  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 ${\sf RCA}_0$ and
axiomatizable by a $\Pi^1_{k+2}$ sentence, and for any $n\geq k+1$, \[ T_0+
\mathrm{RFN}_{\varPi^1_{n+2}}(T) \ = \ T_0 +
\mathrm{TI}_{\varPi^1_n}(\varepsilon_0), \] \[ T_0+
\mathrm{RFN}_{\varSigma^1_{n+1}}(T) \ = \ T_0+
\mathrm{TI}_{\varPi^1_n}(\varepsilon_0)^{-}, \] where $T$ is $T_0$ augmented
with full induction, and $\mathrm{TI}_{\varPi^1_n}(\varepsilon_0)^{-}$ denotes
the schema of transfinite induction up to $\varepsilon_0$ for $\varPi^1_n$
formulas without set parameters.
