Below you'll find a section of page 110, from Gelman's 3rd edition Bayesian Data Analysis (BDA).
In the problem at hand, the observations and parameters are independent, $y_j\sim Bin(n_j,\theta_j)$, and $\theta_j\sim Beta(\alpha,\beta)$.
What's the point of trying to reparameterize first? Do they want a uniform distribution on the new parameters, and then see what they obtain for the old parametrization?
Any help would be appreciated.