  In 1855 H. J. S. Smith proved Fermat's two-square using the notion of
palindromic continuants. In his paper, Smith constructed a proper
representation of a prime number $p$ as a sum of two squares, given a solution
of $z^2+1\equiv0\pmod{p}$, and vice versa. In this paper, we extend the use of
continuants to proper representations by sums of two squares in rings of
polynomials on fields of characteristic different from 2. New deterministic
algorithms for finding the corresponding proper representations are presented.
  Our approach will provide a new constructive proof of the four-square theorem
and new proofs for other representations of integers by quaternary quadratic
forms.
