  Let $Z_2$, $Z_3$, and $Z_4$ denote $2^{\rm nd}$, $3^{\rm rd}$, and $4^{\rm
th}$ order arithmetic, respectively. We let Harrington's Principle, {\sf HP},
denote the statement that there is a real $x$ such that every $x$--admissible
ordinal is a cardinal in $L$. The known proofs of Harrington's theorem
"$Det(\Sigma_1^1)$ implies $0^{\sharp}$ exists" are done in two steps: first
show that $Det(\Sigma_1^1)$ implies {\sf HP}, and then show that {\sf HP}
implies $0^{\sharp}$ exists. The first step is provable in $Z_2$. In this paper
we show that $Z_2 \, + \, {\sf HP}$ is equiconsistent with ${\sf ZFC}$ and that
$Z_3\, + \, {\sf HP}$ is equiconsistent with ${\sf ZFC} \, +$ there exists a
remarkable cardinal. As a corollary, $Z_3\, + \, {\sf HP}$ does not imply
$0^{\sharp}$ exists, whereas $Z_4\, + \, {\sf HP}$ does. We also study
strengthenings of Harrington's Principle over $2^{\rm nd}$ and $3^{\rm rd}$
order arithmetic.
