A note on fragments of uniform reflection in second order arithmetic

We consider fragments of uniform reflection for formulas in the analytic hierarchy over theories of second order arithmetic. The main result is that for any second order arithmetic theory $T_0$ extending ${\mathsf{R}\mathsf{C}\mathsf{A}}_0$ and axiomatizable by a ${\mathrm{\Pi }}_{k+2}^1$ sentence, and for any $n\ge k+1$, $$T_0+{\mathrm{R}\mathrm{F}\mathrm{N}}_{{\Pi }_{n+2}^1}(T)=T_0+{\mathrm{T}\mathrm{I}}_{{\Pi }_n^1}({\epsilon }_0),$$ $$T_0+{\mathrm{R}\mathrm{F}\mathrm{N}}_{{\Sigma }_{n+1}^1}(T)=T_0+{\mathrm{T}\mathrm{I}}_{{\Pi }_n^1}({\epsilon }_0{)}^{-},$$ where $T$ is $T_0$ augmented with full induction, and ${\mathrm{T}\mathrm{I}}_{{\Pi }_n^1}({\epsilon }_0{)}^{-}$ denotes the schema of transfinite induction up to ${\epsilon }_0$ for ${\Pi }_n^1$ formulas without set parameters.