= Solution
{c}
Use the <Joukowski map> with $|c|<a$, in particular its intended range $0\leq c<a/2$. On the original circle,
$$
X=\left(a+\frac{c^2}{a}\right)\cos\theta,\qquad
Y=\left(a-\frac{c^2}{a}\right)\sin\theta.
$$
Define the exterior inverse
$$
f(Z)=\frac{Z+\sqrt{Z^2-4c^2}}2,
$$
where the <holomorphic square root> is chosen so that $\sqrt{Z^2-4c^2}\sim Z$ at infinity. Then $f(Z)\sim Z$ is the branch mapping the exterior of the <ellipse> to $|z|>a$. Composition of the zero-<circulation> <complex potential of a two-dimensional potential flow>[complex potential] from part 4(c) with this inverse gives
$$
\boxed{W(Z)=\beta e^{i\alpha}f(Z)^2
+\frac{\beta a^4e^{-i\alpha}}{f(Z)^2}.}
$$
The <ellipse> is a <streamline> because its inverse image is the circle, and $W(Z)\sim\beta e^{i\alpha}Z^2$ at infinity. An additive constant in the <complex potential of a two-dimensional potential flow>[complex potential] has no effect on the <velocity>.
The transformed <complex velocity> obeys the <chain rule>:
$$
\frac{dW}{dZ}=\frac{dw/dz}{dZ/dz}
=\frac{2\beta e^{i\alpha}z-2\beta a^4e^{-i\alpha}z^{-3}}{1-c^2/z^2}.
$$
Since $|c|<a$, the denominator is nonzero on and outside $|z|=a$. The four <stagnation points> therefore come from
$$
z^4=a^4e^{-2i\alpha},\qquad
z_n=ae^{i\theta_n},\qquad \theta_n=-\frac\alpha2+\frac{n\pi}2,\quad n=0,1,2,3.
$$
Their physical locations are
$$
\boxed{Z_n=\left(a+\frac{c^2}{a}\right)\cos\theta_n
+i\left(a-\frac{c^2}{a}\right)\sin\theta_n,\qquad n=0,1,2,3.}
$$
They all lie on the <ellipse>, and their positions are independent of $\beta>0$, which only sets the overall <speed> scale.
The printed restriction that $c$ is real and $c<a/2$ is insufficient literally: $c=-2a$ satisfies it but makes $dZ/dz=0$ at $z=2a$ in the exterior, so there is no exterior <conformal map> of the claimed type; the stated minor semiaxis would also be negative. The intended positive-parameter range, or more generally $|c|<a$, is needed for the above solution. The reference to a sphere in the source is a wording slip: the geometry is two-dimensional flow around the mapped cylinder's elliptical cross-section.
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