k-Parameter geodesic variations

Suppose $S$ is a semispray on a manifold $M$. We know that the complete lift $S^c$ of $S$ is a semispray on $TM$ with the property that geodesics of $S^c$ correspond to Jacobi fields of $S$. In this note we generalize this result and show how geodesic variations of $k$-variables are related to geodesics of the $k$th iterated complete lift of $S$. Moreover, for sprays (that is, homogeneous semisprays) we show how geodesic variations of $(n-1)$-variables are related to a natural generalisation of Jacobi tensors.