M=1$ is the absolute minimum of $\gamma$ within
our special class of loops, and identify all the stationary points of $\gamma$
in this class.
