  The static and dynamic properties of the anisotropic XY-model $(s=1/2)$ on
the inhomogeneous periodic chain, composed of $N$ cells with $n$ different
exchange interactions and magnetic moments, in a transverse field $h,$ are
determined exactly at arbitrary temperatures. The properties are obtained by
introducing the Jordan-Wigner fermionization and by reducing the problem to a
diagonalization of a finite matrix of $nth$ order. The quantum transitions are
determined exactly by analyzing, as a function of the field, the induced
magnetization $1/n\sum_{m=1}^{n}\mu_{m}\left< S_{j,m}^{z}\right>$ ($j$ denotes
the cell, $m$ the site within the cell, $\mu_{m}$ the magnetic moment at site
$m$ within the cell) and the spontaneous magnetization $1/n\sum_{m=1}^{n}\left<
S_{j,m}^{x},\right>$ which is obtained from the correlations $\left<
S_{j,m}^{x}S_{j+r,m}^{x}\right>$ for large spin separations. These results,
which are obtained for infinite chains, correspond to an extension of the ones
obtained by Tong and Zhong(\textit{Physica B} \textbf{304,}91 (2001)). The
dynamic correlations, $\left<
S_{j,m}^{z}(t)S_{j^{\prime},m^{\prime}}^{z}(0)\right>$, and the dynamic
susceptibility, $\chi_{q}^{zz}(\omega),$ are also obtained at arbitrary
temperatures. Explicit results are presented in the limit T=0, where the
critical behaviour occurs, for the static susceptibility $\chi_{q}^{zz}(0)$ as
a function of the transverse field $h$, and for the frequency dependency of
dynamic susceptibility $\chi_{q}^{zz}(\omega)$.
