  We examine the leading order corrections to the Nambu effective action for
the motion of a cosmic string, which appear at fourth order in the ratio of the
width to radius of curvature of the string. We determine the numerical
coefficients of these extrinsic curvature corrections, and derive the equations
of motion of the worldsheet. Using these equations, we calculate the
corrections to the motion of a collapsing loop, a travelling wave, and a
helical breather. From the numerical coefficients we have calculated, we
discuss whether the string motion can be labelled as `rigid' or `antirigid,'
and hence whether cusp or kink formation might be suppressed or enhanced.
