  Let $f(x)$ be a square free quartic polynomial defined over a quadratic field
$K$ such that its leading coefficient is a square. If the continued fraction
expansion of $\displaystyle \sqrt{f(x)}$ is periodic, then its period $n$ lies
in the set \[\{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,17,18,22,26,30,34\}.\] We
write explicitly all such polynomials for which the period $n$ occurs over $K$
but not over $\Q$ and $\displaystyle n\not\in\{ 13,15,17\}$. Moreover we give
necessary and sufficient conditions for the existence of such continued
fraction expansions with period $13, 15$ or $17$ over $K$.
