= Solution
For the $m$ independent <Bernoulli trials>, the log <likelihood function>, apart from its constant, is $\ell(\theta)=x\log\theta+(m-x)\log(1-\theta)$. Its negative expected second derivative, the <Fisher information>, is
$$
I_m(\theta)=\frac{\mathbb E[X]}{\theta^2}+\frac{m-\mathbb E[X]}{(1-\theta)^2}
=\frac{m}{\theta(1-\theta)}.
$$
For $m>0$, the <Jeffreys prior> is proportional to its square root. Since $\mathrm B(1/2,1/2)=\pi$, its normalized form is
$$
\boxed{\pi_J(\theta)=\frac1{\pi\sqrt{\theta(1-\theta)}},\qquad
\theta\sim\operatorname{Beta}(1/2,1/2).}
$$
The <Jeffreys prior> is invariant under smooth one-to-one reparameterization as a measure. If $\phi=h(\theta)$, then $I_\phi(\phi)=I_\theta(\theta)(d\theta/d\phi)^2$, so $\sqrt{I_\phi}\,d\phi=\sqrt{I_\theta}\,|d\theta|$. Thus recomputing it in the new coordinate gives precisely the transformed prior, rather than a new prior. In several dimensions the corresponding density is $\sqrt{\det I(\theta)}$, with the same Jacobian transformation property.
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