  Let $K$ be an algebraically closed field of characteristic $p\geqslant 0$ and
let $W$ be a finite-dimensional $K$-space of dimension greater than or equal to
$5.$ In this paper, we give the structure of certain Weyl modules for
$G=\mbox{Spin}(W)$ in the case where $p\neq 2,$ as well as the dimension of the
corresponding irreducible, finite-dimensional, rational $KG$-modules. In
addition, we determine the composition factors of the restriction of certain
irreducible, finite-dimensional, rational $K\mbox{SL}(W)$-modules to
$\mbox{SO}(W).$
