  A formula discovered by L. Carlitz in 1935 finds an interesting application
in permutation rational functions of finite fields. It allows us to determine
all rational functions of degree three that permute the projective line $\Bbb
P^1(\Bbb F_q)$ over $\Bbb F_q$, a result previously obtained by Ferraguti and
Micheli through a different method. It also allows us to determine all rational
functions of degree four that permute $\Bbb P^1(\Bbb F_q)$ under a certain
condition. (A complete determination of all rational functions of degree four
that permute $\Bbb P^1(\Bbb F_q)$ without any condition will appear in a
separate forthcoming paper.)
