  Within an expansion in slow-roll inflation parameters, we derive the complete
second-order expressions relating the ratio of tensor to scalar density
perturbations and the spectral index of the scalar spectrum. We find that
``corrections'' to previously derived formulae can dominate if the tensor to
scalar ratio is small. For instance, if $V V''/(V')^2\neq 1$ or if
$m_{Pl}^2/(4\pi) ~|V'''/V'|\ga 1$, where $V(\phi)$ is the inflaton potential
and $m_{Pl}$ is the Planck mass, then the previously used simple relations
between the indices and the tensor to scalar ratio fails. This failure occurs
in particular for natural inflation, Coleman--Weinberg inflation, and
``chaotic'' inflation.
