  We assume a vector bundle $p: E\to M$ with a general linear connection $K$
and a classical linear connection $\Lam$ on $M$. We prove that all classical
linear connections on the total space $E$ naturally given by $(\Lam, K)$ form a
15-parameter family. Further we prove that all connections on $J^1 E$ naturally
given by $(\Lam, K)$ form a 14-parameter family. Both families of connections
are described geometrically.
