Summary:

Let $F_1$ be a natural bundle of order $r_1$; a basis of the $s$th order differential operators of $F_1$ with values in $r_2$th order bundles is an operator $D$ of that type such that any other one is obtained by composing $D$ with a suitable zeroorder operator. In this article a basis is found in the following two cases: for $F_1=\text{semi} F^{r_1}$ (semiholonomic $r_1$th order frame bundle), $s=0$, $r_2<r_1$ and $F_1=F^1$ ($1$st order frame bundle), $r_2\le s$. The author uses here the socalled method of orbit reduction which provides one with a criterion for checking a basis in terms of the $K^{r_1+s,r_2}_n$action on the type fiber of the concerned bundle, where $K^{r_1+s,r_2}_n$ denotes the kernel of the projection of the $(n,r_1+s)$ jet group onto the $(n,r_2)$ jet group [see I. Kolár, P. Michor and J. Slovák, `Natural operations in differential geometry' (SpringerVerlag, Berlin) (1993; Zbl 0782.53013) or D. Krupka, Loc! (English) 