  For any given positive integer $m$ we construct certain totally positive
algebraic integers $\alpha$ of a real bi-quadratic field $K$ and obtain some
necessary conditions for which $m\alpha$ can not be represented as sum of
integral squares. We show this for integers lie in quadratic subfields of $K$
and for integers which are in $K$ but not in any quadratic subfield of $K$. We
provide examples in tabular form for each cases to corroborate the results.
