Consider the ellipse $b^{2} x^{2}+a^{2} y^{2}=a^{2} b^{2}$ rotated about its minor axis: inscribe a cylinder of maximum volume in the resulting ellipsoid. (The axis of the cylinder is the axis of rotation of the ellipsoid, the circumferences of the bases of the cylinder lie on the ellipsoid.) In this cylinder, inscribe a rotational ellipsoid so that the ellipsoid touches the lateral surface, the base, and the top of the cylinder. In the thus obtained ellipsoid, inscribe a cylinder of maximum volume s. i. t. What is the sum of the volumes of the infinitely many such right circular cylinders, and the sum of the volumes of the infinitely many such ellipsoids? What is the ratio of these two sums?
Let $a$ be the semi-major axis of the ellipse, and $b$ the semi-minor axis. The equation of the ellipse, with the principal axes as coordinate axes, is such that the $X$-axis is the major axis. If we rotate the ellipse around its minor axis, i.e., the $X$-axis, the volume of the resulting ellipsoid is:

$$
\begin{aligned}
V_{e}=2 \pi \int_{0}^{b} x^{2} d y & =2 \pi \frac{a^{2}}{b^{2}} \int_{0}^{b}\left(b^{2}-y^{2}\right) d y=2 \pi \frac{a^{2}}{b^{2}}\left[b^{2} y-\frac{y^{3}}{3}\right]_{0}^{b}= \\
& =2 \pi \frac{a^{2}}{b^{2}}\left(b^{3}-\frac{b^{3}}{3}\right)=\frac{4 \pi a^{2} b}{3}
\end{aligned}
$$

1. Inscribing a rectangle in the ellipse; the coordinates of the vertex in the first quadrant are: $a_{1}, b_{1}$. If we rotate this rectangle around the $Y$-axis, the volume of the resulting cylinder is: $a_{1}^{2} \pi \cdot 2 b_{1}$.

$a_{1}, b_{1}$ satisfy the equation of the ellipse:

$$
b^{2} a_{1}^{2}+a^{2} b_{1}^{2}=a^{2} b^{2}, \quad \text { hence } \quad a_{1}^{2}=\frac{a^{2}}{b^{2}}\left(b^{2}-b_{1}^{2}\right)
$$

The volume of the cylinder is: $V_{1}=2 \pi \frac{a^{2}}{b^{2}}\left(b^{2} b_{1}-b_{1}^{3}\right)$.

The volume of the cylinder is a function of $b_{1}$. Its extremum occurs when the derivative with respect to $b_{1}$ is zero, i.e.,

$$
V_{1}^{\prime}=2 \pi \frac{a^{2}}{b^{2}}\left(b^{2}-3 b_{1}^{2}\right)=0 \quad \text { hence } \quad b_{1}=b \sqrt{\frac{1}{3}}
$$

At this value of $b_{1}$, $V_{1}$ has a maximum, as $V_{1}^{\prime}$ transitions from positive to negative values.

For $a_{1}$, we have:

$$
a_{1}^{2}=\frac{a^{2}}{b^{2}}\left(b^{2}-\frac{b^{2}}{3}\right)=\frac{2 a^{2}}{3}, \quad a_{1}=a \sqrt{\frac{2}{3}}
$$

The maximum volume of the cylinder is then:

$$
\begin{gathered}
V_{h}=2 \pi \frac{a^{2}}{b^{2}}\left(b^{3} \sqrt{\frac{1}{3}}-\frac{b^{3}}{3} \sqrt{\frac{1}{3}}\right)=\frac{4 \pi a^{2} b}{3} \sqrt{\frac{1}{3}}=V_{e} \sqrt{\frac{1}{3}} \\
V_{h}: V_{e}=1: \sqrt{3}
\end{gathered}
$$

The ellipsoid inscribed in this cylinder is formed by the rotation of an ellipse whose semi-major axis is $a_{1}=a \sqrt{\frac{2}{3}}$ and semi-minor axis is $b_{1}=b \sqrt{\frac{1}{3}}$. The volume of this ellipsoid is:

$$
\frac{4 \pi a_{1}^{2} b_{1}}{3}=\frac{4 \pi a^{2} b^{2}}{3} \cdot \frac{2}{3} \sqrt{\frac{1}{3}}=V_{e} \cdot \frac{2}{3} \sqrt{\frac{1}{3}}
$$

The volumes of the successive ellipsoids form a geometric progression with a common ratio of $\frac{2}{3} \sqrt{\frac{1}{3}}=\sqrt{\frac{4}{27}}<1$. The infinite geometric series is convergent, and its sum is:

$$
\begin{aligned}
& S_{e}=V_{e} \frac{1}{1-\sqrt{\frac{4}{27}}}=V_{e} \frac{1+\sqrt{\frac{4}{27}}}{1-\frac{4}{27}}=\frac{27}{23} V_{e}\left(1+\sqrt{\frac{4}{27}}\right) \\
& S_{e}=\frac{36}{23} a^{2} b \pi\left(1+\sqrt{\frac{4}{27}}\right)=\frac{36}{23} a^{2} b \pi\left(1+\frac{2 \sqrt{3}}{9}\right)
\end{aligned}
$$

The volumes of the largest cylinders inscribed in the ellipsoids also form a geometric progression with a common ratio of $\sqrt{\frac{4}{27}}$. We have seen that:

$$
V_{h}=V_{e} \sqrt{\frac{1}{3}}
$$

thus,

$$
\begin{aligned}
& \sum V_{h}=\sqrt{\frac{1}{3}} \sum V_{e}, \quad \text { or } \quad S_{h}=S_{e} \sqrt{\frac{1}{3}} \\
& S_{h}: S_{e}=1: \sqrt{3} \quad \text { or } \quad S_{h}: S_{e}=\sqrt{3}: 3
\end{aligned}
$$

Jenő Krisztonosich (Szent László g. VIII. o. Bp. X.)