  We study the effects of curvature in the expansion of the logarithm of the
differential elastic scattering cross section near $t=0$ as
$d\sigma(s,t)/dt=d\sigma(s,0)/dt\,\times\exp(Bt+Ct^2+Dt^3\cdots)$ in an eikonal
model for $pp$ and $\bar{p}p$ scattering, and use the results to discuss the
extrapolation of measured differential cross sections and the slope parameters
$B$ to $t=-q^2=0$. We find that the curvature effects represented by the
parameters $C$ and $D$, while small, lead to significant changes in the forward
slope parameter relative to that determined in a purely exponential fit, and to
smaller but still significant changes in the forward elastic scattering and
total cross sections. Curvature effects should therefore be considered in
future analyses or reanalyses of the elastic scattering data.
