  The motion of a charged particle in a nonuniform straight magnetic field with
a uniform magnetic-field gradient is solved exactly in terms of elliptic
functions. The connection between this problem and the guiding-center
approximation is discussed. It is shown that, for this problem, the predictions
of guiding-center theory agree very well with the orbit-averaged particle
motion and hold well beyond the standard guiding-center limit $\epsilon \equiv
\rho/L \ll 1$, where $\rho$ is the gyromotion length scale and $L$ is the
magnetic-field gradient length scale.
