  For any $k \in \Nat$, we show that the cone of $(k+1)$-secant lines of a
closed subscheme $Z \subset \mathbb{P}^n_K$ over an algebraically closed field
$K$ running through a closed point $p \in \mathbb{P}^n_K$ is defined by the
$k$-th partial elimination ideal of $Z$ with respect to $p$. We use this fact
to give an algorithm for computing secant cones. Also, we show that under
certain conditions partial elimination ideals describe the length of the fibres
of a multiple projection in a way similar to the way they do for simple
projections. Finally, we study some examples illustrating these results,
computed by means of {\sc Singular}.
