= Solution
The only nonzero angular <Fourier mode> is $\cos(2\sigma)$. Its radial equation is $T''-4T=0$. A solution vanishing at $\tau=a$ is proportional to $\sinh[2(\tau-a)]$; making its value one at $b$ fixes the constant. The <Dirichlet problem between bipolar coordinate circles> therefore gives
$$
\boxed{\Phi(\sigma,\tau)=\frac{\sinh[2(\tau-a)]}{\sinh[2(b-a)]}\cos(2\sigma).}
$$
Its second derivatives in $\sigma$ and $\tau$ cancel, and the endpoint values are respectively zero and $\cos(2\sigma)$. The difference of any two solutions would be a <harmonic function> with zero <Dirichlet boundary conditions>, so the <maximum principle> proves <uniqueness>.
\b[Geometric wording.] In the printed <bipolar coordinates>, the circle radius $1/\sinh\tau$ decreases with $\tau$. For $0<a<b$, the $b$ circle is contained inside the $a$ circle. Thus the source’s “inner” and “outer” labels are interchanged geometrically. The solution above follows the explicit coordinate boundary values, which unambiguously specify the problem.
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